text: 8 were separately examined and debugged in collabora- tion with CERN IT, while other errors were captured in server logs and invetigated by developers. 6 Summary We have provided a detailed description of the WMArchive system for distributed workﬂow management agents in the CMS collaboration. The system is designed accord- ing to a speciﬁc set of requirements imposed by CMS data-operations teams and it has been deployed into production since mid 2016. Since then we have collected more than 125 million FWJR documents and have not experienced any signiﬁcant problems with system main- tenance. The introduction of short-term and long-term storage systems has helped us to maintain the injection rate and isolate it from user-based queries. We found that complex search queries are desired by the CMS data-operations teams and we are able to accommodate Fig. 3 The WMArchive Performance Service web interface them via a ﬂexible query language discussed in the pa- gives CMS data operators ﬂexible access to a range of com- monly needed data aggregation queries. per. We plan to expand our system and include a new data steam coming out from CMS CRAB [15] analy- sis facilities which process user-analysis jobs across the globe. This will roughly double the demands on the de- ployed system where we may face new challenges. But the ﬂexible design of the system and almost a year of running in production environment allow us to be op- timistic about such an expansion. Acknowledgements We would like to thank our colleagues Seangchan Ryu (FNAL) and Alan Malta (Univ. of Nebraska) for numerous feedback and guidance across various details of CMS Workﬂow Management System. Special thanks goes to Eric Vaandering (FNAL) for initiating the idea of WMArchive Fig. 4 The Kibana data visualization platform exposes the system in CMS and his constant support along development entire aggregated WMArchive Performance Service dataset. cycle. We also would like to thank Luca Menichetti from CERN IT who provided support for development, mainte- nance and deployment of our scripts on Spark platform. 5.3 Error handling References The error handling is an important component of 1. J. Adelman et al, CMS Computing Operations during Run every system. Apart from usual programming bugs catched 1, Journal of Physics: Conference Series 513 (2014) 032040. via Exception mechanism in the Python code, we mostly 2. M. Giﬀels, Y. Guo, V. Kuznetsov, N. Magini and struggled to handle errors on the Spark side. The PyS- T. Wildish The CMS Data Management System J. Phys.: Conf. Ser. 513 (2014) 042052; doi:10.1088/1742- park framework is a wrapper around Java Spark li- 6596/513/4/042052 braries. As such, the nested, often very long and cum- 3. D. Evans, et al, The CMS workload management system, bersome, errors were hard to debug and understand. Journal of Physics: Conference Series, Volume 396, Part 3 Most of them came from JVM memory issues due to 4. J. Balcas1, S. Belforte, B. Bockelman, D. Colling, O. Gutsche, D. Hufnagel, F. Khan, K. Larson, J. Letts, M. incorrect data handling on a worker node. This was Mascheroni Using the glideinWMS System as a Com- caused by a sequential data model in our code rather mon Resource Provisioning Layer in CMS J. Phys.: Conf. than a distributed lazy processing oﬀered by PySpark Ser. 664 (2015) 062031, http://iopscience.iop.org/1742- APIs. We must admit that a there is learning curve to 6596/664/6/062031 5. D. Thain, T. Tannenbaum and M. Livny (2005) Dis- eﬃciently use PySpark and adapt it to the Python code tributed computing in practice: the Condor experience Con- we were dealing with before implementing this system. currency and Computation: Practice and Experience 17 2-4 Due to this nature, the errors caused by our Spark jobs 323-356 doi:10.1002/cpe.938 ##LTFigure## ##LTFigure## ##LTLine##

text: were

text: separately

text: examined

text: and

text: debugged

text: in

text: collabora-

text: tion

text: with

text: CERN

text: IT,

text: while

text: other

text: errors

text: were

text: captured

text: in

text: server

text: logs

text: and

text: invetigated

text: by

text: developers.

label: section

text: Summary

text: We

text: have

text: provided

text: a

text: detailed

text: description

text: of

text: the

text: WMArchive

text: system

text: for

text: distributed

text: workﬂow

text: management

text: agents

text: in

text: the

text: CMS

text: collaboration.

text: The

text: system

text: is

text: designed

text: accord-

text: ing

text: to

text: a

text: speciﬁc

text: set

text: of

text: requirements

text: imposed

text: by

text: CMS

text: data-operations

text: teams

text: and

text: it

text: has

text: been

text: deployed

text: into

text: production

text: since

text: mid

text: Since

text: then

text: we

text: have

text: collected

text: more

text: than

text: million

text: FWJR

text: documents

text: and

text: have

text: not

text: experienced

text: any

text: signiﬁcant

text: problems

text: with

text: system

text: main-

text: tenance.

text: The

text: introduction

text: of

text: short-term

text: and

text: long-term

text: storage

text: systems

text: has

text: helped

text: us

text: to

text: maintain

text: the

text: injection

text: rate

text: and

text: isolate

text: it

text: from

text: user-based

text: queries.

text: We

text: found

text: that

text: complex

text: search

text: queries

text: are

text: desired

text: by

text: the

text: CMS

text: data-operations

text: teams

text: and

text: we

text: are

text: able

text: to

text: accommodate

text: Fig.

text: The

text: WMArchive

text: Performance

text: Service

text: web

text: interface

text: them

text: via

text: a

text: ﬂexible

text: query

text: language

text: discussed

text: in

text: the

text: pa-

text: gives

text: CMS

text: data

text: operators

text: ﬂexible

text: access

text: to

text: a

text: range

text: of

text: com-

text: monly

text: needed

text: data

text: aggregation

text: queries.

text: per.

text: We

text: plan

text: to

text: expand

text: our

text: system

text: and

text: include

text: a

text: new

text: data

text: steam

text: coming

text: out

text: from

text: CMS

text: CRAB

text: analy-

text: sis

text: facilities

text: which

text: process

text: user-analysis

text: jobs

text: across

text: the

text: globe.

text: This

text: will

text: roughly

text: double

text: the

text: demands

text: on

text: the

text: de-

text: ployed

text: system

text: where

text: we

text: may

text: face

text: new

text: challenges.

text: But

text: the

text: ﬂexible

text: design

text: of

text: the

text: system

text: and

text: almost

text: a

text: year

text: of

text: running

text: in

text: production

text: environment

text: allow

text: us

text: to

text: be

text: op-

text: timistic

text: about

text: such

text: an

text: expansion.

text: Acknowledgements

text: We

text: would

text: like

text: to

text: thank

text: our

text: colleagues

text: Seangchan

text: Ryu

text: (FNAL)

text: and

text: Alan

text: Malta

text: (Univ.

text: of

text: Nebraska)

text: for

text: numerous

text: feedback

text: and

text: guidance

text: across

text: various

text: details

text: of

text: CMS

text: Workﬂow

text: Management

text: System.

text: Special

text: thanks

text: goes

text: to

text: Eric

text: Vaandering

text: (FNAL)

text: for

text: initiating

text: the

text: idea

text: of

text: WMArchive

text: Fig.

text: The

text: Kibana

text: data

text: visualization

text: platform

text: exposes

text: the

text: system

text: in

text: CMS

text: and

text: his

text: constant

text: support

text: along

text: development

text: entire

text: aggregated

text: WMArchive

text: Performance

text: Service

text: dataset.

text: cycle.

text: We

text: also

text: would

text: like

text: to

text: thank

text: Luca

text: Menichetti

text: from

text: CERN

text: IT

text: who

text: provided

text: support

text: for

text: development,

text: mainte-

text: nance

text: and

text: deployment

text: of

text: our

text: scripts

text: on

text: Spark

text: platform.

text: Error

text: handling

text: References

text: The

text: error

text: handling

text: is

text: an

text: important

text: component

text: of

text: J.

text: Adelman

text: et

text: al,

text: CMS

text: Computing

text: Operations

text: during

text: Run

text: every

text: system.

text: Apart

text: from

text: usual

text: programming

text: bugs

text: catched

text: Journal

text: of

text: Physics:

text: Conference

text: Series

text: via

text: Exception

text: mechanism

text: in

text: the

text: Python

text: code,

text: we

text: mostly

text: M.

text: Giﬀels,

text: Y.

text: Guo,

text: V.

text: Kuznetsov,

text: N.

text: Magini

text: and

text: struggled

text: to

text: handle

text: errors

text: on

text: the

text: Spark

text: side.

text: The

text: PyS-

text: T.

text: Wildish

text: The

text: CMS

text: Data

text: Management

text: System

text: J.

text: Phys.:

text: Conf.

text: Ser.

text: park

text: framework

text: is

text: a

text: wrapper

text: around

text: Java

text: Spark

text: li-

text: braries.

text: As

text: such,

text: the

text: nested,

text: often

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text: long

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text: et

text: al,

text: The

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text: system,

text: bersome,

text: errors

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text: hard

text: to

text: debug

text: and

text: understand.

text: Journal

text: of

text: Physics:

text: Conference

text: Series,

text: Volume

text: Part

text: Most

text: of

text: them

text: came

text: from

text: JVM

text: memory

text: issues

text: due

text: to

text: J.

text: S.

text: Belforte,

text: B.

text: Bockelman,

text: D.

text: Colling,

text: O.

text: Gutsche,

text: D.

text: Hufnagel,

text: F.

text: Khan,

text: K.

text: Larson,

text: J.

text: Letts,

text: M.

text: incorrect

text: data

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text: T.

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text: ##LTFigure##

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text: 9 6. I. Sﬁligoi, D. C. Bradley, B. Holzman, P. Mhashilkar, S. Padhi and F. Wurthwein (2009) The Pilot Way to Grid Resources Using glideinWMS Proceedings of the 2009 WRI World Congress on Computer Science and Information En- gineering - Volume 02 CSIE ’09 (Washington, DC, USA: IEEE Computer Society) pp 428–432 ISBN 978-0-7695- 3507-4; http://dx.doi.org/10.1109/CSIE.2009.950 7. MongoDB document-oriented database, https://docs.mongodb.org/ 8. MongoDB indices, https://docs.mongodb.com/manual/tutorial/ensure- indexes-ﬁt-ram/ 9. MongoDB Query Language, https://docs.mongodb.org/manual/tutorial/query- documents 10. Apache Avro, https://www.wikiwand.com/en/Apache Avro 11. WMArchive UI dependencies, https://github.com/dmwm/WMArchive/wiki 12. Kibana, https://www.elastic.co/products/kibana 13. Grafana, https://grafana.com 14. Apache Spark, https://spark.apache.org/ 15. D. Spiga, S. Lacaprara, W. Bacchi, M. Cinquilli, G. Codispoti, M. Corvo, A. Dorigo, A. Fanfani, F. Fanzago, F. Farina, M. Merlo, O. Gutsche, L. Servoli, C. Kavka The CMS Remote Analysis Builder (CRAB) High Performance Computing, HiPC 2007 pp 580-586; doi: 10.1007/978-3-540- 77220-0 52 ##LTLine## ##LTLine## ##LTLine##

label: paragraph

label: reference

text: I.

text: Sﬁligoi,

text: D.

text: C.

text: Bradley,

text: B.

text: Holzman,

text: P.

text: Mhashilkar,

text: S.

text: Padhi

text: and

text: F.

text: Wurthwein

text: The

text: Pilot

text: Way

text: to

text: Grid

text: Resources

text: Using

text: glideinWMS

text: Proceedings

text: of

text: the

text: WRI

text: World

text: Congress

text: on

text: Computer

text: Science

text: and

text: Information

text: En-

text: gineering

text: Volume

text: CSIE

text: (Washington,

text: DC,

text: USA:

text: IEEE

text: Computer

text: Society)

text: pp

text: ISBN

text: MongoDB

text: document-oriented

text: database,

text: MongoDB

text: indices,

text: indexes-ﬁt-ram/

text: MongoDB

text: Query

text: Language,

text: documents

text: Apache

text: Avro,

text: Avro

text: WMArchive

text: UI

text: dependencies,

text: Kibana,

text: Grafana,

text: Apache

text: Spark,

text: D.

text: Spiga,

text: S.

text: Lacaprara,

text: W.

text: Bacchi,

text: M.

text: Cinquilli,

text: G.

text: Codispoti,

text: M.

text: Corvo,

text: A.

text: Dorigo,

text: A.

text: Fanfani,

text: F.

text: Fanzago,

text: F.

text: Farina,

text: M.

text: Merlo,

text: O.

text: Gutsche,

text: L.

text: Servoli,

text: C.

text: Kavka

text: The

text: CMS

text: Remote

text: Analysis

text: Builder

text: (CRAB)

text: High

text: Performance

text: Computing,

text: HiPC

text: pp

text: doi:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Spin-selective metal insulator transition in two dimensions∗ Shashikant Singh Kunwar,1 Prabuddha B. Chakraborty,1 and R. Narayanan1 (Dated: May 30, 2020) The phenomenon of Anderson localization wherein non-interacting electrons are localized by quenched impurities is a subject matter that has been extremely well studied. However, local- ization transition under the combined inﬂuence of interaction and quenched disorder is less well understood. In this context we study the localization transition in a two-dimensional Hubbard model under the inﬂuence of a spin-selective disorder i.e, disorder which is operational on just one of the spin-species. The model is analyzed by laying recourse to a Quantum Monte Carlo based scheme. Using this approach we show the possibility of a metal-insulator transition. However, we will show that this metal-insulator transition is extremely sensitive to the ﬁlling-fraction inherent in the system. INTRODUCTION ever, since the RG trajectories ﬂow to strong coupling, the ultimate fate of the metal insulator transition can- not be ascertained by these ﬂow equations. Later, in a The paradigmatic scaling theory[1] of localization[2] at- signiﬁcant development, a systematic two-loop RG ﬂow tests to the absence of a metallic state in d ≤ 2 dimensions equation was derived in the limit of n → ∞, (where n for a non-interacting electronic system in the presence of v v is the number of valleys in a degenerate semiconductor) arbitrary small amount of disorder. However, this cele- [12]. Using the so derived ﬂow equations, the authors brated gang of four result which applies to certain sym- conclusively demonstrated the existence of an interaction metry classes in the Altland-Zirnbauer scheme [3, 4] is driven MIT in a two dimensional disordered electronic silent on the fate of the transition in the presence of in- system. However, the n → ∞ limit brings forth the teractions. This is particularly intriguing as not only can v question of applicability of this theory to the case of the Coulomb interactions by itself trigger a metal insulator Si- MOSFET heterostructures where it is well known that transition called the Mott transition [5], but also the in- the number of valleys is two. terplay between disorder and interactions could lead to novel states of matter and intervening phase transitions Another oft-used track to theoretically study the ques- between them. In fact, experiments on two dimensional tion of metal-insulator transition in a two dimensional MOSFET devices with extremely high mobility seem to interacting disordered electron system is by laying re- indicate a possible metal to insulator transition as a func- course to Quantum Monte-Carlo based simulation meth- tion of the interaction [6] [7]. Furthermore, advances in ods [13, 14]. However, progress using this technique is the ﬁeld of engineered cold atomic systems has opened up stymied due to the so-called sign-problem[15]. unchartered avenues wherein interaction eﬀects on An- This manuscript falls into the latter category wherein derson localized states could be explored in a controlled by using a Determinant Quantum Monte Carlo (DQMC) setting. scheme we simulate a Hubbard model in the presence Even though, as in the above paragraph [6, 7], there is of a spin-dependent disorder. Spin-dependent disorder plausible experimental evidence to support the scenario of can in principle be realized in cold-atomic systems by us- an interaction driven metal insulator transition, the the- ing laser beams of two diﬀerent polarizations to create a oretical understanding of the same is some what lacking. speckle type of disorder which acts diﬀerently on particles Notable exceptions to the same are ﬁeld theoretical ap- with diﬀerent spin-polarizations. The eﬀect of such spin- proaches that are based on adapting Wegner’s non-linear dependent disorder on a repulsive Hubbard model was sigma model approach to non-interacting disordered elec- investigated within a DMFT framework by [16]. Very in- tron systems near two dimensions. The structure of such triguingly, these investigations pointed to the existence of a theory was ﬁrst guessed by Finkelstein [8]. The deriva- a novel spin-selective localized phase wherein one of the tion of the ﬁeld theory was put on much ﬁrmer grounds spin-species gets localized due to the disorder potential by Belitz and Kirkpatrick [9], [10] who developed a sys- whereas the other spin species retain its itinerant char- tematic technology of keeping only the relevant degrees acter. Apart from the spin-selective localized phase the of freedom whilst integrating out all others. However, it model also hosts the putative correlated metallic phase is by now well known that the ﬁeld theory is pathological: at small values of disorder and a disordered Mott insu- Under Renormalization Group (RG) transformations the lator at large values of the interaction. Furthermore, it amplitude of the spin-triplet channel γ ﬂows to inﬁnity was shown in Ref. [16] that the transition from the cor- t thus signalling the breakdown of the perturbation theory. related metal to the spin-selective localized phase follows This runaway behavior of the RG ﬂows was later shown to the conventional Anderson localization scenario, whereas be a signature of a ferromagnetic instability [11]. How- the transition from the spin-selective localized phase to

text: Spin-selective

label: title

text: metal

text: insulator

text: transition

text: in

text: two

text: dimensions∗

text: Shashikant

label: author

text: Singh

text: Prabuddha

text: B.

text: and

text: R.

text: (Dated:

text: May

label: date

text: The

label: abstract

text: phenomenon

text: of

text: Anderson

text: localization

text: wherein

text: non-interacting

text: electrons

text: are

text: localized

text: by

text: quenched

text: impurities

text: is

text: a

text: subject

text: matter

text: that

text: has

text: been

text: extremely

text: well

text: studied.

text: However,

text: local-

text: ization

text: transition

text: under

text: the

text: combined

text: inﬂuence

text: of

text: interaction

text: and

text: quenched

text: disorder

text: is

text: less

text: well

text: understood.

text: In

text: this

text: context

text: we

text: study

text: the

text: localization

text: transition

text: in

text: a

text: two-dimensional

text: Hubbard

text: model

text: under

text: the

text: inﬂuence

text: of

text: a

text: spin-selective

text: disorder

text: i.e,

text: disorder

text: which

text: is

text: operational

text: on

text: just

text: one

text: of

text: the

text: spin-species.

text: The

text: model

text: is

text: analyzed

text: by

text: laying

text: recourse

text: to

text: a

text: Quantum

text: Monte

text: Carlo

text: based

text: scheme.

text: Using

text: this

text: approach

text: we

text: show

text: the

text: possibility

text: of

text: a

text: metal-insulator

text: transition.

text: However,

text: we

text: will

text: show

text: that

text: this

text: metal-insulator

text: transition

text: is

text: extremely

text: sensitive

text: to

text: the

text: ﬁlling-fraction

text: inherent

text: in

text: the

text: system.

text: INTRODUCTION

text: ever,

text: since

text: the

text: RG

text: trajectories

text: ﬂow

text: to

text: strong

text: coupling,

text: the

text: ultimate

text: fate

text: of

text: the

text: metal

text: insulator

text: transition

text: can-

text: not

text: be

text: ascertained

text: by

text: these

text: ﬂow

text: equations.

text: Later,

text: in

text: a

text: The

text: paradigmatic

text: scaling

text: of

text: at-

text: signiﬁcant

text: development,

text: a

text: systematic

text: two-loop

text: RG

text: ﬂow

text: tests

text: to

text: the

text: absence

text: of

text: a

text: metallic

text: state

text: in

text: d

text: dimensions

text: equation

text: was

text: derived

text: in

text: the

text: limit

text: of

text: n

text: (where

text: n

text: for

text: a

text: non-interacting

text: electronic

text: system

text: in

text: the

text: presence

text: of

text: v

text: v

text: is

text: the

text: number

text: of

text: valleys

text: in

text: a

text: degenerate

text: semiconductor)

text: arbitrary

text: small

text: amount

text: of

text: disorder.

text: However,

text: this

text: cele-

text: Using

text: the

text: so

text: derived

text: ﬂow

text: equations,

text: the

text: authors

text: brated

text: gang

text: of

text: four

text: result

text: which

text: applies

text: to

text: certain

text: sym-

text: conclusively

text: demonstrated

text: the

text: existence

text: of

text: an

text: interaction

text: metry

text: classes

text: in

text: the

text: Altland-Zirnbauer

text: scheme

text: is

text: driven

text: MIT

text: in

text: a

text: two

text: dimensional

text: disordered

text: electronic

text: silent

text: on

text: the

text: fate

text: of

text: the

text: transition

text: in

text: the

text: presence

text: of

text: in-

text: system.

text: However,

text: the

text: n

text: limit

text: brings

text: forth

text: the

text: teractions.

text: This

text: is

text: particularly

text: intriguing

text: as

text: not

text: only

text: can

text: v

text: question

text: of

text: applicability

text: of

text: this

text: theory

text: to

text: the

text: case

text: of

text: the

text: Coulomb

text: interactions

text: by

text: itself

text: trigger

text: a

text: metal

text: insulator

text: Si-

text: MOSFET

text: heterostructures

text: where

text: it

text: is

text: well

text: known

text: that

text: transition

text: called

text: the

text: Mott

text: transition

text: but

text: also

text: the

text: in-

text: the

text: number

text: of

text: valleys

text: is

text: two.

text: terplay

text: between

text: disorder

text: and

text: interactions

text: could

text: lead

text: to

text: novel

text: states

text: of

text: matter

text: and

text: intervening

text: phase

text: transitions

text: Another

text: oft-used

text: track

text: to

text: theoretically

text: study

text: the

text: ques-

text: between

text: them.

text: In

text: fact,

text: experiments

text: on

text: two

text: dimensional

text: tion

text: of

text: metal-insulator

text: transition

text: in

text: a

text: two

text: dimensional

text: MOSFET

text: devices

text: with

text: extremely

text: high

text: mobility

text: seem

text: to

text: interacting

text: disordered

text: electron

text: system

text: is

text: by

text: laying

text: re-

text: indicate

text: a

text: possible

text: metal

text: to

text: insulator

text: transition

text: as

text: a

text: func-

text: course

text: to

text: Quantum

text: Monte-Carlo

text: based

text: simulation

text: meth-

text: tion

text: of

text: the

text: interaction

text: Furthermore,

text: advances

text: in

text: ods

text: However,

text: progress

text: using

text: this

text: technique

text: is

text: the

text: ﬁeld

text: of

text: engineered

text: cold

text: atomic

text: systems

text: has

text: opened

text: up

text: stymied

text: due

text: to

text: the

text: so-called

text: unchartered

text: avenues

text: wherein

text: interaction

text: eﬀects

text: on

text: An-

text: This

text: manuscript

text: falls

text: into

text: the

text: latter

text: category

text: wherein

text: derson

text: localized

text: states

text: could

text: be

text: explored

text: in

text: a

text: controlled

text: by

text: using

text: a

text: Determinant

text: Quantum

text: Monte

text: Carlo

text: (DQMC)

text: setting.

text: scheme

text: we

text: simulate

text: a

text: Hubbard

text: model

text: in

text: the

text: presence

text: Even

text: though,

text: as

text: in

text: the

text: above

text: paragraph

text: there

text: is

text: of

text: a

text: spin-dependent

text: disorder.

text: Spin-dependent

text: disorder

text: plausible

text: experimental

text: evidence

text: to

text: support

text: the

text: scenario

text: of

text: can

text: in

text: principle

text: be

text: realized

text: in

text: cold-atomic

text: systems

text: by

text: us-

text: an

text: interaction

text: driven

text: metal

text: insulator

text: transition,

text: the

text: the-

text: ing

text: laser

text: beams

text: of

text: two

text: diﬀerent

text: polarizations

text: to

text: create

text: a

text: oretical

text: understanding

text: of

text: the

text: same

text: is

text: some

text: what

text: lacking.

text: speckle

text: type

text: of

text: disorder

text: which

text: acts

text: diﬀerently

text: on

text: particles

text: Notable

text: exceptions

text: to

text: the

text: same

text: are

text: ﬁeld

text: theoretical

text: ap-

text: with

text: diﬀerent

text: spin-polarizations.

text: The

text: eﬀect

text: of

text: such

text: spin-

text: proaches

text: that

text: are

text: based

text: on

text: adapting

text: Wegner’s

text: non-linear

text: dependent

text: disorder

text: on

text: a

text: repulsive

text: Hubbard

text: model

text: was

text: sigma

text: model

text: approach

text: to

text: non-interacting

text: disordered

text: elec-

text: investigated

text: within

text: a

text: DMFT

text: framework

text: by

text: Very

text: in-

text: tron

text: systems

text: near

text: two

text: dimensions.

text: The

text: structure

text: of

text: such

text: triguingly,

text: these

text: investigations

text: pointed

text: to

text: the

text: existence

text: of

text: a

text: theory

text: was

text: ﬁrst

text: guessed

text: by

text: Finkelstein

text: The

text: deriva-

text: a

text: novel

text: spin-selective

text: localized

text: phase

text: wherein

text: one

text: of

text: the

text: tion

text: of

text: the

text: ﬁeld

text: theory

text: was

text: put

text: on

text: much

text: ﬁrmer

text: grounds

text: spin-species

text: gets

text: localized

text: due

text: to

text: the

text: disorder

text: potential

text: by

text: Belitz

text: and

text: Kirkpatrick

text: who

text: developed

text: a

text: sys-

text: whereas

text: the

text: other

text: spin

text: species

text: retain

text: its

text: itinerant

text: char-

text: tematic

text: technology

text: of

text: keeping

text: only

text: the

text: relevant

text: degrees

text: acter.

text: Apart

text: from

text: the

text: spin-selective

text: localized

text: phase

text: the

text: of

text: freedom

text: whilst

text: integrating

text: out

text: all

text: others.

text: However,

text: it

text: model

text: also

text: hosts

text: the

text: putative

text: correlated

text: metallic

text: phase

text: is

text: by

text: now

text: well

text: known

text: that

text: the

text: ﬁeld

text: theory

text: is

text: pathological:

text: at

text: small

text: values

text: of

text: disorder

text: and

text: a

text: disordered

text: Mott

text: insu-

text: Under

text: Renormalization

text: Group

text: (RG)

text: transformations

text: the

text: lator

text: at

text: large

text: values

text: of

text: the

text: interaction.

text: Furthermore,

text: it

text: amplitude

text: of

text: the

text: spin-triplet

text: channel

text: γ

text: ﬂows

text: to

text: inﬁnity

text: was

text: shown

text: in

text: Ref.

text: that

text: the

text: transition

text: from

text: the

text: cor-

text: t

text: thus

text: signalling

text: the

text: breakdown

text: of

text: the

text: perturbation

text: theory.

text: related

text: metal

text: to

text: the

text: spin-selective

text: localized

text: phase

text: follows

text: This

text: runaway

text: behavior

text: of

text: the

text: RG

text: ﬂows

text: was

text: later

text: shown

text: to

text: the

text: conventional

text: Anderson

text: localization

text: scenario,

text: whereas

text: be

text: a

text: signature

text: of

text: a

text: ferromagnetic

text: instability

text: How-

text: the

text: transition

text: from

text: the

text: spin-selective

text: localized

text: phase

text: to


text: 2 the Mott Insulator is of the Falicov-Kimball type, [17] In this paper, by using DQMC based simulations we I mΛ(q = 0, ω) report the possibility of a novel interaction driven de- σdc = li→m (2) localization transition in a Hubbard-model in the presence ω 0 ω of a spin-selective disorder. We show that this delocal- ization transition exists for ﬁlling fractions that are away The current-current correlation function in imaginary from half-ﬁlling. We further show that this de-localization time is connected to the dynamical current-current corre- transition resembles the Finkelstein scenario wherein the lation function via t(cid:90)he integral transform given by Eq. (2), ferromagnetic spin-susceptibility gets enhanced close to the MIT. ∞ − PROBLEM AND METHOD Λ(q, τ) = + dω e ω−τ ImΛ(q, ω) (3) −∞ π 1 − e βω The starting point of our analysis is the disordered Hubbard model(cid:88)deﬁned on square latt(cid:88)ice: The correlation function in DQMC are measured in imaginary time τ. Calculating dynamical correlation † function given correlation in imaginary time via invert- (cid:18) (cid:19) (cid:18) (cid:19) H = − t((cid:88)ci,σcj,σ + h.c) + µi,σni,σ ing Eq. (2) is not a well deﬁned problem. The kernel, (cid:104) (cid:105) − i,j ,σ i,σ (1) k = 1−ee−ωτβω decays exponentially with frequency. At +U ni,↑ − 12 ni,↓ − 12 . τ = β2 the kernel k reduces to 2sinh1(β2ω) and it Igets most i of the contribution from the small ω limit .Now mΛ(0, ω) can be replaced by ωσ and the kernel k is replaced by dc Here, in Eq. 1, t is the nearest neighbour hopping am- its value at τ = β . 2 plitude which has been set to unity. The on-sit†e inter- Once these(cid:18)approximatio(cid:19)ns ar(cid:90)e made eq(2) becomes, action is repulsive, i.e., U > 0. Furthermore, c , and i,σ cdie,σﬁnaerde tohnesfietremiioannidc ccraerartyiotnheansdpianninnidheilxatσio.nTohpeereaﬀtoecrst Λ q = 0, τ = β = +∞ dω ωσdc (4) of spin-selective disorder is incorporated by making the 2 −∞ 2π sinh(βω) 2 random chemical potential spin-dependent. In our real- ization of the same, we assume that the σ =↓ feels the ef- After carrying out the i(cid:18)ntegration, (cid:19) fect of a random chemical potential whereas σ =↑ moves (cid:104) (cid:105) in the background of a non-random chemical potential. More speciﬁcally, the chemical potential µ↓ is uniformly β2 β σ = Λ q = 0, τ = . (5) sampled from the range −∆↓ , ∆↓ . dc π 2 2 2 We utilize a DQMC scheme [13],to numerically solve Now σ has been expressed in terms of a current- the Hamiltonian in Eq. 1. The DQMC formalism ini- dc current correlation function calculated in imaginary time. tially entails the slicing of the inverse temperature β into Current-Current correlation function in imaginary time is L slices of width ∆τ each. Further, the grand canoni- τ calculated using DQMC and is given by, cal partition function is decomposed by laying recourse to the Suzuki-Trotter formalism [18]. The interacting elec- tron problem of Eq. 1 is now recast into an eﬀective non- Λ(q, τ) = (cid:104)j (q, τ)j (−q, τ)(cid:105). (6) interacting problem by utilizing a Hubbard-Stratonovich x x [19] transformation in terms of an Ising-type auxillary ﬁeld. The fermionic degrees of freedom are then for- where j (q, τ) is the fourier transform of current density x mally integrated out. Various physical quantities like the operator given in real space by j (r, τ), x Greens function, the electronic density, and the two par- ticle correlator are calculated via Monte-Carlo sampling − of the Ising Hubbard-Stratanovich ﬁeld. The parame- jx(r, τ) = eτhH jx(r)e τhH (7) ter space of our simulation is spanned by the interaction strength U, the disorder strength ∆↓ and temperature T, (cid:88) which are all expressed in terms of the hopping integral t. The simulations are further performed at various ﬁlling ieat † † fractions. j (r) = (c c − c c ). (8) x h r+ex,σ r,σ r,σ r+ex,σ The dc conductivity can be calculated by taking the σ long wavelength(q = 0) and ω → 0 limit of the dynamical current-current correlation function [20]. In rest of the paper we work in natural units. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: Mott

text: Insulator

text: is

text: of

text: the

text: Falicov-Kimball

text: type,

text: In

text: this

text: paper,

text: by

text: using

text: DQMC

text: based

text: simulations

text: we

text: I

label: equation

text: mΛ(q

text: ω)

text: report

text: the

text: possibility

text: of

text: a

text: novel

text: interaction

text: driven

text: de-

text: σdc

text: li→m

text: localization

text: transition

text: in

text: a

text: Hubbard-model

text: in

text: the

text: presence

text: ω

text: ω

text: of

text: a

text: spin-selective

text: disorder.

text: We

text: show

text: that

text: this

text: delocal-

text: ization

text: transition

text: exists

text: for

text: ﬁlling

text: fractions

text: that

text: are

text: away

text: The

text: current-current

text: correlation

text: function

text: in

text: imaginary

text: from

text: half-ﬁlling.

text: We

text: further

text: show

text: that

text: this

text: de-localization

text: time

text: is

text: connected

text: to

text: the

text: dynamical

text: current-current

text: corre-

text: transition

text: resembles

text: the

text: Finkelstein

text: scenario

text: wherein

text: the

text: lation

text: function

text: via

text: integral

text: transform

text: given

text: by

text: Eq.

text: ferromagnetic

text: spin-susceptibility

text: gets

text: enhanced

text: close

text: to

text: the

text: MIT.

text: PROBLEM

text: AND

text: METHOD

text: Λ(q,

text: τ)

text: dω

text: e

text: ω−τ

text: ImΛ(q,

text: ω)

text: π

text: e

text: βω

text: The

text: starting

text: point

text: of

text: our

text: analysis

text: is

text: the

text: disordered

text: Hubbard

text: on

text: square

text: The

text: correlation

text: function

text: in

text: DQMC

text: are

text: measured

text: in

text: imaginary

text: time

text: τ.

text: Calculating

text: dynamical

text: correlation

text: function

text: given

text: correlation

text: in

text: imaginary

text: time

text: via

text: invert-

text: H

text: h.c)

text: µi,σni,σ

text: ing

text: Eq.

text: is

text: not

text: a

text: well

text: deﬁned

text: problem.

text: The

text: kernel,

text: i,j

text: ,σ

text: i,σ

text: k

text: 1−ee−ωτβω

text: decays

text: exponentially

text: with

text: frequency.

text: At

text: +U

text: ni,↑

text: ni,↓

text: τ

text: β2

text: the

text: kernel

text: k

text: reduces

text: to

text: and

text: it

text: Igets

text: most

text: i

text: of

text: the

text: contribution

text: from

text: the

text: small

text: ω

text: limit

text: .Now

text: ω)

text: can

text: be

text: replaced

text: by

text: ωσ

text: and

text: the

text: kernel

text: k

text: is

text: replaced

text: by

text: dc

text: Here,

text: in

text: Eq.

text: t

text: is

text: the

text: nearest

text: neighbour

text: hopping

text: am-

text: its

text: value

text: at

text: τ

text: β

text: plitude

text: which

text: has

text: been

text: set

text: to

text: unity.

text: The

text: on-sit†e

text: inter-

text: Once

text: made

text: becomes,

text: action

text: is

text: repulsive,

text: i.e.,

text: U

text: Furthermore,

text: c

text: and

text: i,σ

text: cdie,σﬁnaerde

text: tohnesfietremiioannidc

text: Λ

text: q

text: τ

text: β

text: dω

text: ωσdc

text: of

text: spin-selective

text: disorder

text: is

text: incorporated

text: by

text: making

text: the

text: 2π

text: sinh(βω)

text: random

text: chemical

text: potential

text: spin-dependent.

text: In

text: our

text: real-

text: ization

text: of

text: the

text: same,

text: we

text: assume

text: that

text: the

text: σ

text: feels

text: the

text: ef-

text: After

text: carrying

text: out

text: the

text: fect

text: of

text: a

text: random

text: chemical

text: potential

text: whereas

text: σ

text: moves

text: in

text: the

text: background

text: of

text: a

text: non-random

text: chemical

text: potential.

text: More

text: speciﬁcally,

text: the

text: chemical

text: potential

text: µ↓

text: is

text: uniformly

text: β2

text: β

text: σ

text: Λ

text: q

text: τ

text: sampled

text: from

text: the

text: range

text: dc

text: π

text: We

text: utilize

text: a

text: DQMC

text: scheme

text: numerically

text: solve

text: Now

text: σ

text: has

text: been

text: expressed

text: in

text: terms

text: of

text: a

text: current-

text: the

text: Hamiltonian

text: in

text: Eq.

text: The

text: DQMC

text: formalism

text: ini-

text: dc

text: current

text: correlation

text: function

text: calculated

text: in

text: imaginary

text: time.

text: tially

text: entails

text: the

text: slicing

text: of

text: the

text: inverse

text: temperature

text: β

text: into

text: Current-Current

text: correlation

text: function

text: in

text: imaginary

text: time

text: is

text: L

text: slices

text: of

text: width

text: ∆τ

text: each.

text: Further,

text: the

text: grand

text: canoni-

text: τ

text: calculated

text: using

text: DQMC

text: and

text: is

text: given

text: by,

text: cal

text: partition

text: function

text: is

text: decomposed

text: by

text: laying

text: recourse

text: to

text: the

text: Suzuki-Trotter

text: formalism

text: The

text: interacting

text: elec-

text: tron

text: problem

text: of

text: Eq.

text: is

text: now

text: recast

text: into

text: an

text: eﬀective

text: non-

text: Λ(q,

text: τ)

text: (q,

text: τ)j

text: (−q,

text: interacting

text: problem

text: by

text: utilizing

text: a

text: Hubbard-Stratonovich

text: x

text: x

text: transformation

text: in

text: terms

text: of

text: an

text: Ising-type

text: auxillary

text: ﬁeld.

text: The

text: fermionic

text: degrees

text: of

text: freedom

text: are

text: then

text: for-

text: where

text: j

text: (q,

text: τ)

text: is

text: the

text: fourier

text: transform

text: of

text: current

text: density

text: x

text: mally

text: integrated

text: out.

text: Various

text: physical

text: quantities

text: like

text: the

text: operator

text: given

text: in

text: real

text: space

text: by

text: j

text: (r,

text: τ),

text: x

text: Greens

text: function,

text: the

text: electronic

text: density,

text: and

text: the

text: two

text: par-

text: ticle

text: correlator

text: are

text: calculated

text: via

text: Monte-Carlo

text: sampling

text: of

text: the

text: Ising

text: Hubbard-Stratanovich

text: ﬁeld.

text: The

text: parame-

text: jx(r,

text: τ)

text: eτhH

text: jx(r)e

text: τhH

text: ter

text: space

text: of

text: our

text: simulation

text: is

text: spanned

text: by

text: the

text: interaction

text: strength

text: U,

text: the

text: disorder

text: strength

text: and

text: temperature

text: T,

text: which

text: are

text: all

text: expressed

text: in

text: terms

text: of

text: the

text: hopping

text: integral

text: t.

text: The

text: simulations

text: are

text: further

text: performed

text: at

text: various

text: ﬁlling

text: ieat

text: fractions.

text: j

text: (r)

text: (c

text: c

text: c

text: c

text: x

text: h

text: r+ex,σ

text: r,σ

text: r,σ

text: r+ex,σ

text: The

text: dc

text: conductivity

text: can

text: be

text: calculated

text: by

text: taking

text: the

text: σ

text: long

text: wavelength(q

text: and

text: ω

text: limit

text: of

text: the

text: dynamical

text: current-current

text: correlation

text: function

text: In

text: rest

text: of

text: the

text: paper

text: we

text: work

text: in

text: natural

text: units.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: PATHS TO UNDERSTANDING BIRATIONAL ROWMOTION ON PRODUCTS OF TWO CHAINS GREGG MUSIKER AND TOM ROBY Abstract. Birational rowmotion is an action on the space of assignments of rational functions to the elements of a ﬁnite partially-ordered set (poset). It is lifted from the well- studied rowmotion map on order ideals (equivariantly on antichains) of a poset P [AST11, BrS74, CaFl95, Pan09, PR15, RuSh12, RuWa15+, StWi11, ThWi17, Yil17], which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) [PR15]. In this context, rowmotion appears to be related to Auslander-Reiten translation on certain × quivers [Yil17], and birational rowmotion to Y -systems of type A A described in § m n Zamolodchikov periodicity [Rob16, 4.4]. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths r and s is r + s + 2 (ﬁrst proved by D. Grinberg and the second author [GrRo15]), as well as the ﬁrst proof of the birational analogue of homomesy along ﬁles for such posets. 1. Introduction The rowmotion map ρ, deﬁned on the set J(P ) of order ideals (equivariantly on antichains) of a poset P , has been thoroughly studied by a number of combinatorialists and representation theorists. When iterated on special posets, particularly root posets and (co)minuscule posets associated with representations of ﬁnite-dimensional Lie algebras, ρ has unexpected nice properties in terms of periodicity, cyclic sieving, and homomesy [AST11, BrS74, CaFl95, Pan09, PR15, RuSh12, RuWa15+, StWi11, ThWi17]. Excellent summaries of the history of this map and further references are available in the introductions of Striker-Williams [StWi11] and Thomas-Williams [ThWi17]. (cid:80) Armstrong, Stump, and Thomas [AST11] proved a conjecture of Panyushev [Pan09, Conj. O 2.1(iii)] that under the action of rowmotion on antichains of root posets, for any orbit , O the value 1O ∈O #A is a constant, independent of the choice of . This was one of the # A ﬁrst explicit statements of a type later isolated by Propp and the second author as the much more widespread homomesy phenomenon [PR15] (see Deﬁnition 2.6). (In particular, the antichain-cardinality is homomesic with respect to rowmotion on antichains of root posets.) Propp and Roby’s result that cardinality is a homomesic statistic for rowmotion acting on × J([a] [b]) was generalized by Rush and Wang to all minuscule posets [RuWa15+]. In the cominuscule (equivalently minuscule) context, there is a still mysterious connection between rowmotion on certain posets and Auslander-Reiten translation on related quivers [Yil17]. Date: 30 May 2020. Key words and phrases. birational rowmotion, dynamical algebraic combinatorics, homomesy, periodicity, toggling, 1 ##LTLine## ##LTLine##

text: PATHS

text: TO

text: UNDERSTANDING

text: BIRATIONAL

text: ROWMOTION

text: ON

text: PRODUCTS

text: OF

text: TWO

text: CHAINS

text: GREGG

text: MUSIKER

text: AND

text: TOM

text: ROBY

text: Abstract.

text: Birational

text: rowmotion

text: is

text: an

text: action

text: on

text: the

text: space

text: of

text: assignments

text: of

text: rational

text: functions

text: to

text: the

text: elements

text: of

text: a

text: ﬁnite

text: partially-ordered

text: set

text: (poset).

text: It

text: is

text: lifted

text: from

text: the

text: well-

text: studied

text: rowmotion

text: map

text: on

text: order

text: ideals

text: (equivariantly

text: on

text: antichains)

text: of

text: a

text: poset

text: P

text: which

text: when

text: iterated

text: on

text: special

text: posets,

text: has

text: unexpectedly

text: nice

text: properties

text: in

text: terms

text: of

text: periodicity,

text: cyclic

text: sieving,

text: and

text: homomesy

text: (statistics

text: whose

text: averages

text: over

text: each

text: orbit

text: are

text: constant)

text: In

text: this

text: context,

text: rowmotion

text: appears

text: to

text: be

text: related

text: to

text: Auslander-Reiten

text: translation

text: on

text: certain

text: quivers

text: and

text: birational

text: rowmotion

text: to

text: Y

text: -systems

text: of

text: type

text: A

text: A

text: described

text: in

text: m

text: n

text: Zamolodchikov

text: periodicity

text: We

text: give

text: a

text: formula

text: in

text: terms

text: of

text: families

text: of

text: non-intersecting

text: lattice

text: paths

text: for

text: iterated

text: actions

text: of

text: the

text: birational

text: rowmotion

text: map

text: on

text: a

text: product

text: of

text: two

text: chains.

text: This

text: allows

text: us

text: to

text: give

text: a

text: much

text: simpler

text: direct

text: proof

text: of

text: the

text: key

text: fact

text: that

text: the

text: period

text: of

text: this

text: map

text: on

text: a

text: product

text: of

text: chains

text: of

text: lengths

text: r

text: and

text: s

text: is

text: r

text: s

text: (ﬁrst

text: proved

text: by

text: D.

text: Grinberg

text: and

text: the

text: second

text: author

text: as

text: well

text: as

text: the

text: ﬁrst

text: proof

text: of

text: the

text: birational

text: analogue

text: of

text: homomesy

text: along

text: ﬁles

text: for

text: such

text: posets.

text: Introduction

text: The

text: rowmotion

text: map

text: ρ,

text: deﬁned

text: on

text: the

text: set

text: J(P

text: of

text: order

text: ideals

text: (equivariantly

text: on

text: antichains)

text: of

text: a

text: poset

text: P

text: has

text: been

text: thoroughly

text: studied

text: by

text: a

text: number

text: of

text: combinatorialists

text: and

text: representation

text: theorists.

text: When

text: iterated

text: on

text: special

text: posets,

text: particularly

text: root

text: posets

text: and

text: (co)minuscule

text: posets

text: associated

text: with

text: representations

text: of

text: ﬁnite-dimensional

text: Lie

text: algebras,

text: ρ

text: has

text: unexpected

text: nice

text: properties

text: in

text: terms

text: of

text: periodicity,

text: cyclic

text: sieving,

text: and

text: homomesy

text: Excellent

text: summaries

text: of

text: the

text: history

text: of

text: this

text: map

text: and

text: further

text: references

text: are

text: available

text: in

text: the

text: introductions

text: of

text: Striker-Williams

text: and

text: Thomas-Williams

text: Armstrong,

text: Stump,

text: and

text: Thomas

text: proved

text: a

text: conjecture

text: of

text: Panyushev

text: Conj.

text: O

text: that

text: under

text: the

text: action

text: of

text: rowmotion

text: on

text: antichains

text: of

text: root

text: posets,

text: for

text: any

text: orbit

text: O

text: the

text: value

text: ∈O

text: #A

text: is

text: a

text: constant,

text: independent

text: of

text: the

text: choice

text: of

text: This

text: was

text: one

text: of

text: the

text: A

text: ﬁrst

text: explicit

text: statements

text: of

text: a

text: type

text: later

text: isolated

text: by

text: Propp

text: and

text: the

text: second

text: author

text: as

text: the

text: much

text: more

text: widespread

text: homomesy

text: phenomenon

text: (see

text: Deﬁnition

text: (In

text: particular,

text: the

text: antichain-cardinality

text: is

text: homomesic

text: with

text: respect

text: to

text: rowmotion

text: on

text: antichains

text: of

text: root

text: posets.)

text: Propp

text: and

text: Roby’s

text: result

text: that

text: cardinality

text: is

text: a

text: homomesic

text: statistic

text: for

text: rowmotion

text: acting

text: on

text: J([a]

text: [b])

text: was

text: generalized

text: by

text: Rush

text: and

text: Wang

text: to

text: all

text: minuscule

text: posets

text: In

text: the

text: cominuscule

text: (equivalently

text: minuscule)

text: context,

text: there

text: is

text: a

text: still

text: mysterious

text: connection

text: between

text: rowmotion

text: on

text: certain

text: posets

text: and

text: Auslander-Reiten

text: translation

text: on

text: related

text: quivers

text: Date:

text: May

text: Key

text: words

text: and

text: phrases.

text: birational

text: rowmotion,

text: dynamical

text: algebraic

text: combinatorics,

text: homomesy,

text: periodicity,

text: toggling,

text: ##LTLine##

text: ##LTLine##


text: ρ G. MUSIKER AND T. ROBY PATHS TO UNDERSTANDING B path can only end at the top of the poset if it does not meet a red path that it can follow downward. It follows that one of these two bounce paths ends at the top of the interval, at ◦ the on top untouched by the red paths, and the other bounce path ends at the bottom of × the interval, at the on the bottom not covered by a twig. We call the former a vertical bounce path and the latter a horizontal bounce path. (Note that in the case that all ◦ blue paths point northeast (resp. northwest) starting from the bottom ’s, the horizontal bounce path turns out to be a twig as well. The remainder of our procedure is consistent whether or not we treat this as a twig or as a horizontal bounce path.) We proceed by interchanging the colors of the edges along the horizontal bounce path, along all × ◦ the twigs, and swap the and endpoints at the bottom, while leaving the remaining edges B ∪ R of unchanged (also leaving the colors of the vertical bounce path unchanged). We then truncate the vertical bounce path by deleting the bottommost edge. These transformations B(cid:48) R(cid:48) result in a new pair of lattice path families which we denote as ( , ). The bottom endpoints B(cid:48) for will be one step either to the northeast or northwest of the original ones, indicating respectively whether it is contributing to the ﬁrst or second summand on the right-hand R(cid:48) side of Lemma 10. The bottom endpoints of are skewed in the other direction, i.e., the southwest or the southeast, respectively. ∈ B ∈ R Furthermore, if the lattice paths L and L did not originally intersect, then their B R edges would not lie along any bounce path. Consequently, L would be a lattice path again B(cid:48) R(cid:48) B in unchanged3, and the same is true for L in . They would again not intersect. On R the other hand, if L and L did originally intersect, then they could meet along a bounce B R B path. Being part of larger NILPs, L would not intersect any path in and L would R B R not intersect any path in . Swapping colors of individual edges along L and L might B R break this intersection-free property, but since all colors of edges along a horizontal bounce B(cid:48) R(cid:48) path are swapped simultaneously, we ensure that each collection of paths, and , is still intersection-free. B(cid:48) R(cid:48) B(cid:48) Hence, the result is a new pair of NILPs ( , ) with the lower endpoints of on the second rank from the bottom of the interval skewed left (resp. right) while the lower endpoints of R(cid:48) are on the bottom rank of the interval and skewed right (resp. left). By construction, B(cid:48) ◦ ◦ R(cid:48) this map is well-deﬁned, and is a collection of k lattice paths from ’s to ’s, and is − × × a collection of (k 1) lattice paths from ’s to ’s. Thus the new pair represents a pair − − − − of monomials counted by ϕ (i k, j k + 1)ϕ − (i k + 1, j k) in the former case, and − −k − − k 1 counted by ϕ (i k + 1, j k)ϕ − (i k, j k + 1) in the latter case. k k 1 Finally this procedure is reversible, yielding the desired bijection. In particular, given a pair B(cid:48) R(cid:48) B(cid:48) of NILPs ( , ), which has the lower endpoints of skewed left (resp. right) while the lower R(cid:48) endpoints of are skewed right (resp. left), we build a vertical bounce path starting from the B(cid:48) leftmost (resp. rightmost) lower endpoint of and a horizontal bounce path starting from R(cid:48) the rightmost (resp. leftmost) lower endpoint of . Swapping colors along the horizontal bounce path and the twigs (deﬁned similarly to as above) yields a centrally symmetric pair B R of NILPs ( , ). The validity of this construction follows by the same argument which we used above. 3With the small exception of possibly truncating the bottommost leftmost or rightmost edge. However, even this change would not aﬀect intersections. 17 ##LTLine## ##LTLine##

text: ρ

text: G.

text: MUSIKER

text: AND

text: T.

text: ROBY

text: PATHS

text: TO

text: UNDERSTANDING

text: B

text: path

text: can

text: only

text: end

text: at

text: the

text: top

text: of

text: the

text: poset

text: if

text: it

text: does

text: not

text: meet

text: a

text: red

text: path

text: that

text: it

text: can

text: follow

text: downward.

text: It

text: follows

text: that

text: one

text: of

text: these

text: two

text: bounce

text: paths

text: ends

text: at

text: the

text: top

text: of

text: the

text: interval,

text: at

text: the

text: on

text: top

text: untouched

text: by

text: the

text: red

text: paths,

text: and

text: the

text: other

text: bounce

text: path

text: ends

text: at

text: the

text: bottom

text: of

text: the

text: interval,

text: at

text: the

text: on

text: the

text: bottom

text: not

text: covered

text: by

text: a

text: twig.

text: We

text: call

text: the

text: former

text: a

text: vertical

text: bounce

text: path

text: and

text: the

text: latter

text: a

text: horizontal

text: bounce

text: path.

text: (Note

text: that

text: in

text: the

text: case

text: that

text: all

text: blue

text: paths

text: point

text: northeast

text: (resp.

text: northwest)

text: starting

text: from

text: the

text: bottom

text: ’s,

text: the

text: horizontal

text: bounce

text: path

text: turns

text: out

text: to

text: be

text: a

text: twig

text: as

text: well.

text: The

text: remainder

text: of

text: our

text: procedure

text: is

text: consistent

text: whether

text: or

text: not

text: we

text: treat

text: this

text: as

text: a

text: twig

text: or

text: as

text: a

text: horizontal

text: bounce

text: path.)

text: We

text: proceed

text: by

text: interchanging

text: the

text: colors

text: of

text: the

text: edges

text: along

text: the

text: horizontal

text: bounce

text: path,

text: along

text: all

text: the

text: twigs,

text: and

text: swap

text: the

text: and

text: endpoints

text: at

text: the

text: bottom,

text: while

text: leaving

text: the

text: remaining

text: edges

text: B

text: R

text: of

text: unchanged

text: (also

text: leaving

text: the

text: colors

text: of

text: the

text: vertical

text: bounce

text: path

text: unchanged).

text: We

text: then

text: truncate

text: the

text: vertical

text: bounce

text: path

text: by

text: deleting

text: the

text: bottommost

text: edge.

text: These

text: transformations

text: result

text: in

text: a

text: new

text: pair

text: of

text: lattice

text: path

text: families

text: which

text: we

text: denote

text: as

text: The

text: bottom

text: endpoints

text: for

text: will

text: be

text: one

text: step

text: either

text: to

text: the

text: northeast

text: or

text: northwest

text: of

text: the

text: original

text: ones,

text: indicating

text: respectively

text: whether

text: it

text: is

text: contributing

text: to

text: the

text: ﬁrst

text: or

text: second

text: summand

text: on

text: the

text: right-hand

text: side

text: of

text: Lemma

text: The

text: bottom

text: endpoints

text: of

text: are

text: skewed

text: in

text: the

text: other

text: direction,

text: i.e.,

text: the

text: southwest

text: or

text: the

text: southeast,

text: respectively.

text: B

text: R

text: Furthermore,

text: if

text: the

text: lattice

text: paths

text: L

text: and

text: L

text: did

text: not

text: originally

text: intersect,

text: then

text: their

text: B

text: R

text: edges

text: would

text: not

text: lie

text: along

text: any

text: bounce

text: path.

text: Consequently,

text: L

text: would

text: be

text: a

text: lattice

text: path

text: again

text: B

text: in

text: and

text: the

text: same

text: is

text: true

text: for

text: L

text: in

text: They

text: would

text: again

text: not

text: intersect.

text: On

text: R

text: the

text: other

text: hand,

text: if

text: L

text: and

text: L

text: did

text: originally

text: intersect,

text: then

text: they

text: could

text: meet

text: along

text: a

text: bounce

text: B

text: R

text: B

text: path.

text: Being

text: part

text: of

text: larger

text: NILPs,

text: L

text: would

text: not

text: intersect

text: any

text: path

text: in

text: and

text: L

text: would

text: R

text: B

text: R

text: not

text: intersect

text: any

text: path

text: in

text: Swapping

text: colors

text: of

text: individual

text: edges

text: along

text: L

text: and

text: L

text: might

text: B

text: R

text: break

text: this

text: intersection-free

text: property,

text: but

text: since

text: all

text: colors

text: of

text: edges

text: along

text: a

text: horizontal

text: bounce

text: path

text: are

text: swapped

text: simultaneously,

text: we

text: ensure

text: that

text: each

text: collection

text: of

text: paths,

text: and

text: is

text: still

text: intersection-free.

text: Hence,

text: the

text: result

text: is

text: a

text: new

text: pair

text: of

text: NILPs

text: with

text: the

text: lower

text: endpoints

text: of

text: on

text: the

text: second

text: rank

text: from

text: the

text: bottom

text: of

text: the

text: interval

text: skewed

text: left

text: (resp.

text: right)

text: while

text: the

text: lower

text: endpoints

text: of

text: are

text: on

text: the

text: bottom

text: rank

text: of

text: the

text: interval

text: and

text: skewed

text: right

text: (resp.

text: left).

text: By

text: construction,

text: this

text: map

text: is

text: well-deﬁned,

text: and

text: is

text: a

text: collection

text: of

text: k

text: lattice

text: paths

text: from

text: ’s

text: to

text: ’s,

text: and

text: is

text: a

text: collection

text: of

text: (k

text: lattice

text: paths

text: from

text: ’s

text: to

text: ’s.

text: Thus

text: the

text: new

text: pair

text: represents

text: a

text: pair

text: of

text: monomials

text: counted

text: by

text: ϕ

text: (i

text: k,

text: j

text: k

text: (i

text: k

text: j

text: k)

text: in

text: the

text: former

text: case,

text: and

text: −k

text: k

text: counted

text: by

text: ϕ

text: (i

text: k

text: j

text: k)ϕ

text: (i

text: k,

text: j

text: k

text: in

text: the

text: latter

text: case.

text: k

text: k

text: Finally

text: this

text: procedure

text: is

text: reversible,

text: yielding

text: the

text: desired

text: bijection.

text: In

text: particular,

text: given

text: a

text: pair

text: of

text: NILPs

text: which

text: has

text: the

text: lower

text: endpoints

text: of

text: skewed

text: left

text: (resp.

text: right)

text: while

text: the

text: lower

text: endpoints

text: of

text: are

text: skewed

text: right

text: (resp.

text: left),

text: we

text: build

text: a

text: vertical

text: bounce

text: path

text: starting

text: from

text: the

text: leftmost

text: (resp.

text: rightmost)

text: lower

text: endpoint

text: of

text: and

text: a

text: horizontal

text: bounce

text: path

text: starting

text: from

text: the

text: rightmost

text: (resp.

text: leftmost)

text: lower

text: endpoint

text: of

text: Swapping

text: colors

text: along

text: the

text: horizontal

text: bounce

text: path

text: and

text: the

text: twigs

text: (deﬁned

text: similarly

text: to

text: as

text: above)

text: yields

text: a

text: centrally

text: symmetric

text: pair

text: B

text: R

text: of

text: NILPs

text: The

text: validity

text: of

text: this

text: construction

text: follows

text: by

text: the

text: same

text: argument

text: which

text: we

text: used

text: above.

text: the

text: small

text: exception

text: of

text: possibly

text: truncating

text: the

text: bottommost

text: leftmost

text: or

text: rightmost

text: edge.

text: However,

text: even

text: this

text: change

text: would

text: not

text: aﬀect

text: intersections.

text: ##LTLine##

text: ##LTLine##


text: (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) ρ G. MUSIKER AND T. ROBY PATHS TO UNDERSTANDING B AD + BC BD F D A + B C = = E E BD DG + BH E D G + B H F F where − − − − − A = µ([k j]+,[k 1 i]+)ϕ − − (i k + 1 + M , j k + M ) k 1 M10 10 10 − − − − − B = µ([k j]+,[k 1 i]+)ϕ − (i k + 1 + M , j k + M ) k M10 10 10 − − − − − C = µ([k 1 j]+,[k i]+)ϕ − − (i k + M , j k + 1 + M ) k 1 M01 01 01 − − − − − D = µ([k i]+,[k 1 j]+)ϕ − (i k + M , j k + 1 + M ) k M01 01 01 − − − − − − E = µ([k 1 j]+,[k 1 i]+)ϕ − − (i k + 1 + M , j k + 1 + M ) k 1 M11 11 11 − − − − − − F = µ([k 1 j]+,[k 1 i]+)ϕ − (i k + 1 + M , j k + 1 + M ) k M11 11 11 − − − − − G = µ([k j]+,[k 1 i]+)ϕ − (i k + 1 + M , j k + M ) k+1 M10 10 10 − − − − − H = µ([k 1 j]+,[k i]+)ϕ − (i k + M , j k + 1 + M ) − − − −k+1 M01 ∈ { 01} 01 using M = [k i (cid:15) ] + [k j (cid:15) ] for (cid:15) , (cid:15) 0, 1 . (cid:15)i,(cid:15)j i + j + i j To prove Theorem 2.2 (a), it therefore suﬃces to prove, using this above shorthand, that A C − − − − D + B = µ([k j]+,[k i]+)ϕ − (i k + M , j k + M ) E E k M00 00 00 and G H − − − − D + B = µ([k j]+,[k i]+)ϕ − (i k + M , j k + M ). F F k+1 M00 00 00 − − − − Letting α = µ([k j]+,[k i]+)ϕ − (i k + M , j k + M ) and − − k −M00 −00 00 β = µ([k j]+,[k i]+)ϕ − (i k + M , j k + M ), we note that these two equations, i.e., k+1 M00 00 00 αE = BC + DA and βF = GD + HB, both follow from two applications4 of Lemma 12. Remark 14. The proof is analogous in cases where the element (i, j) covers (or is covered by) only a single element, with some of the terms in the above expression being replaced with a 1 or a 0. 4 The second equation follows from substituting k by k + 1, i by i + 1, and j by j + 1. 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ρ

text: G.

text: MUSIKER

text: AND

text: T.

text: ROBY

text: PATHS

text: TO

text: UNDERSTANDING

text: B

text: AD

text: BC

text: BD

text: F

text: D

text: A

text: B

text: C

text: E

text: E

text: BD

text: DG

text: BH

text: E

text: D

text: G

text: B

text: H

text: F

text: F

text: where

text: A

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: k

text: B

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: k

text: C

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: k

text: D

text: µ([k

text: i]+,[k

text: j]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: k

text: E

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: k

text: F

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: k

text: G

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: H

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: using

text: M

text: [k

text: i

text: [k

text: j

text: for

text: i

text: j

text: i

text: j

text: To

text: prove

text: Theorem

text: (a),

text: it

text: therefore

text: suﬃces

text: to

text: prove,

text: using

text: this

text: above

text: shorthand,

text: that

text: A

text: C

text: D

text: B

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: E

text: E

text: k

text: and

text: G

text: H

text: D

text: B

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: F

text: F

text: Letting

text: α

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: and

text: k

text: β

text: µ([k

text: j]+,[k

text: i]+)ϕ

text: (i

text: k

text: M

text: j

text: k

text: M

text: we

text: note

text: that

text: these

text: two

text: equations,

text: i.e.,

text: αE

text: BC

text: DA

text: and

text: βF

text: GD

text: HB,

text: both

text: follow

text: from

text: two

text: of

text: Lemma

text: Remark

text: The

text: proof

text: is

text: analogous

text: in

text: cases

text: where

text: the

text: element

text: (i,

text: j)

text: covers

text: (or

text: is

text: covered

text: by)

text: only

text: a

text: single

text: element,

text: with

text: some

text: of

text: the

text: terms

text: in

text: the

text: above

text: expression

text: being

text: replaced

text: with

text: a

text: or

text: a

text: The

text: second

text: equation

text: follows

text: from

text: substituting

text: k

text: by

text: k

text: i

text: by

text: i

text: and

text: j

text: by

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: ρ G. MUSIKER AND T. ROBY PATHS TO UNDERSTANDING B − We next consider the case when M = k + 1 (i.e., i + j = k 1). We start with the degenerate (cid:122) (cid:122) 1 1 1 case ρ2B(0, 0) = ρ2B(1,ρ01B(cid:18))(0,ρ02B)(0,1) = xr−1,sx1rsxr,s−1 = (cid:19)xr(cid:18)−1,sx+1rsxr,s−1 = A1rs . Note here th(cid:19)at we used Claim 15 to simplify the calculations. Continuing by induction5, − − (cid:122) ρk (i, j 1) + ρk (i 1, j) ρk+1(i + 1, j) ρk+1(i, j + 1) B B B B ρk+1(i, j) = . B ρk (i, j) (cid:18) B (cid:19)(cid:18) (cid:19) Using Claim 15 and the inductive hypothesis, the right-hand side simpliﬁes to (cid:122) ρkB−1−(i+j−11)(cid:18)(r−i,s−j+1) + ρkB−1−(i+j−11)(r−i+1,s−j) 1/xr−i−1(cid:19),s−j 1/xr−i,s−j−1 = 1/x − − r i,s j 1 1 1 − − + − − . (cid:18) ρ1B(cid:88)(r i, s j + 1) ρ1B(r i + 1, s (cid:88)j) Ar−i,s−j − (cid:19) Using Equation (10), we can expand(cid:81)this out further as (assuming i + j(cid:81)= k 1) 1 1 1 ρk+1(i, j) = + . B paths L from (r−i,s−j+1) (p,q)∈L Apq paths L from (r−i+1,s−j) (p,q)∈L Apq Ar−i,s−j − − − − Any lattice path connecting (r i, s j) to (r, s) either goes through (r i, s j + 1) or − − (cid:88) through (r i + 1, s j). Combinin(cid:81)g these into a single sum over lattice paths, we get − − 1 ϕ (r i, s j) 1 ρk+1(i, j) = = 1 − − = − − , B paths L from (r−i,s−j) (p,q)∈L Apq ϕ0(r i, s j) ρ1B(r i, s j) agreeing with Theorem 2.2 (b) when k = i + j + 1. (cid:18) (cid:19)(cid:18) (cid:19) Lastly, when M > k + 1, we use Theorem 2.2 (b) inductively to obtain − − (cid:122) ρk (i, j 1) + ρk (i 1, j) ρk+1(i + 1, j) ρk+1(i, j + 1) B B B B (cid:18)ρk+1(i, j) = (cid:19)(cid:18) (cid:19) B ρk (i, j) B (cid:122) − − −1 − + − − −1 − − − − 1− − − − − − 1− − − ρk i j(r i,s j+1) ρk i j(r i+1,s j) ρk 1 i j(r i 1,s j) ρk 1 i j(r i,s j 1) (cid:18)= B B (cid:19)(cid:18) B B (cid:19) 1 (cid:32) − − − − − (cid:33) ρk 1 i j(r i,s j) B (cid:122) − − ρK+1(I, J + 1) ρK+1(I + 1, J) ρK(I 1, J) + ρK(I, J 1) −1 B B B B 1 = = ρK(I, J) ρK+1(I, J) B B − − − − − where K = k 1 i j, I = r i, J = s j. This ﬁnishes the proof. 5As above, if (i, j) only covers one element, we have a single summand rather than two inside the left parenthesis. 25 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ρ

text: G.

text: MUSIKER

text: AND

text: T.

text: ROBY

text: PATHS

text: TO

text: UNDERSTANDING

text: B

text: We

text: next

text: consider

text: the

text: case

text: when

text: M

text: k

text: (i.e.,

text: i

text: j

text: k

text: We

text: start

text: with

text: the

text: degenerate

text: case

text: Note

text: here

text: we

text: used

text: Claim

text: to

text: simplify

text: the

text: calculations.

text: Continuing

text: by

text: ρk

text: (i,

text: j

text: ρk

text: (i

text: j)

text: j)

text: j

text: B

text: B

text: B

text: B

text: j)

text: B

text: ρk

text: (i,

text: j)

text: B

text: Using

text: Claim

text: and

text: the

text: inductive

text: hypothesis,

text: the

text: right-hand

text: side

text: simpliﬁes

text: to

text: r

text: i,s

text: j

text: i,

text: s

text: j

text: i

text: s

text: Ar−i,s−j

text: Using

text: Equation

text: we

text: can

text: out

text: further

text: as

text: (assuming

text: i

text: k

text: j)

text: B

text: paths

text: L

text: from

text: (p,q)∈L

text: Apq

text: paths

text: L

text: from

text: (p,q)∈L

text: Apq

text: Ar−i,s−j

text: Any

text: lattice

text: path

text: connecting

text: (r

text: i,

text: s

text: j)

text: to

text: (r,

text: s)

text: either

text: goes

text: through

text: (r

text: i,

text: s

text: j

text: or

text: through

text: (r

text: i

text: s

text: j).

text: these

text: into

text: a

text: single

text: sum

text: over

text: lattice

text: paths,

text: we

text: get

text: ϕ

text: (r

text: i,

text: s

text: j)

text: j)

text: B

text: paths

text: L

text: from

text: (r−i,s−j)

text: (p,q)∈L

text: Apq

text: i,

text: s

text: j)

text: i,

text: s

text: j)

text: agreeing

text: with

text: Theorem

text: (b)

text: when

text: k

text: i

text: j

text: Lastly,

text: when

text: M

text: k

text: we

text: use

text: Theorem

text: (b)

text: inductively

text: to

text: obtain

text: ρk

text: (i,

text: j

text: ρk

text: (i

text: j)

text: j)

text: j

text: B

text: B

text: B

text: B

text: j)

text: B

text: ρk

text: (i,

text: j)

text: B

text: −1

text: −1

text: 1−

text: 1−

text: ρk

text: i

text: j(r

text: i,s

text: ρk

text: i

text: j(r

text: j)

text: ρk

text: i

text: j(r

text: i

text: j)

text: ρk

text: i

text: j(r

text: i,s

text: j

text: B

text: B

text: B

text: B

text: ρk

text: i

text: j(r

text: i,s

text: j)

text: B

text: J

text: J)

text: ρK(I

text: J)

text: ρK(I,

text: J

text: −1

text: B

text: B

text: B

text: B

text: ρK(I,

text: J)

text: J)

text: B

text: B

text: where

text: K

text: k

text: i

text: j,

text: I

text: r

text: i,

text: J

text: s

text: j.

text: This

text: ﬁnishes

text: the

text: proof.

text: above,

text: if

text: (i,

text: j)

text: only

text: covers

text: one

text: element,

text: we

text: have

text: a

text: single

text: summand

text: rather

text: than

text: two

text: inside

text: the

text: left

text: parenthesis.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ρ PATHS TO UNDERSTANDING B G. MUSIKER AND T. ROBY Figure 5. Illustrating the double-counting argument as in Example 19. (cid:20) (cid:21) − − − − ϕ − − (k + c s, k + c d) − ≤ ≤ − µ(k+c s,k+c d) s+d k 2c − − for d c k d + s 2c. ϕ − − (k + c s, k + c d) −s+d−k 2c+1 − − ϕ − − (r + s + 1 (cid:20)c k, d + s + 1(cid:21) c k) − ≤ ≤ − k d s+2c − − − − for d + s + 1 2c k d + s + 1 c, ϕ − − − (r + s + 1 c k, d + s + 1 c k) k d s+2c 1 − − − − ϕ (r d, 0) − ≤ ≤ − µ(k s 1+c d(cid:20),0) c+1 − for d + s + 1 c (cid:21) k r + s + 1 c, ϕ (r d, 0) c − − − − − − − − − − − − ϕ − (k + c s 1 d, k + c r s 1) − ≤ ≤ µ(k+c s 1 d,k+c r s 1) r+s+2 k − − − − − − for r + s + 1 c k r + s + 1. ϕ − (k + c s 1 d, k + c r s 1) r+s+1 k This third case includes the possibility of the middle-most ﬁle if r + s = 2d where the antipodal map sends elements from the top half of the ﬁle to the bottom half of the same ﬁle. Either way, the same analysis utilizing cancellations of numerators and denominators applies. { } Example 19. Let (r, s) = (4, 3), and d = 2, with corresponding ﬁle F = (4, 2), (3, 1), (2, 0) . The left-hand side of Figure 5 shows the contributions corresponding to Equation (13), while the right-hand side shows the contributions corresponding to the reciprocal of Equation (14). For example, in the left picture, the 6-element order ﬁlter at (3, 1) represents ϕ (3, 1), while 0 the 6-element interval [(0, 0), (1, 2)] represents µ(3,1)ϕ (3, 1). Either way, the full product can 0 also be expressed as a product built up ﬁle-by-ﬁle as · · · (A )1(A A )2(A A A A A A A A A A A A A A )3(A A )2(A )1. 40 41 30 42 31 20 40 32 21 10 33 22 11 00 23 12 01 13 02 03 4. Connections to other works and future directions We have noticed the following connections to our work and some open problems for further exploration. We went through a large number of candidate bijections, none of which worked, before ﬁnding the colorful combinatorial bijections (proofs of Lemma 10–12) at the heart of our proof of 30 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ρ

text: PATHS

text: TO

text: UNDERSTANDING

text: B

text: G.

text: MUSIKER

text: AND

text: T.

text: ROBY

text: Figure

text: Illustrating

label: caption

text: the

text: double-counting

text: argument

text: as

text: in

text: Example

text: ϕ

text: (k

text: c

text: s,

text: k

text: c

text: d)

text: µ(k+c

text: s,k+c

text: d)

text: s+d

text: k

text: for

text: d

text: c

text: k

text: d

text: s

text: ϕ

text: (k

text: c

text: s,

text: k

text: c

text: d)

text: −s+d−k

text: ϕ

text: (r

text: s

text: k,

text: d

text: s

text: c

text: k)

text: k

text: d

text: for

text: d

text: s

text: k

text: d

text: s

text: c,

text: ϕ

text: (r

text: s

text: c

text: k,

text: d

text: s

text: c

text: k)

text: k

text: d

text: ϕ

text: (r

text: d,

text: µ(k

text: s

text: for

text: d

text: s

text: c

text: k

text: r

text: s

text: c,

text: ϕ

text: (r

text: d,

text: c

text: ϕ

text: (k

text: c

text: s

text: d,

text: k

text: c

text: r

text: s

text: µ(k+c

text: s

text: d,k+c

text: r

text: s

text: k

text: for

text: r

text: s

text: c

text: k

text: r

text: s

text: ϕ

text: (k

text: c

text: s

text: d,

text: k

text: c

text: r

text: s

text: k

text: This

text: third

text: case

text: includes

text: the

text: possibility

text: of

text: the

text: middle-most

text: ﬁle

text: if

text: r

text: s

text: where

text: the

text: antipodal

text: map

text: sends

text: elements

text: from

text: the

text: top

text: half

text: of

text: the

text: ﬁle

text: to

text: the

text: bottom

text: half

text: of

text: the

text: same

text: ﬁle.

text: Either

text: way,

text: the

text: same

text: analysis

text: utilizing

text: cancellations

text: of

text: numerators

text: and

text: denominators

text: applies.

text: Example

text: Let

text: (r,

text: s)

text: and

text: d

text: with

text: corresponding

text: ﬁle

text: F

text: The

text: left-hand

text: side

text: of

text: Figure

text: shows

text: the

text: contributions

text: corresponding

text: to

text: Equation

text: while

text: the

text: right-hand

text: side

text: shows

text: the

text: contributions

text: corresponding

text: to

text: the

text: reciprocal

text: of

text: Equation

text: For

text: example,

text: in

text: the

text: left

text: picture,

text: the

text: order

text: ﬁlter

text: at

text: represents

text: ϕ

text: while

text: the

text: interval

text: represents

text: Either

text: way,

text: the

text: full

text: product

text: can

text: also

text: be

text: expressed

text: as

text: a

text: product

text: built

text: up

text: ﬁle-by-ﬁle

text: as

text: (A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: Connections

text: to

text: other

text: works

text: and

text: future

text: directions

text: We

text: have

text: noticed

text: the

text: following

text: connections

text: to

text: our

text: work

text: and

text: some

text: open

text: problems

text: for

text: further

text: exploration.

text: We

text: went

text: through

text: a

text: large

text: number

text: of

text: candidate

text: bijections,

text: of

text: which

text: worked,

text: before

text: ﬁnding

text: the

text: colorful

text: combinatorial

text: bijections

text: (proofs

text: of

text: Lemma

text: at

text: the

text: heart

text: of

text: our

text: proof

text: of

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: resulting spatiotemporal EEG responses to measure their algorithmic complex- ity, based on the Lempel-Ziv compression. For a given segment of EEG data, the Lempel-Ziv algorithm quantiﬁes complexity by counting the number of dis- tinct patterns in the data. For example, this can be proportional to the size of a computer ﬁle after applying a data compression algorithm. Computing the Lempel-Ziv compressibility requires binarizing the time-series data, based either on event-related potentials or with respect to a given threshold. Using PCI, [18] were able to discriminate levels of consciousness during wakefulness, sleep, and anesthesia, as well as in patients who had emerged from coma and recovered a minimal level of consciousness. Later, the Lempel-Ziv complexity was also used by [31] on spontaneous high-density EEG data recorded from subjects under- going propofol-induced anesthesia. Once again, a robust decline in complexity was observed during anesthesia. These are complexity measures based on data compression algorithms. A qualitative comparison between a data compression measure inspired by PCI and Φ was made in [38]. While compression-based measures do seem to capture certain aspects of Φ, the exact relationship be- tween the two is not completely clear. Nonetheless, these empirical measures have been useful for clinical purposes, in terms of broadly discriminating dis- orders of consciousness. Another relevant complexity measure is the weighted symbolic mutual information (wSMI), introduced by [25]. This is a measure of global information sharing across brain areas. It evaluates the extent to which two EEG channels present nonrandom joint ﬂuctuations, suggesting that they share common sources. This is done by ﬁrst transforming continuous signals into discrete symbols, and subsequently computing the joint probabilities of symbol pairs between two EEG channels. Before computing the symbolic mutual infor- mation between two time-series signals, a weighting is introduced to disregard conjunctions of identical or opposite-sign symbols from the two signal trains as that could potentially arise from common-source artifacts. In [25] wSMI was estimated for 181 EEG recordings from awake but noncommunicating patients diagnosed in various disorders of consciousness (including 143 from patients in vegetative and minimally conscious states). This measure of information sharing was found to systematically increases with consciousness. In particular, it was able to distinguish patients in the vegetative state, minimally conscious state, and fully conscious state. In Table 2 we summarize the above empirical mea- sures along with their domains of application. 4 Discussion The paradigm-shifting proposal that consciousness might be measurable in terms of the information generated by causal dynamics of the brain as a whole, over the sum of its parts, has led to precise quantitative formulations of information- theoretic complexity measures. These measures seek to operationalize the intu- ition that the complexity associated to consciousness arises from simultaneous integration and diﬀerentiation of the brain’s structural and dynamical hierar- chies. However, progress in this direction has faced practical challenges such as high computational cost upon scaling with network size. This is especially true with regard to realistic neuroimaging or physiological datasets. Even in the

text: resulting

text: spatiotemporal

text: EEG

text: responses

text: to

text: measure

text: their

text: algorithmic

text: complex-

text: ity,

text: based

text: on

text: the

text: Lempel-Ziv

text: compression.

text: For

text: a

text: given

text: segment

text: of

text: EEG

text: data,

text: the

text: Lempel-Ziv

text: algorithm

text: quantiﬁes

text: complexity

text: by

text: counting

text: the

text: number

text: of

text: dis-

text: tinct

text: patterns

text: in

text: the

text: data.

text: For

text: example,

text: this

text: can

text: be

text: proportional

text: to

text: the

text: size

text: of

text: a

text: computer

text: ﬁle

text: after

text: applying

text: a

text: data

text: compression

text: algorithm.

text: Computing

text: the

text: Lempel-Ziv

text: compressibility

text: requires

text: binarizing

text: the

text: time-series

text: data,

text: based

text: either

text: on

text: event-related

text: potentials

text: or

text: with

text: respect

text: to

text: a

text: given

text: threshold.

text: Using

text: PCI,

text: were

text: able

text: to

text: discriminate

text: levels

text: of

text: consciousness

text: during

text: wakefulness,

text: sleep,

text: and

text: anesthesia,

text: as

text: well

text: as

text: in

text: patients

text: who

text: had

text: emerged

text: from

text: coma

text: and

text: recovered

text: a

text: minimal

text: level

text: of

text: consciousness.

text: Later,

text: the

text: Lempel-Ziv

text: complexity

text: was

text: also

text: used

text: by

text: on

text: spontaneous

text: high-density

text: EEG

text: data

text: recorded

text: from

text: subjects

text: under-

text: going

text: propofol-induced

text: anesthesia.

text: Once

text: again,

text: a

text: robust

text: decline

text: in

text: complexity

text: was

text: observed

text: during

text: anesthesia.

text: These

text: are

text: complexity

text: measures

text: based

text: on

text: data

text: compression

text: algorithms.

text: A

text: qualitative

text: comparison

text: between

text: a

text: data

text: compression

text: measure

text: inspired

text: by

text: PCI

text: and

text: Φ

text: was

text: made

text: in

text: While

text: compression-based

text: measures

text: do

text: seem

text: to

text: capture

text: certain

text: aspects

text: of

text: Φ,

text: the

text: exact

text: relationship

text: be-

text: tween

text: the

text: two

text: is

text: not

text: completely

text: clear.

text: Nonetheless,

text: these

text: empirical

text: measures

text: have

text: been

text: useful

text: for

text: clinical

text: purposes,

text: in

text: terms

text: of

text: broadly

text: discriminating

text: dis-

text: orders

text: of

text: consciousness.

text: Another

text: relevant

text: complexity

text: measure

text: is

text: the

text: weighted

text: symbolic

text: mutual

text: information

text: (wSMI),

text: introduced

text: by

text: This

text: is

text: a

text: measure

text: of

text: global

text: information

text: sharing

text: across

text: brain

text: areas.

text: It

text: evaluates

text: the

text: extent

text: to

text: which

text: two

text: EEG

text: channels

text: present

text: nonrandom

text: joint

text: ﬂuctuations,

text: suggesting

text: that

text: they

text: share

text: common

text: sources.

text: This

text: is

text: done

text: by

text: ﬁrst

text: transforming

text: continuous

text: signals

text: into

text: discrete

text: symbols,

text: and

text: subsequently

text: computing

text: the

text: joint

text: probabilities

text: of

text: symbol

text: pairs

text: between

text: two

text: EEG

text: channels.

text: Before

text: computing

text: the

text: symbolic

text: mutual

text: infor-

text: mation

text: between

text: two

text: time-series

text: signals,

text: a

text: weighting

text: is

text: introduced

text: to

text: disregard

text: conjunctions

text: of

text: identical

text: or

text: opposite-sign

text: symbols

text: from

text: the

text: two

text: signal

text: trains

text: as

text: that

text: could

text: potentially

text: arise

text: from

text: common-source

text: artifacts.

text: In

text: wSMI

text: was

text: estimated

text: for

text: EEG

text: recordings

text: from

text: awake

text: but

text: noncommunicating

text: patients

text: diagnosed

text: in

text: various

text: disorders

text: of

text: consciousness

text: (including

text: from

text: patients

text: in

text: vegetative

text: and

text: minimally

text: conscious

text: states).

text: This

text: measure

text: of

text: information

text: sharing

text: was

text: found

text: to

text: systematically

text: increases

text: with

text: consciousness.

text: In

text: particular,

text: it

text: was

text: able

text: to

text: distinguish

text: patients

text: in

text: the

text: vegetative

text: state,

text: minimally

text: conscious

text: state,

text: and

text: fully

text: conscious

text: state.

text: In

text: Table

text: we

text: summarize

text: the

text: above

text: empirical

text: mea-

text: sures

text: along

text: with

text: their

text: domains

text: of

text: application.

text: Discussion

text: The

text: paradigm-shifting

text: proposal

text: that

text: consciousness

text: might

text: be

text: measurable

text: in

text: terms

text: of

text: the

text: information

text: generated

text: by

text: causal

text: dynamics

text: of

text: the

text: brain

text: as

text: a

text: whole,

text: over

text: the

text: sum

text: of

text: its

text: parts,

text: has

text: led

text: to

text: precise

text: quantitative

text: formulations

text: of

text: information-

text: theoretic

text: complexity

text: measures.

text: These

text: measures

text: seek

text: to

text: operationalize

text: the

text: intu-

text: ition

text: that

text: the

text: complexity

text: associated

text: to

text: consciousness

text: arises

text: from

text: simultaneous

text: integration

text: and

text: diﬀerentiation

text: of

text: the

text: brain’s

text: structural

text: and

text: dynamical

text: hierar-

text: chies.

text: However,

text: progress

text: in

text: this

text: direction

text: has

text: faced

text: practical

text: challenges

text: such

text: as

text: high

text: computational

text: cost

text: upon

text: scaling

text: with

text: network

text: size.

text: This

text: is

text: especially

text: true

text: with

text: regard

text: to

text: realistic

text: neuroimaging

text: or

text: physiological

text: datasets.

text: Even

text: in

text: the


text: 14 YIANNIS SAKELLARIDIS Part 1. General considerations 2. CALCULUS OF EQUIVARIANT FOURIER TRANSFORMS Before we embark on trace formula comparisons, we need to introduce a basic family of transformations that will appear all the time, the equivariant Fourier transforms. Here, “equivariant” means with respect to the action of a multiplicative group or, more generally, a torus. We ﬁx throughout a non-trivial unitary character ψ of the additive group F . At non-Archimedean places, we will be assuming that its conductor is the ring of integers o. We also ﬁx a measure dx on F which is self-dual with | | respect to ψ; this induces a measure ω on X(F ), for every volume form ω × on a smooth variety X over F . We will also use the measure d x := |dx| on × x the multiplicative group F . M Let T be a torus, and (T ) a space of measures on T = T (F ) (with 0 ∈ C properties to be speciﬁed). For any s , consider the distribution | | × D := x sψ(x)d x (2.1) s on F ×. Any cocharacter λˇ : G → T induces, by push-forward, a distri- m F bution λˇ∗D on T . The equivariant Fourier transform is deﬁned as the s λˇ,s M operator of convolution by λˇ∗D , on the given space of measures (T ). If M S s 0 (T ) = (T ), the convolution is convergent. In the general case, we will 0 × M S typically need to regularize it. For example, if T = F and (T ) = (F ) × ⊂ 0 (considered by(cid:90) restriction as measures on F (cid:90) F ), with λˇ the identity cocharacter, we formally have: F | | − × | |− | | − − f(ξ) = x sψ(x)f(x 1ξ)d x = ξ s x s 1f(x 1)ψ(xξ)dx, λˇ,s × × F F and the integral on the right makes sense as the Fourier transform of the (cid:55)→ | | − − distribution x x s 1f(x 1), whenever the latter has a “natural” inter- pretation as a tempered distribution. “Fourier transform”, here, maps dis- tributions to distributions, since we have ﬁxed the self-dual measure dx. In general, I will leave “natural” interpretations of such regularizations to the reader to ﬁgure out, unless there is something highly non-trivial to be pointed out; the typical example of a distribution that needs to be under- | | ∈ S stood as a tempered distribution is of the form x sf(x), where f (F ) is a (cid:60) − Schwartz measure; it is well-known that this is a measure when (s) > 1, and makes sense by analytic continuation as a distribution for all but a countable set of values of s. Finally, we discuss Mellin transforms on tori and functional equations under the equivariant Fourier transforms. Let f be a measure on a torus T . Its Mellin transform is a function on the character group of T , typically ##LTLine##

font: JSSNYA+URWPalladioL-Roma

text: YIANNIS

text: SAKELLARIDIS

text: Part

font: OLNZRV+URWPalladioL-Bold

text: General

text: considerations

text: CALCULUS

text: OF

text: EQUIVARIANT

text: FOURIER

text: TRANSFORMS

text: Before

text: we

text: embark

text: on

text: trace

text: formula

text: comparisons,

text: we

text: need

text: to

text: introduce

text: a

text: basic

text: family

text: of

text: transformations

text: that

text: will

text: appear

text: all

text: the

text: time,

text: the

text: equivariant

font: ZZYUIZ+URWPalladioL-Ital

text: Fourier

text: transforms.

text: Here,

text: “equivariant”

text: means

text: with

text: respect

text: to

text: the

text: action

text: of

text: a

text: multiplicative

text: group

text: or,

text: more

text: generally,

text: a

text: torus.

text: We

text: ﬁx

text: throughout

text: a

text: non-trivial

text: unitary

text: character

text: ψ

text: of

text: the

text: additive

text: group

text: F

text: At

text: non-Archimedean

text: places,

text: we

text: will

text: be

text: assuming

text: that

text: its

text: conductor

text: is

text: the

text: ring

text: of

text: integers

text: o.

text: We

text: also

text: ﬁx

text: a

text: measure

text: dx

text: on

text: F

text: which

text: is

text: self-dual

text: with

text: respect

text: to

text: ψ;

text: this

text: induces

text: a

text: measure

text: ω

text: on

text: X(F

text: for

text: every

text: volume

text: form

text: ω

text: on

text: a

text: smooth

text: variety

text: X

text: over

text: F

text: We

text: will

text: also

text: use

text: the

text: measure

text: d

text: x

text: |dx|

text: on

text: x

text: the

text: multiplicative

text: group

text: F

text: M

text: Let

text: T

text: be

text: a

text: torus,

text: and

text: (T

text: a

text: space

text: of

text: measures

text: on

text: T

text: T

text: (F

text: (with

text: C

text: properties

text: to

text: be

text: speciﬁed).

text: For

text: any

text: s

text: consider

text: the

text: distribution

text: D

text: x

text: sψ(x)d

text: x

text: s

text: on

text: F

text: Any

text: cocharacter

text: λˇ

text: G

text: T

text: induces,

text: by

text: push-forward,

text: a

text: distri-

text: m

text: F

text: bution

text: λˇ∗D

text: on

text: T

text: The

text: equivariant

text: Fourier

text: transform

text: is

text: deﬁned

text: as

text: the

text: s

text: λˇ,s

text: M

text: operator

text: of

text: convolution

text: by

text: λˇ∗D

text: on

text: the

text: given

text: space

text: of

text: measures

text: (T

text: If

text: M

text: S

text: s

text: (T

text: (T

text: the

text: convolution

text: is

text: convergent.

text: In

text: the

text: general

text: case,

text: we

text: will

text: M

text: S

text: typically

text: need

text: to

text: regularize

text: it.

text: For

text: example,

text: if

text: T

text: F

text: and

text: (T

text: (F

text: (considered

text: restriction

text: as

text: measures

text: on

text: F

text: F

text: with

text: λˇ

text: the

text: identity

text: cocharacter,

text: we

text: formally

text: have:

text: F

text: f(ξ)

text: x

text: sψ(x)f(x

text: x

text: ξ

text: s

text: x

text: s

text: λˇ,s

text: F

text: F

text: and

text: the

text: integral

text: on

text: the

text: right

text: makes

text: sense

text: as

text: the

text: Fourier

text: transform

text: of

text: the

text: distribution

text: x

text: x

text: s

text: whenever

text: the

text: latter

text: has

text: a

text: “natural”

text: inter-

text: pretation

text: as

text: a

text: tempered

text: distribution.

text: “Fourier

text: transform”,

text: here,

text: maps

text: dis-

text: tributions

text: to

text: distributions,

text: since

text: we

text: have

text: ﬁxed

text: the

text: self-dual

text: measure

text: dx.

text: In

text: general,

text: I

text: will

text: leave

text: “natural”

text: interpretations

text: of

text: such

text: regularizations

text: to

text: the

text: reader

text: to

text: ﬁgure

text: out,

text: unless

text: there

text: is

text: something

text: highly

text: non-trivial

text: to

text: be

text: pointed

text: out;

text: the

text: typical

text: example

text: of

text: a

text: distribution

text: that

text: needs

text: to

text: be

text: under-

text: S

text: stood

text: as

text: a

text: tempered

text: distribution

text: is

text: of

text: the

text: form

text: x

text: sf(x),

text: where

text: f

text: (F

text: is

text: a

text: Schwartz

text: measure;

text: it

text: is

text: well-known

text: that

text: this

text: is

text: a

text: measure

text: when

text: (s)

text: and

text: makes

text: sense

text: by

text: analytic

text: continuation

text: as

text: a

text: distribution

text: for

text: all

text: but

text: a

text: countable

text: set

text: of

text: values

text: of

text: s.

text: Finally,

text: we

text: discuss

text: Mellin

text: transforms

text: on

text: tori

text: and

text: functional

text: equations

text: under

text: the

text: equivariant

text: Fourier

text: transforms.

text: Let

text: f

text: be

text: a

text: measure

text: on

text: a

text: torus

text: T

text: Its

text: Mellin

text: transform

text: is

text: a

text: function

text: on

text: the

text: character

text: group

text: of

text: T

text: typically

text: ##LTLine##