text: 7 F. Khalvati, A. Wong, and M. A. Haider, “Automated prostate cancer detection via compre- hensive multi-parametric magnetic resonance imaging texture feature models,” BMC medical imaging 15(1), 27 (2015). 8 A. Cameron, F. Khalvati, A. Wong, et al., “Maps: A quantitative radiomics approach for prostate cancer detection,” IEEE Transactions on Biomedical Engineering 63(6), 1145–1156 (2015). 9 R. Anirudh, J. J. Thiagarajan, T. Bremer, et al., “Lung nodule detection using 3d convolu- tional neural networks trained on weakly labeled data,” in SPIE Medical Imaging, 978532– 978532, International Society for Optics and Photonics (2016). 10 H. M. Orozco, O. Villegas, V. G. C. Sa´nchez, et al., “Automated system for lung nodules classiﬁcation based on wavelet feature descriptor and support vector machine,” Biomedical engineering online 14(1), 9 (2015). 11 W. Shen, M. Zhou, F. Yang, et al., “Multi-scale convolutional neural networks for lung nodule classiﬁcation,” 588–599 (2015). 12 W. Shen, M. Zhou, F. Yang, et al., “Multi-crop convolutional neural networks for lung nodule malignancy suspiciousness classiﬁcation,” Pattern Recognition 61, 663–673 (2017). 13 D. Kumar, A. Wong, and D. A. Clausi, “Lung nodule classiﬁcation using deep features in ct images,” Computer and Robot Vision (CRV), 2015 12th Conference on , 133–138 (2015). 14 W. Shen, M. Zhou, F. Yang, et al., “Learning from experts: Developing transferable deep features for patient-level lung cancer prediction,” in International Conference on Medical Image Computing and Computer-Assisted Intervention, 124–131, Springer (2016). 15 A.-H. Karimi, A. G. Chung, M. J. Shaﬁee, et al., “Discovery radiomics via a mixture of deep 22

label: reference

text: F.

text: Khalvati,

text: A.

text: Wong,

text: and

text: M.

text: A.

text: Haider,

text: “Automated

text: prostate

text: cancer

text: detection

text: via

text: compre-

text: hensive

text: multi-parametric

text: magnetic

text: resonance

text: imaging

text: texture

text: feature

text: models,”

text: BMC

text: medical

text: imaging

text: A.

text: Cameron,

text: F.

text: Khalvati,

text: A.

text: Wong,

text: et

text: al.,

text: “Maps:

text: A

text: quantitative

text: radiomics

text: approach

text: for

text: prostate

text: cancer

text: detection,”

text: IEEE

text: Transactions

text: on

text: Biomedical

text: Engineering

text: R.

text: Anirudh,

text: J.

text: J.

text: Thiagarajan,

text: T.

text: Bremer,

text: et

text: al.,

text: “Lung

text: nodule

text: detection

text: using

text: convolu-

text: tional

text: neural

text: networks

text: trained

text: on

text: weakly

text: labeled

text: data,”

text: in

text: SPIE

text: Medical

text: Imaging,

text: International

text: Society

text: for

text: Optics

text: and

text: Photonics

text: H.

text: M.

text: Orozco,

text: O.

text: Villegas,

text: V.

text: G.

text: C.

text: Sa´nchez,

text: et

text: al.,

text: “Automated

text: system

text: for

text: lung

text: nodules

text: classiﬁcation

text: based

text: on

text: wavelet

text: feature

text: descriptor

text: and

text: support

text: vector

text: machine,”

text: Biomedical

text: engineering

text: online

text: W.

text: Shen,

text: M.

text: Zhou,

text: F.

text: Yang,

text: et

text: al.,

text: “Multi-scale

text: convolutional

text: neural

text: networks

text: for

text: lung

text: nodule

text: classiﬁcation,”

text: W.

text: Shen,

text: M.

text: Zhou,

text: F.

text: Yang,

text: et

text: al.,

text: “Multi-crop

text: convolutional

text: neural

text: networks

text: for

text: lung

text: nodule

text: malignancy

text: suspiciousness

text: classiﬁcation,”

text: Pattern

text: Recognition

text: D.

text: Kumar,

text: A.

text: Wong,

text: and

text: D.

text: A.

text: Clausi,

text: “Lung

text: nodule

text: classiﬁcation

text: using

text: deep

text: features

text: in

text: ct

text: images,”

text: Computer

text: and

text: Robot

text: Vision

text: (CRV),

text: Conference

text: on

text: W.

text: Shen,

text: M.

text: Zhou,

text: F.

text: Yang,

text: et

text: al.,

text: “Learning

text: from

text: experts:

text: Developing

text: transferable

text: deep

text: features

text: for

text: patient-level

text: lung

text: cancer

text: prediction,”

text: in

text: International

text: Conference

text: on

text: Medical

text: Image

text: Computing

text: and

text: Computer-Assisted

text: Intervention,

text: Springer

text: A.-H.

text: Karimi,

text: A.

text: G.

text: Chung,

text: M.

text: J.

text: Shaﬁee,

text: et

text: al.,

text: “Discovery

text: radiomics

text: via

text: a

text: mixture

text: of

text: deep


text: Schnyder woods, SLE , and Liouville quantum gravity 16 Yiting Li Xin Sun Samuel S. Watson Brandeis University MIT Brown University Abstract In 1990, Schnyder used a 3-spanning-tree decomposition of a simple triangulation, now known as the Schnyder wood, to give a fundamental grid-embedding algorithm for planar maps. We show that a uniformly sampled Schnyder-wood-decorated triangulation with n vertices converges as n → ∞ to the Liouville quantum gravity with parameter 1, decorated with a triple of SLE ’s of angle diﬀerence 2π/3 in the imaginary geometry 16 sense. Our convergence result provides a description of the continuum limit of Schnyder’s embedding algorithm via Liouville quantum gravity and imaginary geometry. 1 Introduction A planar map is an embedding of a connected planar graph in the plane, considered modulo orientation-preserving homeomorphism. A planar map is said to be simple if it does not have any edges from a vertex to itself or multiple edges between the same pair of vertices. We say that a planar map is a triangulation if every face is bounded by three edges. A spanning tree of a graph G is a connected, cycle-free subgraph of G including all vertices of G. In this paper we will study wooded triangulations, which are simple triangulations equipped with a certain 3-spanning-tree decomposition known as a Schnyder wood. The Schnyder wood, also referred to as a graph realizer in the literature, was introduced by Walter Schnyder [Sch89] to prove a characterization of graph planarity. He later used Schnyder woods to describe an algorithm for embedding an order-n planar graph in such a way that its edges are straight lines and its vertices lie on an (n − 2) × (n − 2) grid [Sch90]. Schnyder’s celebrated construction has continued to play an important role in graph theory and enumerative combinatorics [BB09, FZ08, MRST16]. In the present article, we are primarily interested in the n → ∞ behavior of wooded triangulations and their Schnyder embeddings. Consider a simple plane triangulation M with unbounded face (cid:52)A A A where the b r g outer vertices A , A , and A —which we think of as blue, red, and green—are arranged in b r g counterclockwise order. We denote by A A the edge connecting A and A , and similarly r g r g for A A and A A . Vertices, edges, and faces of M other than A , A , A , A A , A A , A A g b b r b r g r g g b b r and (cid:52)A A A are called inner vertices, edges, and faces respectively. We deﬁne the size b r g of M to be the number of interior vertices. Euler’s formula implies that a simple plane triangulation of size n has n + 3 vertices, 3n + 3 edges, and 2n + 2 faces. 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Schnyder

label: title

text: woods,

text: SLE

text: and

text: Liouville

text: quantum

text: gravity

text: Yiting

label: author

text: Li

text: Xin

text: Sun

text: Samuel

text: S.

text: Watson

text: Brandeis

text: University

text: MIT

text: Brown

text: University

text: Abstract

label: paragraph

text: In

label: abstract

text: Schnyder

text: used

text: a

text: decomposition

text: of

text: a

text: simple

text: triangulation,

text: now

text: known

text: as

text: the

text: Schnyder

text: wood,

text: to

text: give

text: a

text: fundamental

text: grid-embedding

text: algorithm

text: for

text: planar

text: maps.

text: We

text: show

text: that

text: a

text: uniformly

text: sampled

text: Schnyder-wood-decorated

text: triangulation

text: with

text: n

text: vertices

text: converges

text: as

text: n

text: to

text: the

text: Liouville

text: quantum

text: gravity

text: with

text: parameter

text: decorated

text: with

text: a

text: triple

text: of

text: SLE

text: ’s

text: of

text: angle

text: diﬀerence

text: in

text: the

text: imaginary

text: geometry

text: sense.

text: Our

text: convergence

text: result

text: provides

text: a

text: description

text: of

text: the

text: continuum

text: limit

text: of

text: Schnyder’s

text: embedding

text: algorithm

text: via

text: Liouville

text: quantum

text: gravity

text: and

text: imaginary

text: geometry.

text: Introduction

text: A

text: planar

text: map

text: is

text: an

text: embedding

text: of

text: a

text: connected

text: planar

text: graph

text: in

text: the

text: plane,

text: considered

text: modulo

text: orientation-preserving

text: homeomorphism.

text: A

text: planar

text: map

text: is

text: said

text: to

text: be

text: simple

text: if

text: it

text: does

text: not

text: have

text: any

text: edges

text: from

text: a

text: vertex

text: to

text: itself

text: or

text: multiple

text: edges

text: between

text: the

text: same

text: pair

text: of

text: vertices.

text: We

text: say

text: that

text: a

text: planar

text: map

text: is

text: a

text: triangulation

text: if

text: every

text: face

text: is

text: bounded

text: by

text: three

text: edges.

text: A

text: spanning

text: tree

text: of

text: a

text: graph

text: G

text: is

text: a

text: connected,

text: cycle-free

text: subgraph

text: of

text: G

text: including

text: all

text: vertices

text: of

text: G.

text: In

text: this

text: paper

text: we

text: will

text: study

text: wooded

text: triangulations,

text: which

text: are

text: simple

text: triangulations

text: equipped

text: with

text: a

text: certain

text: decomposition

text: known

text: as

text: a

text: Schnyder

text: wood.

text: The

text: Schnyder

text: wood,

text: also

text: referred

text: to

text: as

text: a

text: graph

text: realizer

text: in

text: the

text: literature,

text: was

text: introduced

text: by

text: Walter

text: Schnyder

text: to

text: prove

text: a

text: characterization

text: of

text: graph

text: planarity.

text: He

text: later

text: used

text: Schnyder

text: woods

text: to

text: describe

text: an

text: algorithm

text: for

text: embedding

text: an

text: order-n

text: planar

text: graph

text: in

text: such

text: a

text: way

text: that

text: its

text: edges

text: are

text: straight

text: lines

text: and

text: its

text: vertices

text: lie

text: on

text: an

text: (n

text: (n

text: grid

text: Schnyder’s

text: celebrated

text: construction

text: has

text: continued

text: to

text: play

text: an

text: important

text: role

text: in

text: graph

text: theory

text: and

text: enumerative

text: combinatorics

text: In

text: the

text: present

text: article,

text: we

text: are

text: primarily

text: interested

text: in

text: the

text: n

text: behavior

text: of

text: wooded

text: triangulations

text: and

text: their

text: Schnyder

text: embeddings.

text: Consider

text: a

text: simple

text: plane

text: triangulation

text: M

text: with

text: unbounded

text: face

text: A

text: A

text: where

text: the

text: b

text: r

text: g

text: outer

text: vertices

text: A

text: A

text: and

text: A

text: —which

text: we

text: think

text: of

text: as

text: blue,

text: red,

text: and

text: green—are

text: arranged

text: in

text: b

text: r

text: g

text: counterclockwise

text: order.

text: We

text: denote

text: by

text: A

text: A

text: the

text: edge

text: connecting

text: A

text: and

text: A

text: and

text: similarly

text: r

text: g

text: r

text: g

text: for

text: A

text: A

text: and

text: A

text: A

text: Vertices,

text: edges,

text: and

text: faces

text: of

text: M

text: other

text: than

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: A

text: g

text: b

text: b

text: r

text: b

text: r

text: g

text: r

text: g

text: g

text: b

text: b

text: r

text: and

text: A

text: A

text: are

text: called

text: inner

text: vertices,

text: edges,

text: and

text: faces

text: respectively.

text: We

text: deﬁne

text: the

text: size

text: b

text: r

text: g

text: of

text: M

text: to

text: be

text: the

text: number

text: of

text: interior

text: vertices.

text: Euler’s

text: formula

text: implies

text: that

text: a

text: simple

text: plane

text: triangulation

text: of

text: size

text: n

text: has

text: n

text: vertices,

text: edges,

text: and

text: faces.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:16) (cid:17) Var[L ] = Var[R ] = t and Cov[L , R ] = − cos πγ2 t. (1) t t t t 4 Then condition Z | on the event that both L and R stay positive in (0, 1) and L = [0,1] 1 R = 0. Moreover, Z determines (µ , η) a.s. 1 h The singular conditioning referred to in this theorem statement can be made rigorous by a limiting procedure [MS15b, Section 3]. In the present article we do not give detailed deﬁnitions of SLE, GFF, and LQG, because Theorem 1.2 (and its inﬁnite-volume version, Theorem 4.1) allow us to work entirely on the Z Brownian motion side of the (µ , η) ↔ correspondence. Thus most of our results and h proofs can be understood by assuming these two theorems and skipping over the underlying SLE/GFF/LQG background. In light of the Mullin-Bernardi bijection and Theorem 1.2, we can say that the spanning-tree- √ decorated random planar map (M , T ) converges to 2-LQG decorated with an independent n n SLE , in the peanosphere sense. The same type of convergence has been established for several 8 models; see [She16b, GMS15, GS15a, GS15b, KMSW15, GHS16, GKMW16, HS16]. In many cases, the topology of convergence can be further improved by using special properties of the model of interest [GMS15, GS15b, GHS16]. 1.2 A Schnyder wood mating-of-trees encoding We consider a uniform sample (M , O ) from S , which we call a uniform wooded n n n triangulation of size n. We view (M , O ) as a decorated random planar map as in Section . n n Under the marginal law of M , the probability of each triangulation is proportional to the n number of Schnyder woods it admits. Conditioned on M , the law of S is uniform on the n n set of Schnyder woods on M . We are interested in the scaling limit of (M , O ) as n → ∞. n n n Our starting point is a bijection similar to the Mullin-Bernardi bijection for spanning-tree- decorated planar maps in Section . Suppose S ∈ S . We deﬁne a path P (see Figure 2(b)) n S which starts in the outer face, enters an inner face through A A , crosses all edges incident to r g A , enters the outer face through A A , enters an inner face through A A , explores T (S) g g b g b b clockwise, enters the outer face through A A , enters an inner face through A A , crosses the b r b r edges incident to A , enters the outer face through A A , and returns at the starting point. r r g The path P crosses A A , A A and A A and each red and green edge twice and traverses S r g g b b r each blue edge twice. Z Deﬁne ϕ(S) = Zb to be a walk on 2 as follows. The walk starts at (0, 0). When P S traverses a blue edge for the second time, Zb takes a (1, −1)-step. When P crosses a red S edge for the second time, Zb takes a (−1, 0)-step. When P crosses a green edge for the S second time, Zb takes a (0, 1)-step. See Figure 9 for an example of a pair (S, ϕ(S)). Z Deﬁnition 1.3. Deﬁne W to be the set of walks on 2 satisfying the following conditions. n 1. The walk starts and ends at (0, 0) and stays in the closed ﬁrst quadrant. 2. The walk has 3n steps, of three types: (0, 1), (−1, 0) and (1, −1). 3. No (1, −1)-step is immediately preceded by a (−1, 0)-step. 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: Var[L

text: Var[R

text: t

text: and

text: Cov[L

text: R

text: cos

text: πγ2

text: t.

text: t

text: t

text: t

text: t

text: Then

text: condition

text: Z

text: on

text: the

text: event

text: that

text: both

text: L

text: and

text: R

text: stay

text: positive

text: in

text: and

text: L

text: R

text: Moreover,

text: Z

text: determines

text: (µ

text: η)

text: a.s.

text: h

text: The

text: singular

text: conditioning

text: referred

text: to

text: in

text: this

text: theorem

text: statement

text: can

text: be

text: made

text: rigorous

text: by

text: a

text: limiting

text: procedure

text: Section

text: In

text: the

text: present

text: article

text: we

text: do

text: not

text: give

text: detailed

text: deﬁnitions

text: of

text: SLE,

text: GFF,

text: and

text: LQG,

text: because

text: Theorem

text: (and

text: its

text: inﬁnite-volume

text: version,

text: Theorem

text: allow

text: us

text: to

text: work

text: entirely

text: on

text: the

text: Z

text: Brownian

text: motion

text: side

text: of

text: the

text: (µ

text: η)

text: correspondence.

text: Thus

text: most

text: of

text: our

text: results

text: and

text: h

text: proofs

text: can

text: be

text: understood

text: by

text: assuming

text: these

text: two

text: theorems

text: and

text: skipping

text: over

text: the

text: underlying

text: SLE/GFF/LQG

text: background.

text: In

text: light

text: of

text: the

text: Mullin-Bernardi

text: bijection

text: and

text: Theorem

text: we

text: can

text: say

text: that

text: the

text: spanning-tree-

text: decorated

text: random

text: planar

text: map

text: (M

text: T

text: converges

text: to

text: decorated

text: with

text: an

text: independent

text: n

text: n

text: SLE

text: in

text: the

text: peanosphere

text: sense.

text: The

text: same

text: type

text: of

text: convergence

text: has

text: been

text: established

text: for

text: several

text: models;

text: see

text: In

text: many

text: cases,

text: the

text: topology

text: of

text: convergence

text: can

text: be

text: further

text: improved

text: by

text: using

text: special

text: properties

text: of

text: the

text: model

text: of

text: interest

text: A

text: Schnyder

text: wood

text: mating-of-trees

text: encoding

text: We

text: consider

text: a

text: uniform

text: sample

text: (M

text: O

text: from

text: S

text: which

text: we

text: call

text: a

text: uniform

text: wooded

text: n

text: n

text: n

text: triangulation

text: of

text: size

text: n.

text: We

text: view

text: (M

text: O

text: as

text: a

text: decorated

text: random

text: planar

text: map

text: as

text: in

text: Section

text: n

text: n

text: Under

text: the

text: marginal

text: law

text: of

text: M

text: the

text: probability

text: of

text: each

text: triangulation

text: is

text: proportional

text: to

text: the

text: n

text: number

text: of

text: Schnyder

text: woods

text: it

text: admits.

text: Conditioned

text: on

text: M

text: the

text: law

text: of

text: S

text: is

text: uniform

text: on

text: the

text: n

text: n

text: set

text: of

text: Schnyder

text: woods

text: on

text: M

text: We

text: are

text: interested

text: in

text: the

text: scaling

text: limit

text: of

text: (M

text: O

text: as

text: n

text: n

text: n

text: n

text: Our

text: starting

text: point

text: is

text: a

text: bijection

text: similar

text: to

text: the

text: Mullin-Bernardi

text: bijection

text: for

text: spanning-tree-

text: decorated

text: planar

text: maps

text: in

text: Section

text: Suppose

text: S

text: S

text: We

text: deﬁne

text: a

text: path

text: P

text: (see

text: Figure

text: n

text: S

text: which

text: starts

text: in

text: the

text: outer

text: face,

text: enters

text: an

text: inner

text: face

text: through

text: A

text: A

text: crosses

text: all

text: edges

text: incident

text: to

text: r

text: g

text: A

text: enters

text: the

text: outer

text: face

text: through

text: A

text: A

text: enters

text: an

text: inner

text: face

text: through

text: A

text: A

text: explores

text: T

text: (S)

text: g

text: g

text: b

text: g

text: b

text: b

text: clockwise,

text: enters

text: the

text: outer

text: face

text: through

text: A

text: A

text: enters

text: an

text: inner

text: face

text: through

text: A

text: A

text: crosses

text: the

text: b

text: r

text: b

text: r

text: edges

text: incident

text: to

text: A

text: enters

text: the

text: outer

text: face

text: through

text: A

text: A

text: and

text: returns

text: at

text: the

text: starting

text: point.

text: r

text: r

text: g

text: The

text: path

text: P

text: crosses

text: A

text: A

text: A

text: A

text: and

text: A

text: A

text: and

text: each

text: red

text: and

text: green

text: edge

text: twice

text: and

text: traverses

text: S

text: r

text: g

text: g

text: b

text: b

text: r

text: each

text: blue

text: edge

text: twice.

text: Z

text: Deﬁne

text: ϕ(S)

text: Zb

text: to

text: be

text: a

text: walk

text: on

text: as

text: follows.

text: The

text: walk

text: starts

text: at

text: When

text: P

text: S

text: traverses

text: a

text: blue

text: edge

text: for

text: the

text: second

text: time,

text: Zb

text: takes

text: a

text: When

text: P

text: crosses

text: a

text: red

text: S

text: edge

text: for

text: the

text: second

text: time,

text: Zb

text: takes

text: a

text: When

text: P

text: crosses

text: a

text: green

text: edge

text: for

text: the

text: S

text: second

text: time,

text: Zb

text: takes

text: a

text: See

text: Figure

text: for

text: an

text: example

text: of

text: a

text: pair

text: (S,

text: ϕ(S)).

text: Z

text: Deﬁnition

text: Deﬁne

text: W

text: to

text: be

text: the

text: set

text: of

text: walks

text: on

text: satisfying

text: the

text: following

text: conditions.

text: n

text: The

text: walk

text: starts

text: and

text: ends

text: at

text: and

text: stays

text: in

text: the

text: closed

text: ﬁrst

text: quadrant.

text: The

text: walk

text: has

text: steps,

text: of

text: three

text: types:

text: and

text: No

text: is

text: immediately

text: preceded

text: by

text: a

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: start end A A A A g r g r A A b b (a) (b) Figure 2: (a) a triangulation equipped with a Schnyder wood, and (b) the path P , which S traces out the blue tree clockwise. Theorem 1.4. We have ϕ(S) ∈ W for all S ∈ S , and ϕ : S → W is a bijection. n n n n By symmetry, we may also deﬁne Zr and Zg similarly to Zb, based on clockwise explorations of T and T , respectively. Theorem 1.4 implies that the Zr and Zg encodings are also bijective. r g 1.3 SLE -decorated 1-LQG as the scaling limit 16 Given a whole plane Gaussian free ﬁeld h, we may construct the ﬂow lines of the vector ﬁeld ei(2h/3+θ) using so-called imaginary geometry (see Section for a review of of this construction). These ﬂow lines are called the angle-θ ﬂow lines and are distributed as SLE curves. Moreover, C 1 h determines a unique Peano curve η(cid:48)θ on so that the left and right boundaries of η(cid:48)θ are ﬂow lines of angle θ ± π. The space-ﬁlling curve η(cid:48)θ is distributed as an SLE . By analogy 2 16 2π 4π to the discrete setting, we abbreviate (η(cid:48)0, η(cid:48) 3 , η(cid:48) 3 ) to (ηb, ηr, ηg). Let h be a unit-area 1-LQG sphere which is independent of (ηb, ηr, ηg). Let S be a uniform n wooded triangulation of size n. We conjecture that, under a conformal embedding and a suitable normalization, the volume measure of S converges to µ , and that the clockwise n h exploration curves of the three trees in S jointly converge to ηb, ηr, and ηg. n We do not prove this conjecture in the present article, but we do prove its peanosphere version: Theorem 1.5. Suppose (Zb, Zr, Zg) is the triple of random walks encoding of the three trees in a uniformly sampled wooded triangulation of size n. For c ∈ {b, r, g}, let Z c be the Brownian excur(cid:16)sion associated with (η(cid:17)c, µ ) in the sense of Theorem 1.2. Write Zc = (Lc, Rc). h Then √1 Lc , √1 Rc i−n →law (Z c) for c ∈ {b, r, g}. (2) 4n (cid:98)3nt(cid:99) 2n (cid:98)3nt(cid:99) t∈[0,1] t∈[0,1] Furthermore, the three convergence statements in (2) hold jointly. 6 ##LTFigure## ##LTFigure## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: start

font: default

text: end

text: A

text: A

text: A

text: A

text: g

text: r

text: g

text: r

text: A

text: A

text: b

text: b

text: (a)

text: (b)

text: Figure

text: (a)

label: caption

text: a

text: triangulation

text: equipped

text: with

text: a

text: Schnyder

text: wood,

text: and

text: (b)

text: the

text: path

text: P

text: which

text: S

text: traces

text: out

text: the

text: blue

text: tree

text: clockwise.

text: Theorem

text: We

text: have

text: ϕ(S)

text: W

text: for

text: all

text: S

text: S

text: and

text: ϕ

text: S

text: W

text: is

text: a

text: bijection.

text: n

text: n

text: n

text: n

text: By

text: symmetry,

text: we

text: may

text: also

text: deﬁne

text: Zr

text: and

text: Zg

text: similarly

text: to

text: Zb,

text: based

text: on

text: clockwise

text: explorations

text: of

text: T

text: and

text: T

text: respectively.

text: Theorem

text: implies

text: that

text: the

text: Zr

text: and

text: Zg

text: encodings

text: are

text: also

text: bijective.

text: r

text: g

text: SLE

text: -decorated

text: as

text: the

text: scaling

text: limit

text: Given

text: a

text: whole

text: plane

text: Gaussian

text: free

text: ﬁeld

text: h,

text: we

text: may

text: construct

text: the

text: ﬂow

text: lines

text: of

text: the

text: vector

text: ﬁeld

text: using

text: so-called

text: imaginary

text: geometry

text: (see

text: Section

text: for

text: a

text: review

text: of

text: of

text: this

text: construction).

text: These

text: ﬂow

text: lines

text: are

text: called

text: the

text: angle-θ

text: ﬂow

text: lines

text: and

text: are

text: distributed

text: as

text: SLE

text: curves.

text: Moreover,

text: C

text: h

text: determines

text: a

text: unique

text: Peano

text: curve

text: on

text: so

text: that

text: the

text: left

text: and

text: right

text: boundaries

text: of

text: are

text: ﬂow

text: lines

text: of

text: angle

text: θ

text: π.

text: The

text: space-ﬁlling

text: curve

text: is

text: distributed

text: as

text: an

text: SLE

text: By

text: analogy

text: 2π

text: 4π

text: to

text: the

text: discrete

text: setting,

text: we

text: abbreviate

text: to

text: (ηb,

text: ηr,

text: ηg).

text: Let

text: h

text: be

text: a

text: unit-area

text: sphere

text: which

text: is

text: independent

text: of

text: (ηb,

text: ηr,

text: ηg).

text: Let

text: S

text: be

text: a

text: uniform

text: n

text: wooded

text: triangulation

text: of

text: size

text: n.

text: We

text: conjecture

text: that,

text: under

text: a

text: conformal

text: embedding

text: and

text: a

text: suitable

text: normalization,

text: the

text: volume

text: measure

text: of

text: S

text: converges

text: to

text: µ

text: and

text: that

text: the

text: clockwise

text: n

text: h

text: exploration

text: curves

text: of

text: the

text: three

text: trees

text: in

text: S

text: jointly

text: converge

text: to

text: ηb,

text: ηr,

text: and

text: ηg.

text: n

text: We

text: do

text: not

text: prove

text: this

text: conjecture

text: in

text: the

text: present

text: article,

text: but

text: we

text: do

text: prove

text: its

text: peanosphere

text: version:

text: Theorem

text: Suppose

text: (Zb,

text: Zr,

text: Zg)

text: is

text: the

text: triple

text: of

text: random

text: walks

text: encoding

text: of

text: the

text: three

text: trees

text: in

text: a

text: uniformly

text: sampled

text: wooded

text: triangulation

text: of

text: size

text: n.

text: For

text: c

text: {b,

text: r,

text: g},

text: let

text: Z

text: c

text: be

text: the

text: Brownian

text: associated

text: with

text: µ

text: in

text: the

text: sense

text: of

text: Theorem

text: Write

text: Zc

text: (Lc,

text: Rc).

text: h

text: Then

text: √1

text: Lc

text: √1

text: Rc

text: i−n

text: →law

text: (Z

text: c)

text: for

text: c

text: {b,

text: r,

text: g}.

text: Furthermore,

text: the

text: three

text: convergence

text: statements

text: in

text: hold

text: jointly.

text: ##LTFigure##

text: ##LTFigure##

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: ##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: Figure 5: A uniformly sampled 1,000,000-vertex Schnyder-wood-decorated triangulation, Schnyder-embedded in an equilateral triangle. Each face is colored with the average (in RGB color space) of the colors of its three bounding edges. 1.5 Relation to other models and works Bipolar orientations A bipolar orientation on a planar map is an acyclic orientation with a unique source and (cid:112) sink. A mating-of-trees bijection for bipolar-oriented maps was found in [KMSW15], and the authors used it to prove that bipolar-orientated planar maps converge to SLE decorated (cid:112) 12 4/3-LQG, in the peanosphere sense. A bipolar orientation also induces a bipolar orientation on the dual map. In [KMSW15], the authors conjectured that the bipolar-decorated map and its dual jointly converge to 4/3-LQG decorated with two SLE curves which are 12 perpendicular in the imaginary geometry sense. In [GHS16], this conjecture was proved in the triangulation case. This was the ﬁrst time that a pair of imaginary-geometry-coupled Peano curves was proved to be the scaling limit of a natural discrete model. The present article provides the ﬁrst example for a triple of Peano curves. Our work will make use of some results on the coupling of multiple imaginary geometry trees from [GHS16]. We will review these results in Section . Given S ∈ S , let M(cid:48) = T (S) ∪ T (S) ∪ A A , and reverse the orientation of each blue n b g g b edge. It is proved in [FFNO11, Proposition 7.1] that (i) this operation gives a bipolar oriented map on n + 2 vertices with the property that the right side of every bounded face is of length two, and (ii) this procedure gives a bijection between S and the set of bipolar oriented maps n with that property. Applying the mating-of-trees bijection in [KMSW15], it can be proved that the resulting walk coincides with the walk Zn in Lemma 3.1. Furthermore, both blue and green ﬂow lines in S can be thought of as ﬂow lines in the bipolar orientation on the 9 ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: A

text: uniformly

text: sampled

text: Schnyder-wood-decorated

text: triangulation,

text: Schnyder-embedded

text: in

text: an

text: equilateral

text: triangle.

text: Each

text: face

text: is

text: colored

text: with

text: the

text: average

text: (in

text: RGB

text: color

text: space)

text: of

text: the

text: colors

text: of

text: its

text: three

text: bounding

text: edges.

label: section

text: Relation

text: to

text: other

text: models

text: and

text: works

text: Bipolar

text: orientations

text: A

text: bipolar

text: orientation

text: on

text: a

text: planar

text: map

text: is

text: an

text: acyclic

text: orientation

text: with

text: a

text: unique

text: source

text: and

text: sink.

text: A

text: mating-of-trees

text: bijection

text: for

text: bipolar-oriented

text: maps

text: was

text: found

text: in

text: and

text: the

text: authors

text: used

text: it

text: to

text: prove

text: that

text: bipolar-orientated

text: planar

text: maps

text: converge

text: to

text: SLE

text: decorated

text: in

text: the

text: peanosphere

text: sense.

text: A

text: bipolar

text: orientation

text: also

text: induces

text: a

text: bipolar

text: orientation

text: on

text: the

text: dual

text: map.

text: In

text: the

text: authors

text: conjectured

text: that

text: the

text: bipolar-decorated

text: map

text: and

text: its

text: dual

text: jointly

text: converge

text: to

text: decorated

text: with

text: two

text: SLE

text: curves

text: which

text: are

text: perpendicular

text: in

text: the

text: imaginary

text: geometry

text: sense.

text: In

text: this

text: conjecture

text: was

text: proved

text: in

text: the

text: triangulation

text: case.

text: This

text: was

text: the

text: ﬁrst

text: time

text: that

text: a

text: pair

text: of

text: imaginary-geometry-coupled

text: Peano

text: curves

text: was

text: proved

text: to

text: be

text: the

text: scaling

text: limit

text: of

text: a

text: natural

text: discrete

text: model.

text: The

text: present

text: article

text: provides

text: the

text: ﬁrst

text: example

text: for

text: a

text: triple

text: of

text: Peano

text: curves.

text: Our

text: work

text: will

text: make

text: use

text: of

text: some

text: results

text: on

text: the

text: coupling

text: of

text: multiple

text: imaginary

text: geometry

text: trees

text: from

text: We

text: will

text: review

text: these

text: results

text: in

text: Section

text: Given

text: S

text: S

text: let

text: T

text: (S)

text: T

text: (S)

text: A

text: A

text: and

text: reverse

text: the

text: orientation

text: of

text: each

text: blue

text: n

text: b

text: g

text: g

text: b

text: edge.

text: It

text: is

text: proved

text: in

text: Proposition

text: that

text: (i)

text: this

text: operation

text: gives

text: a

text: bipolar

text: oriented

text: map

text: on

text: n

text: vertices

text: with

text: the

text: property

text: that

text: the

text: right

text: side

text: of

text: every

text: bounded

text: face

text: is

text: of

text: length

text: two,

text: and

text: (ii)

text: this

text: procedure

text: gives

text: a

text: bijection

text: between

text: S

text: and

text: the

text: set

text: of

text: bipolar

text: oriented

text: maps

text: n

text: with

text: that

text: property.

text: Applying

text: the

text: mating-of-trees

text: bijection

text: in

text: it

text: can

text: be

text: proved

text: that

text: the

text: resulting

text: walk

text: coincides

text: with

text: the

text: walk

text: Zn

text: in

text: Lemma

text: Furthermore,

text: both

text: blue

text: and

text: green

text: ﬂow

text: lines

text: in

text: S

text: can

text: be

text: thought

text: of

text: as

text: ﬂow

text: lines

text: in

text: the

text: bipolar

text: orientation

text: on

text: the

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 FIG. 3. (a) Binary regularized DPDC PhC with fabrication-friendly features. (b) The corresponding band structure shows two overlapping TM (solid lines) and TE (dashed lines) Dirac cones. (c) FDTD analysis and the farﬁeld patterns through the prism test [Fig. 3(a)] show orthogonally emerging beams at the prism facet (θ = 0), validating the effective zero index behavior for both TM- and TE-polarized incident waves. (d) Also shown are the TM and TE retrieved effective indices. cone zero-index media [33]. pushes certain extraneous modes away from the Dirac degen- While Fig. 1 demonstrates a perfect DPDC, the gray- eracy. We ﬁnd that relaxing this constraint leads to a simple scale dielectric proﬁle poses a signiﬁcant fabrication chal- DPDC structure with regular geometric features (Fig. 3a) lenge (though it may be implemented in the radio frequency although the proximity of an unintentional TM quadrupole regime). Here, we examine a binary regularized version bet- mode in the band structure engenders an anti-crossing (aka ter suited for experimental realization. The modiﬁed structure mode mixing [36]) off the Γ point near the Dirac frequency (Fig. 3b). (Fig. 2a) is obtained via two stages: ﬁrst, we apply thresh- Nevertheless, FDTD analyses of Fig. 3(a) clearly show the old projection ﬁlters [34] during the optimization process to near-zero-index behavior (Fig. 3c,d) for both polarizations. produce a binary design; then, we feed the intermediate bi- Note that the simple “four-hole” structure (Fig. 3a) has a rel- nary design into a post-optimization pixel-averaging routine ative permittivity (cid:15) = 3.3, making it suitable for fabrication to weed out the ﬁne features and regularize the structure (see with common polymer materials. also the supplemental material). The associated band struc- Dirac Cones at the K point.— To demonstrate the versa- ture (Fig. 2b) shows a DPDC albeit with a spoiled overlap due tility of our approach, we proceed to design DPDCs based ∼ to a small frequency gap of 1% between the TE and TM on a hexagonal lattice with symmetry properties distinct from DCs. Figure 2(c,d) show the corresponding prism tests and those found on a square lattice. In particular, we focus on index retrieval analyses. Arguably, the optimally ﬁne-tuned the K point of the Brillouin zone, where two dipolar eigen- gray-scale structure shows better performance than the com- modes that transform according to the E irreducible repre- pletely binary version. This is due to following reasons: for sentations of C point group form a deterministic DC, i.e, a 3v the gray-scale version, the zero-index crossing is perfectly lin- DC that arises as a consequence of the symmetry of the lat- ear and virtually degenerate for both polarizations; for the bi- tice [27, 28]. We show that we can overlap two such DCs, one nary version, the crossings are separated by about 1% and real with TM polarization and the other with TE polarization, thus part of the effective index shows a constant zero value while restoring electromagnetic duality [12] in the vicinity of the the imaginary part depicts a bump around the zero crossing, four-fold degenerate Dirac point. Speciﬁcally, we employ the which corresponds to a small bandgap near the Dirac-point LDOS TO formulation to design degenerate TM and TE dipo- frequency [3]. Nevertheless, the modiﬁed structures clearly lar modes while imposing C symmetry via suitable transfor- 3v feature a range of wavelengths where near-zero-index be- mations which ensure the concurrence of the corresponding havior is observed for both polarizations, which make them degenerate partner for each polarization, leading to DPDCs. realistic candidates for practical applications. We note that Figure 4 shows complex geometries discovered by TO and an approach to realize DPDCs and polarization-independent the corresponding band structure with overlaid TE and TM zero-index behavior was recently proposed [35], which neces- DCs at K point. The gap between the two Dirac points is sitates the use of complex meta-crystals based on patterning as small as < 0.1%, only limited by numerical discretiza- an anisotropic elliptic metamaterial. In contrast, we identify tion errors. To our knowledge, this structure is the ﬁrst proof- DPDCs by virtue of unconventional geometries that can be of-principle 2D design, based on ordinary isotropic dielectric imprinted on simple ordinary isotropic dielectrics. materials, that hosts overlaid TE/TM DCs at a non-Γ point of The appearance of complicated features in the DPDC ge- a PhC. Moreover, this structure stands in contrast to more so- ometry (Fig. 1) can be attributed to numerous stringent condi- phisticated recent designs using 2D metacrystals [12] or 3D tions imposed upon the optimization process. As noted above, hexagonal PhC [21]. Since DPDCs at the K point of a hexag- one such condition is the mode separation constraint which onal lattice are important precursors to non-trivial topological ##LTFigure##

text: FIG.

text: (a)

text: Binary

text: regularized

text: DPDC

text: PhC

text: with

text: fabrication-friendly

text: features.

text: (b)

text: The

text: corresponding

text: band

text: structure

text: shows

text: two

text: overlapping

text: TM

text: (solid

text: lines)

text: and

text: TE

text: (dashed

text: lines)

text: Dirac

text: cones.

text: (c)

text: FDTD

text: analysis

text: and

text: the

text: farﬁeld

text: patterns

text: through

text: the

text: prism

text: test

text: [Fig.

text: show

text: orthogonally

text: emerging

text: beams

text: at

text: the

text: prism

text: facet

text: (θ

text: validating

text: the

text: effective

text: zero

text: index

text: behavior

text: for

text: both

text: TM-

text: and

text: TE-polarized

text: incident

text: waves.

text: (d)

text: Also

text: shown

text: are

text: the

text: TM

text: and

text: TE

text: retrieved

text: effective

text: indices.

text: cone

text: zero-index

text: media

text: pushes

text: certain

text: extraneous

text: modes

text: away

text: from

text: the

text: Dirac

text: degen-

text: While

text: Fig.

text: demonstrates

text: a

text: perfect

text: DPDC,

text: the

text: gray-

text: eracy.

text: We

text: ﬁnd

text: that

text: relaxing

text: this

text: constraint

text: leads

text: to

text: a

text: simple

text: scale

text: dielectric

text: proﬁle

text: poses

text: a

text: signiﬁcant

text: fabrication

text: chal-

text: DPDC

text: structure

text: with

text: regular

text: geometric

text: features

text: (Fig.

text: lenge

text: (though

text: it

text: may

text: be

text: implemented

text: in

text: the

text: radio

text: frequency

text: although

text: the

text: proximity

text: of

text: an

text: unintentional

text: TM

text: quadrupole

text: regime).

text: Here,

text: we

text: examine

text: a

text: binary

text: regularized

text: version

text: bet-

text: mode

text: in

text: the

text: band

text: structure

text: engenders

text: an

text: anti-crossing

text: (aka

text: ter

text: suited

text: for

text: experimental

text: realization.

text: The

text: modiﬁed

text: structure

text: mode

text: mixing

text: off

text: the

text: Γ

text: point

text: near

text: the

text: Dirac

text: frequency

text: (Fig.

text: (Fig.

text: is

text: obtained

text: via

text: two

text: stages:

text: ﬁrst,

text: we

text: apply

text: thresh-

text: Nevertheless,

text: FDTD

text: analyses

text: of

text: Fig.

text: clearly

text: show

text: the

text: old

text: projection

text: ﬁlters

text: during

text: the

text: optimization

text: process

text: to

text: near-zero-index

text: behavior

text: (Fig.

text: for

text: both

text: polarizations.

text: produce

text: a

text: binary

text: design;

text: then,

text: we

text: feed

text: the

text: intermediate

text: bi-

text: Note

text: that

text: the

text: simple

text: “four-hole”

text: structure

text: (Fig.

text: has

text: a

text: rel-

text: nary

text: design

text: into

text: a

text: post-optimization

text: pixel-averaging

text: routine

text: ative

text: permittivity

text: making

text: it

text: suitable

text: for

text: fabrication

text: to

text: weed

text: out

text: the

text: ﬁne

text: features

text: and

text: regularize

text: the

text: structure

text: (see

text: with

text: common

text: polymer

text: materials.

text: also

text: the

text: supplemental

text: material).

text: The

text: associated

text: band

text: struc-

text: Dirac

text: Cones

text: at

text: the

text: K

text: point.—

text: To

text: demonstrate

text: the

text: versa-

text: ture

text: (Fig.

text: shows

text: a

text: DPDC

text: albeit

text: with

text: a

text: spoiled

text: overlap

text: due

text: tility

text: of

text: our

text: approach,

text: we

text: proceed

text: to

text: design

text: DPDCs

text: based

text: to

text: a

text: small

text: frequency

text: gap

text: of

text: between

text: the

text: TE

text: and

text: TM

text: on

text: a

text: hexagonal

text: lattice

text: with

text: symmetry

text: properties

text: distinct

text: from

text: DCs.

text: Figure

text: show

text: the

text: corresponding

text: prism

text: tests

text: and

text: those

text: found

text: on

text: a

text: square

text: lattice.

text: In

text: particular,

text: we

text: focus

text: on

text: index

text: retrieval

text: analyses.

text: Arguably,

text: the

text: optimally

text: ﬁne-tuned

text: the

text: K

text: point

text: of

text: the

text: Brillouin

text: zone,

text: where

text: two

text: dipolar

text: eigen-

text: gray-scale

text: structure

text: shows

text: better

text: performance

text: than

text: the

text: com-

text: modes

text: that

text: transform

text: according

text: to

text: the

text: E

text: irreducible

text: repre-

text: pletely

text: binary

text: version.

text: This

text: is

text: due

text: to

text: following

text: reasons:

text: for

text: sentations

text: of

text: C

text: point

text: group

text: form

text: a

text: deterministic

text: DC,

text: i.e,

text: a

text: the

text: gray-scale

text: version,

text: the

text: zero-index

text: crossing

text: is

text: perfectly

text: lin-

text: DC

text: that

text: arises

text: as

text: a

text: consequence

text: of

text: the

text: symmetry

text: of

text: the

text: lat-

text: ear

text: and

text: virtually

text: degenerate

text: for

text: both

text: polarizations;

text: for

text: the

text: bi-

text: tice

text: We

text: show

text: that

text: we

text: can

text: overlap

text: two

text: such

text: DCs,

text: one

text: nary

text: version,

text: the

text: crossings

text: are

text: separated

text: by

text: about

text: and

text: real

text: with

text: TM

text: polarization

text: and

text: the

text: other

text: with

text: TE

text: polarization,

text: thus

text: part

text: of

text: the

text: effective

text: index

text: shows

text: a

text: constant

text: zero

text: value

text: while

text: restoring

text: electromagnetic

text: duality

text: in

text: the

text: vicinity

text: of

text: the

text: the

text: imaginary

text: part

text: depicts

text: a

text: bump

text: around

text: the

text: zero

text: crossing,

text: four-fold

text: degenerate

text: Dirac

text: point.

text: Speciﬁcally,

text: we

text: employ

text: the

text: which

text: corresponds

text: to

text: a

text: small

text: bandgap

text: near

text: the

text: Dirac-point

text: LDOS

text: TO

text: formulation

text: to

text: design

text: degenerate

text: TM

text: and

text: TE

text: dipo-

text: frequency

text: Nevertheless,

text: the

text: modiﬁed

text: structures

text: clearly

text: lar

text: modes

text: while

text: imposing

text: C

text: symmetry

text: via

text: suitable

text: transfor-

text: feature

text: a

text: range

text: of

text: wavelengths

text: where

text: near-zero-index

text: be-

text: mations

text: which

text: ensure

text: the

text: concurrence

text: of

text: the

text: corresponding

text: havior

text: is

text: observed

text: for

text: both

text: polarizations,

text: which

text: make

text: them

text: degenerate

text: partner

text: for

text: each

text: polarization,

text: leading

text: to

text: DPDCs.

text: realistic

text: candidates

text: for

text: practical

text: applications.

text: We

text: note

text: that

text: Figure

text: shows

text: complex

text: geometries

text: discovered

text: by

text: TO

text: and

text: an

text: approach

text: to

text: realize

text: DPDCs

text: and

text: polarization-independent

text: the

text: corresponding

text: band

text: structure

text: with

text: overlaid

text: TE

text: and

text: TM

text: zero-index

text: behavior

text: was

text: recently

text: proposed

text: which

text: neces-

text: DCs

text: at

text: K

text: point.

text: The

text: gap

text: between

text: the

text: two

text: Dirac

text: points

text: is

text: sitates

text: the

text: use

text: of

text: complex

text: meta-crystals

text: based

text: on

text: patterning

text: as

text: small

text: as

text: only

text: limited

text: by

text: numerical

text: discretiza-

text: an

text: anisotropic

text: elliptic

text: metamaterial.

text: In

text: contrast,

text: we

text: identify

text: tion

text: errors.

text: To

text: our

text: knowledge,

text: this

text: structure

text: is

text: the

text: ﬁrst

text: proof-

text: DPDCs

text: by

text: virtue

text: of

text: unconventional

text: geometries

text: that

text: can

text: be

text: of-principle

text: design,

text: based

text: on

text: ordinary

text: isotropic

text: dielectric

text: imprinted

text: on

text: simple

text: ordinary

text: isotropic

text: dielectrics.

text: materials,

text: that

text: hosts

text: overlaid

text: TE/TM

text: DCs

text: at

text: a

text: non-Γ

text: point

text: of

text: The

text: appearance

text: of

text: complicated

text: features

text: in

text: the

text: DPDC

text: ge-

text: a

text: PhC.

text: Moreover,

text: this

text: structure

text: stands

text: in

text: contrast

text: to

text: more

text: so-

text: ometry

text: (Fig.

text: can

text: be

text: attributed

text: to

text: numerous

text: stringent

text: condi-

text: phisticated

text: recent

text: designs

text: using

text: metacrystals

text: or

text: tions

text: imposed

text: upon

text: the

text: optimization

text: process.

text: As

text: noted

text: above,

text: hexagonal

text: PhC

text: Since

text: DPDCs

text: at

text: the

text: K

text: point

text: of

text: a

text: hexag-

text: one

text: such

text: condition

text: is

text: the

text: mode

text: separation

text: constraint

text: which

text: onal

text: lattice

text: are

text: important

text: precursors

text: to

text: non-trivial

text: topological

text: ##LTFigure##


text: 7 band bloch modes in a 2d photonic crystal with dirac cones. Zheltikov. Fabrication of three-dimensional periodic mi- Optics express, 23(8):10444–10452, 2015. crostructures by means of two-photon polymerization. Applied [34] Fengwen Wang, Boyan Stefanov Lazarov, and Ole Sigmund. Physics B, 67(6):765–767, 1998. On projection methods, convergence and robust formulations [38] Anders Clausen, Fengwen Wang, Jakob S. Jensen, Ole Sig- in topology optimization. Structural and Multidisciplinary Op- mund, and Jennifer A. Lewis. Topology optimized architec- timization, 43(6):767–784, 2011. tures with programmable poisson’s ratio over large deforma- [35] Jia-Rong Wang, Xiao-Dong Chen, Fu-Li Zhao, and Jian-Wen tions. Advanced Materials, 27(37):5523–5527, 2015. Dong. Full polarization conical dispersion and zero-refractive- [39] Caroline Pouya, Johannes T. B. Overvelde, Mathias Kolle, index in two-dimensional photonic hypercrystals. Scientiﬁc re- Joanna Aizenberg, Katia Bertoldi, James C. Weaver, and Pete ports, 6, 2016. Vukusic. Characterization of a mechanically tunable gyroid [36] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. photonic crystal inspired by the butterﬂy parides sesostris. Ad- Meade. Photonic crystals: molding the ﬂow of light. Princeton vanced Optical Materials, 4(1):99–105, 2016. university press, 2011. [40] Paul W. K. Rothemund. Folding dna to create nanoscale shapes [37] R.A. Borisov, G.N. Dorojkina, N.I. Koroteev, V.M. Kozenkov, and patterns. Nature, 440:297–302, March 2006. S.A. Magnitskii, D.V. Malakhov, A.V. Tarasishin, and A.M.

text: band

text: bloch

text: modes

text: in

text: a

text: photonic

text: crystal

text: with

text: dirac

text: cones.

text: Zheltikov.

text: Fabrication

text: of

text: three-dimensional

text: periodic

text: mi-

text: Optics

text: express,

text: crostructures

text: by

text: means

text: of

text: two-photon

text: polymerization.

text: Applied

text: Fengwen

text: Wang,

text: Boyan

text: Stefanov

text: Lazarov,

text: and

text: Ole

text: Sigmund.

text: Physics

text: B,

text: On

text: projection

text: methods,

text: convergence

text: and

text: robust

text: formulations

text: Anders

text: Clausen,

text: Fengwen

text: Wang,

text: Jakob

text: S.

text: Jensen,

text: Ole

text: Sig-

text: in

text: topology

text: optimization.

text: Structural

text: and

text: Multidisciplinary

text: Op-

text: mund,

text: and

text: Jennifer

text: A.

text: Lewis.

text: Topology

text: optimized

text: architec-

text: timization,

text: tures

text: with

text: programmable

text: poisson’s

text: ratio

text: over

text: large

text: deforma-

text: Jia-Rong

text: Wang,

text: Xiao-Dong

text: Chen,

text: Fu-Li

text: Zhao,

text: and

text: Jian-Wen

text: tions.

text: Advanced

text: Materials,

text: Dong.

text: Full

text: polarization

text: conical

text: dispersion

text: and

text: zero-refractive-

text: Caroline

text: Pouya,

text: Johannes

text: T.

text: B.

text: Overvelde,

text: Mathias

text: Kolle,

text: index

text: in

text: two-dimensional

text: photonic

text: hypercrystals.

text: Scientiﬁc

text: re-

text: Joanna

text: Aizenberg,

text: Katia

text: Bertoldi,

text: James

text: C.

text: Weaver,

text: and

text: Pete

text: ports,

text: Vukusic.

text: Characterization

text: of

text: a

text: mechanically

text: tunable

text: gyroid

text: J.

text: D.

text: Joannopoulos,

text: S.

text: G.

text: Johnson,

text: J.

text: N.

text: Winn,

text: and

text: R.

text: D.

text: photonic

text: crystal

text: inspired

text: by

text: the

text: butterﬂy

text: parides

text: sesostris.

text: Ad-

text: Meade.

text: Photonic

text: crystals:

text: molding

text: the

text: ﬂow

text: of

text: light.

text: Princeton

text: vanced

text: Optical

text: Materials,

text: university

text: press,

text: Paul

text: W.

text: K.

text: Rothemund.

text: Folding

text: dna

text: to

text: create

text: nanoscale

text: shapes

text: R.A.

text: Borisov,

text: G.N.

text: Dorojkina,

text: N.I.

text: Koroteev,

text: V.M.

text: Kozenkov,

text: and

text: patterns.

text: Nature,

text: March

text: S.A.

text: Magnitskii,

text: D.V.

text: Malakhov,

text: A.V.

text: Tarasishin,

text: and

text: A.M.


text: Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). MR1364901 (97c:58008) [Pem91] R. Pemantle. Choosing a spanning tree for the integer lattice uniformly. Ann. Probab., 19(4):1559–1574, 1991. MR1127715 (92g:60014) [Wil96] D. B. Wilson. Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pages 296–303. ACM, New York, 1996. MR1427525 15

text: Proceedings

text: of

text: the

text: Conference

text: in

text: Honor

text: of

text: Jean-Pierre

text: Kahane

text: (Orsay,

text: R.

text: Pemantle.

text: Choosing

text: a

text: spanning

text: tree

text: for

text: the

text: integer

text: lattice

text: uniformly.

text: Ann.

text: Probab.,

text: D.

text: B.

text: Wilson.

text: Generating

text: random

text: spanning

text: trees

text: more

text: quickly

text: than

text: the

text: cover

text: time.

text: In

text: Proceedings

text: of

text: the

text: Twenty-Eighth

text: Annual

text: ACM

text: Symposium

text: on

text: the

text: Theory

text: of

text: Computing

text: (Philadelphia,

text: PA,

text: pages

text: ACM,

text: New

text: York,


text: by identifying all the vertices outside G to one new vertex and µW be the n n uniform spanning tree measure on GW . Then µW also has a limit µW , which n n we call the wired spanning forest. These results are due to [Pem91]. The free and wired spanning forests are the same if G is of polynomial growth or, more generally, amenable [Pem91, BLPS01]. They can be diﬀerent, such as on the Cayley graph of a free group. See [BLPS01, LP16] for more details. On Cayley graphs, the wired spanning forest (WSF) has a single component if the graph has at most quartic growth; otherwise, there are inﬁnitely many components in the WSF [Pem91]. In the latter case, the geometry of the (cid:2) (cid:3) T WSF has intriguing behaviors. Let be the component containing o in (cid:2) (cid:3) o the WSF of G. For Cayley graphs with polynomial growth of order at E |T ∩ | (cid:16) least quartic, B(o, r) r4, whereas nonamenable Cayley graphs E |T ∩ o | (cid:16) (cid:16) satisfy B(o, r) r2 [BLPS01, Section 13]. Here, a(t) b(t) means (cid:46) o (cid:46) a(t) b(t) and b(t) a(t). In [BKPS04], the authors provided a detailed Z ≥ analysis of the geometry of the WSF on d (d 5). Among other results, they showed that the tree components have “stochastic dimension” 4. In this paper, we extend the investigation of the volume-growth property of the WSF to general Cayley graphs (Theorems 1.3 and 1.4). Using a similar method as we use to prove Theorem 1.2, we show the following upper bound: (cid:2) (cid:3) | | Theorem 1.3. Let G be a Cayley graph and V (r) = B(o, r) . Then (cid:2) (cid:3) E |T ∩ | (cid:46) B(o, r) r4 log2 V (r) . (2) o E |T ∩ | (cid:46) In particular, B(o, r) r6. o T ∩ Let C(o, r) be the connected component of B(o, r) containing o. This o provides another way to measure the growth of the WSF. We show the following upper bound in terms of the exit time τ for random walk: r Theorem 1.4. Given a Cayley graph G of superpolynomial growth, let C(o, r) (cid:2) (cid:3) be deﬁned as above. Then there exists r such that 0 E | | ≤ E C(o, r) 4 2[τ ] for r > r . (3) 6r 0 Remark 1.5. As will be clear from our proof of (3), the constants involved are not optimal. 3

text: by

text: identifying

text: all

text: the

text: vertices

text: outside

text: G

text: to

text: one

text: new

text: vertex

text: and

text: µW

text: be

text: the

text: n

text: n

text: uniform

text: spanning

text: tree

text: measure

text: on

text: GW

text: Then

text: µW

text: also

text: has

text: a

text: limit

text: µW

text: which

text: n

text: n

text: we

text: call

text: the

text: wired

text: spanning

text: forest.

text: These

text: results

text: are

text: due

text: to

text: The

text: free

text: and

text: wired

text: spanning

text: forests

text: are

text: the

text: same

text: if

text: G

text: is

text: of

text: polynomial

text: growth

text: or,

text: more

text: generally,

text: amenable

text: They

text: can

text: be

text: diﬀerent,

text: such

text: as

text: on

text: the

text: Cayley

text: graph

text: of

text: a

text: free

text: group.

text: See

text: for

text: more

text: details.

text: On

text: Cayley

text: graphs,

text: the

text: wired

text: spanning

text: forest

text: (WSF)

text: has

text: a

text: single

text: component

text: if

text: the

text: graph

text: has

text: at

text: most

text: quartic

text: growth;

text: otherwise,

text: there

text: are

text: inﬁnitely

text: many

text: components

text: in

text: the

text: WSF

text: In

text: the

text: latter

text: case,

text: the

text: geometry

text: of

text: the

text: T

text: WSF

text: has

text: intriguing

text: behaviors.

text: Let

text: be

text: the

text: component

text: containing

text: o

text: in

text: o

text: the

text: WSF

text: of

text: G.

text: For

text: Cayley

text: graphs

text: with

text: polynomial

text: growth

text: of

text: order

text: at

text: E

text: |T

text: least

text: quartic,

text: B(o,

text: r)

text: whereas

text: nonamenable

text: Cayley

text: graphs

text: E

text: |T

text: o

text: satisfy

text: B(o,

text: r)

text: Section

text: Here,

text: a(t)

text: b(t)

text: means

text: o

text: a(t)

text: b(t)

text: and

text: b(t)

text: a(t).

text: In

text: the

text: authors

text: provided

text: a

text: detailed

text: Z

text: analysis

text: of

text: the

text: geometry

text: of

text: the

text: WSF

text: on

text: d

text: (d

text: Among

text: other

text: results,

text: they

text: showed

text: that

text: the

text: tree

text: components

text: have

text: “stochastic

text: dimension”

text: In

text: this

text: paper,

text: we

text: extend

text: the

text: investigation

text: of

text: the

text: volume-growth

text: property

text: of

text: the

text: WSF

text: to

text: general

text: Cayley

text: graphs

text: (Theorems

text: and

text: Using

text: a

text: similar

text: method

text: as

text: we

text: use

text: to

text: prove

text: Theorem

text: we

text: show

text: the

text: following

text: upper

text: bound:

text: Theorem

text: Let

text: G

text: be

text: a

text: Cayley

text: graph

text: and

text: V

text: (r)

text: B(o,

text: r)

text: Then

text: E

text: |T

text: B(o,

text: r)

text: V

text: (r)

text: o

text: E

text: |T

text: In

text: particular,

text: B(o,

text: r)

text: o

text: T

text: Let

text: C(o,

text: r)

text: be

text: the

text: connected

text: component

text: of

text: B(o,

text: r)

text: containing

text: o.

text: This

text: o

text: provides

text: another

text: way

text: to

text: measure

text: the

text: growth

text: of

text: the

text: WSF.

text: We

text: show

text: the

text: following

text: upper

text: bound

text: in

text: terms

text: of

text: the

text: exit

text: time

text: τ

text: for

text: random

text: walk:

text: r

text: Theorem

text: Given

text: a

text: Cayley

text: graph

text: G

text: of

text: superpolynomial

text: growth,

text: let

text: C(o,

text: r)

text: be

text: deﬁned

text: as

text: above.

text: Then

text: there

text: exists

text: r

text: such

text: that

text: E

text: E

text: C(o,

text: r)

text: for

text: r

text: r

text: Remark

text: As

text: will

text: be

text: clear

text: from

text: our

text: proof

text: of

text: the

text: constants

text: involved

text: are

text: not

text: optimal.


text: ∈ N (cid:48)(cid:48) c > 0 such that for all k +, there is a constant c > 0 such that for all ∈ N ∈ (cid:0) (cid:1) r, m + and all x, y Γ, ≤ (cid:48)(cid:48) − − p (x, y) c m k/2rk/V (r) + e c2m/r2 . (6) m (cid:90) Proof. Choose c as in (4). From the preceding two lemmas, we have (cid:0) (cid:1) − ≤ (cid:46) (cid:90) 1 e λ√m (cid:90) p (x, y) p (o, o) m dλ m m V c/(cid:0) λ (cid:1) (cid:0) (cid:1) 0 − − c2(cid:90)/r2 e λ√m 1 (cid:90) e λ√m = m dλ + m dλ V c/ λ V c/ λ 0 c2/r2 (cid:46) m c2/r2 − 1 − λk/2rke λm dλ + m e λm dλ V (r) 0 c2/r2 (cid:46) − − m k/2rk/V (r) + e c2m/r2 , where in the last line, we use the change of variable u := mλ. The implied constants depend on k. This proves (6). Proof of Theorem 1.2. We may clearly assume that the support of the walk generates the group, as otherwise we simply take the subgroup it generates together with a Cayley graph of the subgroup. We may also assume that S is ≥ lazy, i.e., p (o, o) 1/2. W(cid:88)e w(cid:2)ish to show th(cid:3)at 1 ∞ E P ∈ (cid:46) [L ] = S B(o, r) r2 log V (r) . (7) r m m=0 Since the result is known for groups of polynomial growth, we assume G is of superpolynomial grow(cid:2)th. Write ϕ(m(cid:3)) for the right-hand side of (6). Then ∀ ∈ N m and r > 0, P ∈ ≤ S B(o, r) ϕ(m)V (r) . m − Set α := c 2, where c is as deﬁn(cid:88)ed in (6)(cid:2). Put (cid:3) (cid:98) (cid:99) αr2 log V (r) P ∈ Σ(1) := S B(o, r) r m m=0 and 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: c

text: such

text: that

text: for

text: all

text: k

text: there

text: is

text: a

text: constant

text: c

text: such

text: that

text: for

text: all

text: N

text: r,

text: m

text: and

text: all

text: x,

text: y

text: Γ,

text: p

text: (x,

text: y)

text: c

text: m

text: (r)

text: e

text: m

text: Proof.

text: Choose

text: c

text: as

text: in

text: From

text: the

text: preceding

text: two

text: lemmas,

text: we

text: have

text: e

text: λ√m

text: p

text: (x,

text: y)

text: p

text: (o,

text: o)

text: m

text: dλ

text: m

text: m

text: V

text: λ

text: e

text: λ√m

text: e

text: λ√m

text: m

text: dλ

text: m

text: dλ

text: V

text: c/

text: λ

text: V

text: c/

text: λ

text: m

text: λm

text: dλ

text: m

text: e

text: λm

text: dλ

text: V

text: (r)

text: m

text: (r)

text: e

text: where

text: in

text: the

text: last

text: line,

text: we

text: use

text: the

text: change

text: of

text: variable

text: u

text: mλ.

text: The

text: implied

text: constants

text: depend

text: on

text: k.

text: This

text: proves

text: Proof

text: of

text: Theorem

text: We

text: may

text: clearly

text: assume

text: that

text: the

text: support

text: of

text: the

text: walk

text: generates

text: the

text: group,

text: as

text: otherwise

text: we

text: simply

text: take

text: the

text: subgroup

text: it

text: generates

text: together

text: with

text: a

text: Cayley

text: graph

text: of

text: the

text: subgroup.

text: We

text: may

text: also

text: assume

text: that

text: S

text: is

text: lazy,

text: i.e.,

text: p

text: (o,

text: o)

text: to

text: show

text: E

text: P

text: [L

text: S

text: B(o,

text: r)

text: log

text: V

text: (r)

text: r

text: m

text: Since

text: the

text: result

text: is

text: known

text: for

text: groups

text: of

text: polynomial

text: growth,

text: we

text: assume

text: G

text: is

text: of

text: superpolynomial

text: Write

text: for

text: the

text: right-hand

text: side

text: of

text: Then

text: N

text: m

text: and

text: r

text: P

text: S

text: B(o,

text: r)

text: ϕ(m)V

text: (r)

text: m

text: Set

text: α

text: c

text: where

text: c

text: is

text: as

text: in

text: Put

text: log

text: V

text: (r)

text: P

text: S

text: B(o,

text: r)

text: r

text: m

text: and

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:88) Σ(2) := ϕ(m)V (r) . r m>αr2 log V (r) (cid:88) (cid:2) (cid:3) Since ∞ (cid:46) P ∈ (cid:46) Σ(1) r2 log V (r) and S B(o, r) Σ(1) + Σ(2) , r m r r m=0 (cid:46) to prove (7), it suﬃces to show that Σ(2) r2. Choose k > 2 with Corollary r 2.5 in mind. Now(cid:88) (cid:0) (cid:1) − (cid:46) − (cid:46) m k/2rk r2 log V (r) k/2+1rk r2 . (8) m>αr2 log V (r) On the other(cid:88)hand, − (cid:46) − V (r)e c2m/r2 V (r)r2e αc2 log V (r) = r2 . (9) m>αr2 log V (r) (cid:46) Therefore, Σ(2) r2, as claimed. r Remark 2.6. If G has polynomial growth, then we can separate the sum in (7) at αr2 instead of at αr2 log V (r). The same argument as above combined (cid:16) (cid:16) − with the bounds V (r) rD and p (o, o) m D/2 then gives a proof of the 2m quadratic bound on occupation time. 3 Volume growth of the WSF P (cid:104) (cid:105) Given a ﬁnite path = v , v , . . . , v in a graph G, we deﬁne the forward P 0 1 P n P loop erasure of (denoted by LE[ ]) by erasing cycles in chronologically. P More precisely, LE[ ] is deﬁned inductively as follows. The ﬁrst vertex u P P 0 of LE[ ] is the vertex v of . Supposing that u has been set, let k be 0 j the last index such that v = u . Set u := v if k < n; otherwise, let P (cid:104) (cid:105) k j j+1 k+1 LE[ ] := u , . . . , u . If S is a simple random walk on a Cayley graph G, 0 j then LE[S] is called the loop-erased random walk (LERW). There is no trouble deﬁning the forward loop erasure of S a.s. if G is transient. For recurrent 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ϕ(m)V

text: (r)

text: r

text: log

text: V

text: (r)

text: Since

text: P

text: log

text: V

text: (r)

text: and

text: S

text: B(o,

text: r)

text: r

text: m

text: r

text: r

text: to

text: prove

text: it

text: suﬃces

text: to

text: show

text: that

text: Choose

text: k

text: with

text: Corollary

text: r

text: in

text: mind.

text: m

text: log

text: V

text: (r)

text: log

text: V

text: (r)

text: On

text: the

text: V

text: (r)e

text: V

text: log

text: V

text: (r)

text: log

text: V

text: (r)

text: Therefore,

text: as

text: claimed.

text: r

text: Remark

text: If

text: G

text: has

text: polynomial

text: growth,

text: then

text: we

text: can

text: separate

text: the

text: sum

text: in

text: at

text: instead

text: of

text: at

text: log

text: V

text: (r).

text: The

text: same

text: argument

text: as

text: above

text: combined

text: with

text: the

text: bounds

text: V

text: (r)

text: rD

text: and

text: p

text: (o,

text: o)

text: m

text: then

text: gives

text: a

text: proof

text: of

text: the

text: quadratic

text: bound

text: on

text: occupation

text: time.

text: Volume

text: growth

text: of

text: the

text: WSF

text: P

text: Given

text: a

text: ﬁnite

text: path

text: v

text: v

text: v

text: in

text: a

text: graph

text: G,

text: we

text: deﬁne

text: the

text: forward

text: P

text: P

text: n

text: P

text: loop

text: erasure

text: of

text: (denoted

text: by

text: LE[

text: by

text: erasing

text: cycles

text: in

text: chronologically.

text: P

text: More

text: precisely,

text: LE[

text: is

text: deﬁned

text: inductively

text: as

text: follows.

text: The

text: ﬁrst

text: vertex

text: u

text: P

text: P

text: of

text: LE[

text: is

text: the

text: vertex

text: v

text: of

text: Supposing

text: that

text: u

text: has

text: been

text: set,

text: let

text: k

text: be

text: j

text: the

text: last

text: index

text: such

text: that

text: v

text: u

text: Set

text: u

text: v

text: if

text: k

text: n;

text: otherwise,

text: let

text: P

text: k

text: j

text: LE[

text: u

text: u

text: If

text: S

text: is

text: a

text: simple

text: random

text: walk

text: on

text: a

text: Cayley

text: graph

text: G,

text: j

text: then

text: LE[S]

text: is

text: called

text: the

text: loop-erased

text: random

text: walk

text: (LERW).

text: There

text: is

text: no

text: trouble

text: deﬁning

text: the

text: forward

text: loop

text: erasure

text: of

text: S

text: a.s.

text: if

text: G

text: is

text: transient.

text: For

text: recurrent

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: · 10- 3 1 true step aic-o 0.2 4 wide aic avg 0.5 3 0 0 2 - 0.5 1 - 0.2 - 1 0 0 2 4 6 8 10 0 2 4 6 8 10 S / B S / B 1 1 0.9 10- 1 0.8 10- 2 0.7 10- 3 0 2 4 6 8 10 0 2 4 6 8 10 S / B S / B √ (cid:104) (cid:105) Figure 3. Results for the toy-model example with bkg = 106 versus the true signal S relative to B, ± where B is the background yield in a 2σ window around the signal mass (se√e Table 1 for legend label deﬁnitions). (top left) The mean bias in signal estimator√Sˆ relative to (left axis) B and (right axis) B. (top right) The mean full length of the 68.3% CI relative to B as reported by the proﬁle likelihood, where the dashed lines show the aic and aic-o results without accounting for model-selection uncertainty. (bottom left) Coverage reported at 95% CL by each method, where the solid black line shows the expected 95% and the dashed black lines show the approximate uncertainty in all results due to ensemble size. (bottom right) The fraction of data sets that report a p-value corresponding to a signiﬁcance > 3σ for the NULL hypothesis (the expected value is 0.27% when S = 0 as a two-sided test is performed due to the limited number of data sets). the p-values to be close to those obtained by simply using the wide model. This is expected since if one does not expect all components of the wide model to be relevant at the 5σ level then why include them? An alternative approach for assigning local p-values is to take those obtained using the wide model with no model selection and, therefore, no model-selection uncertainty, which pro- vides a mildly conservative signiﬁcance estimate relative to the model-selection-based one. Either way, following the procedures outlined above should yield valid results—provided that a proper wide model is chosen. 2.4 Alternative Methods In the previous subsection, I considered only the cases of c = 0 and c = 2, i.e. no likelihood penalty and the AIC penalty. One alternative approach is to instead choose c for each data set using cross validation. Figure 6 shows the distribution of optimal c values chosen using 10-fold cross validation for the toy-model problem. In 10-fold cross validation, the data sample is divided into – 9 – æ - Æ LC %59 @ egarevoc B / S S Æ S - S æ / B æ L D Æ s B / 1 < 3 evoba stes atad fo noitcarf ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: true

text: step

text: aic-o

text: wide

text: aic

text: avg

text: S

text: B

text: S

text: B

text: S

text: B

text: S

text: B

text: Figure

text: Results

text: for

text: the

text: toy-model

text: example

text: with

text: bkg

text: versus

text: the

text: true

text: signal

text: S

text: relative

text: to

text: B,

text: where

text: B

text: is

text: the

text: background

text: yield

text: in

text: a

text: 2σ

text: window

text: around

text: the

text: signal

text: mass

text: (se√e

text: Table

text: for

text: legend

text: label

text: deﬁnitions).

text: (top

text: left)

text: The

text: mean

text: bias

text: in

text: signal

text: estimator√Sˆ

text: relative

text: to

text: (left

text: axis)

text: B

text: and

text: (right

text: axis)

text: B.

text: (top

text: right)

text: The

text: mean

text: full

text: length

text: of

text: the

text: CI

text: relative

text: to

text: B

text: as

text: reported

text: by

text: the

text: proﬁle

text: likelihood,

text: where

text: the

text: dashed

text: lines

text: show

text: the

text: aic

text: and

text: aic-o

text: results

text: without

text: accounting

text: for

text: model-selection

text: uncertainty.

text: (bottom

text: left)

text: Coverage

text: reported

text: at

text: CL

text: by

text: each

text: method,

text: where

text: the

text: solid

text: black

text: line

text: shows

text: the

text: expected

text: and

text: the

text: dashed

text: black

text: lines

text: show

text: the

text: approximate

text: uncertainty

text: in

text: all

text: results

text: due

text: to

text: ensemble

text: size.

text: (bottom

text: right)

text: The

text: fraction

text: of

text: data

text: sets

text: that

text: report

text: a

text: p-value

text: corresponding

text: to

text: a

text: signiﬁcance

text: 3σ

text: for

text: the

text: hypothesis

text: (the

text: expected

text: value

text: is

text: when

text: S

text: as

text: a

text: two-sided

text: test

text: is

text: performed

text: due

text: to

text: the

text: limited

text: number

text: of

text: data

text: sets).

text: the

text: p-values

text: to

text: be

text: close

text: to

text: those

text: obtained

text: by

text: simply

text: using

text: the

text: wide

text: model.

text: This

text: is

text: expected

text: since

text: if

text: one

text: does

text: not

text: expect

text: all

text: components

text: of

text: the

text: wide

text: model

text: to

text: be

text: relevant

text: at

text: the

text: 5σ

text: level

text: then

text: why

text: include

text: them?

text: An

text: alternative

text: approach

text: for

text: assigning

text: local

text: p-values

text: is

text: to

text: take

text: those

text: obtained

text: using

text: the

text: wide

text: model

text: with

text: no

text: model

text: selection

text: and,

text: therefore,

text: no

text: model-selection

text: uncertainty,

text: which

text: pro-

text: vides

text: a

text: mildly

text: conservative

text: signiﬁcance

text: estimate

text: relative

text: to

text: the

text: model-selection-based

text: one.

text: Either

text: way,

text: following

text: the

text: procedures

text: outlined

text: above

text: should

text: yield

text: valid

text: results—provided

text: that

text: a

text: proper

text: wide

text: model

text: is

text: chosen.

text: Alternative

text: Methods

text: In

text: the

text: previous

text: subsection,

text: I

text: considered

text: only

text: the

text: cases

text: of

text: c

text: and

text: c

text: i.e.

text: no

text: likelihood

text: penalty

text: and

text: the

text: AIC

text: penalty.

text: One

text: alternative

text: approach

text: is

text: to

text: instead

text: choose

text: c

text: for

text: each

text: data

text: set

text: using

text: cross

text: validation.

text: Figure

text: shows

text: the

text: distribution

text: of

text: optimal

text: c

text: values

text: chosen

text: using

text: cross

text: validation

text: for

text: the

text: toy-model

text: problem.

text: In

text: cross

text: validation,

text: the

text: data

text: sample

text: is

text: divided

text: into

text: æ

text: Æ

text: LC

text: egarevoc

text: B

text: S

text: S

text: Æ

text: S

text: S

text: æ

text: B

text: æ

text: L

text: D

text: Æ

text: s

text: B

text: evoba

text: stes

text: atad

text: fo

text: noitcarf

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: We can now bound the ﬁrst term by Pr[∃p ∈ [‘], B˜p ∈ S ∧ x 6= ⊥ | E] (cid:54) PXr[∃p ∈ [‘], B˜p ∈ S | x 6= ⊥ ∧ E] i i i i (cid:54) ‘ Pr[B˜p ∈ S | x 6= ⊥ ∧ E] X i i p=1 (Since W(A, j) = ∅) = ‘ Pr[B˜p ∈ S | A ∈/ S] i i p=1 (From (3)) (cid:54) (2 + 2 ) (6) ‘ ε δ ηδ . V Consider the other term in (5). We can rearrange it as follows. Pr[∃p ∈ [‘], B˜p ∈ S ∧ x = ⊥ ∧ i ∈ D | E] i i = εT P(cid:0) r[∃p ∈ [‘], B˜ip ∈ S ∧ xi = ⊥ | i ∈ D ∧ E] (cid:1) (cid:54) εT P(cid:0) r[∃p ∈ [‘], B˜ip ∈ S | xi = ⊥ ∧ i ∈ D ∧ E] (cid:1) (cid:54) ε Pr[A ∈ S | x = ⊥ ∧ i ∈ D ∧ E] + Pr[∃p ∈ [‘], B˜p ∈ S | A ∈/ S ∧ x = ⊥ ∧ i ∈ D ∧ E] T i i i i i = ε Pr[A ∈ S | x = ⊥] + Pr[∃p ∈ [‘], B˜p ∈ S | A ∈/ S] T i i i i (From (3)) (cid:54) ( + 2 + 2 ) (7) ε δ ε δ‘ ηδ‘ . T V Combining (5), (6) and (7) and from δ (cid:54) k , we have βR Pr[ | ] (cid:54) 1 ( ) (2 + 2 + + 2 + 2 ) (cid:54) ( ) + ( ) + ( ) G E / βδ ε δ‘ ηδ‘ ε δ ε δ‘ ηδ‘ O ε /β O η‘/β O ε ‘/β . j V T V T V 2. | ( )| 1. Let ∗ and ∗ be two diﬀerent (arbitrary) elements of ( ). Again, for convenient, wWe uAse, xE, jto d>enote thie|1condiit2ional event j∗(A, x) > j ∧ ¬G1 ∧ · · · ∧ ¬GWj−1A∧, x{,i∗1j, i∗2} ⊆ W(A, x, j). Now, let us ﬁrst split Pr[ ] as follows. G E j Pr[ | ] (cid:54) Pr[ ∗ ∈ ] + Pr[ ∗ ∈ ] + Pr[ | ∧ ∗ ∗ ∈ ] Gj E i1 D i2 D Gj E i1, i2 / D = 2 + Pr[ | ∧ ∗ ∗ ∈ ] (8) εT Gj E i1, i2 / D . Observe that, when ∗ ∗ ∈ , 0 0 6= ⊥ for every 0 ∈ ( ). Hence, for to occur, there must be p ∈ [‘] such that ait1,lie2as/t Donexoi∗1f,Bx˜ip∗2∗, B˜p∗ is not in Sx. In CotDhexr words, Gj i1 i2 Pr[Gj | E ∧ i∗1, i∗2 ∈/ D] (cid:54) PXr[∃(cid:16)p ∈ [‘], B˜ip∗1 ∈/ S ∨ B˜ip∗2 ∈/ S | E ∧ i∗1, i∗2 ∈/ D] (cid:17) (cid:54) X‘ (cid:16)Pr[B˜ip∗1 ∈/ S | E ∧ i∗1, i∗2 ∈/ D] + Pr[B˜ip∗2 ∈/ S | E ∧(cid:17) i∗1, i∗2 ∈/ D] p=1 (From i∗1, i∗2 ∈ W(A, x, j)) = ‘ Pr[B˜ip∗1 ∈/ S | Ai∗1 ∈ S] + Pr[B˜ip∗2 ∈/ S | Ai∗2 ∈ S] p=1 (From (4)) (cid:54) 2 + (9) ε ‘ η‘. V Combining this with (8), we have Pr[ | ] (cid:54) ( ) + ( ) + ( ). G E O ε O η‘ O ε ‘ j T V As a result, we can conclude that is at most ( ) + ( ) + ( ). c2 O εT /β O η‘/β O εV ‘/β 11 ##LTLine##

text: We

label: list

text: can

text: now

text: bound

text: the

text: ﬁrst

text: term

text: by

text: Pr[∃p

text: B˜p

text: S

text: x

text: E]

text: PXr[∃p

text: B˜p

text: S

text: x

text: E]

text: i

text: i

text: i

text: i

text: Pr[B˜p

text: S

text: x

text: E]

text: X

text: i

text: i

text: (Since

text: W(A,

text: j)

text: Pr[B˜p

text: S

text: A

text: S]

text: i

text: i

text: (From

text: ε

text: δ

text: ηδ

text: V

text: Consider

text: the

text: other

text: term

text: in

text: We

text: can

text: rearrange

text: it

text: as

text: follows.

text: Pr[∃p

text: B˜p

text: S

text: x

text: i

text: D

text: E]

text: i

text: i

text: εT

text: r[∃p

text: B˜ip

text: S

text: xi

text: i

text: D

text: E]

text: εT

text: r[∃p

text: B˜ip

text: S

text: xi

text: i

text: D

text: E]

text: ε

text: Pr[A

text: S

text: x

text: i

text: D

text: E]

text: Pr[∃p

text: B˜p

text: S

text: A

text: S

text: x

text: i

text: D

text: E]

text: T

text: i

text: i

text: i

text: i

text: i

text: ε

text: Pr[A

text: S

text: x

text: Pr[∃p

text: B˜p

text: S

text: A

text: S]

text: T

text: i

text: i

text: i

text: i

text: (From

text: ε

text: δ

text: ε

text: δ‘

text: ηδ‘

text: T

text: V

text: Combining

text: and

text: and

text: from

text: δ

text: k

text: we

text: have

text: βR

text: Pr[

text: G

text: E

text: βδ

text: ε

text: δ‘

text: ηδ‘

text: ε

text: δ

text: ε

text: δ‘

text: ηδ‘

text: O

text: ε

text: /β

text: O

text: η‘/β

text: O

text: ε

text: ‘/β

text: j

text: V

text: T

text: V

text: T

text: V

text: Let

text: and

text: be

text: two

text: diﬀerent

text: (arbitrary)

text: elements

text: of

text: Again,

text: for

text: convenient,

text: wWe

text: uAse,

text: xE,

text: jto

text: d>enote

text: event

text: j∗(A,

text: x)

text: j

text: W(A,

text: x,

text: j).

text: Now,

text: let

text: us

text: ﬁrst

text: split

text: Pr[

text: as

text: follows.

text: G

text: E

text: j

text: Pr[

text: Pr[

text: Pr[

text: Pr[

text: Gj

text: E

text: D

text: D

text: Gj

text: E

text: D

text: Pr[

text: εT

text: Gj

text: E

text: D

text: Observe

text: that,

text: when

text: for

text: every

text: Hence,

text: for

text: to

text: occur,

text: there

text: must

text: be

text: p

text: such

text: that

text: B˜p∗

text: is

text: not

text: in

text: Sx.

text: In

text: CotDhexr

text: words,

text: Gj

text: Pr[Gj

text: E

text: D]

text: S

text: S

text: E

text: D]

text: X‘

text: S

text: E

text: D]

text: S

text: E

text: D]

text: (From

text: W(A,

text: x,

text: j))

text: S

text: S]

text: S

text: S]

text: (From

text: ε

text: η‘.

text: V

text: Combining

text: this

text: with

text: we

text: have

text: Pr[

text: G

text: E

text: O

text: ε

text: O

text: η‘

text: O

text: ε

text: j

text: T

text: V

text: As

text: a

text: result,

text: we

text: can

text: conclude

text: that

text: is

text: at

text: most

text: O

text: εT

text: /β

text: O

text: η‘/β

text: O

text: εV

text: ‘/β

text: ##LTLine##