text: 12 A.Burden et al. − − smoothing scale for CMASS is 15 h 1 Mpc and 10 h 1 Mpc Table 3. BAO scale errors recovered varying the percentage of the survey that lies along an edge. for LOWZ. 7.2 Number of randoms (cid:104) (cid:105) (cid:104) (cid:105) Sample % edge galaxies αpost σα,post Sα,post The random catalogue serves a dual purpose; it is com- Cut 0 1.0002 0.0114 0.0113 pared to the galaxy density to estimate the overdensity ﬁeld 2 1.0000 0.0114 0.0114 and it is moved in the reconstruction process where it be- comes the shifted ﬁeld (δ ). As it is a discrete ﬁeld, it is Full 67 1.0005 0.0125 0.0134 s desirable to have a large number of data points to reduce 2 1.0002 0.0112 0.0110 the shot noise contribution to both of these measurements. However, the reconstruction process requires a unique set of shifted randoms for each mock and as such, data storage can be a problem if these ﬁles are large. In this section we in theory, if the measured overdensity ﬁeld were the linear vary the number of randoms used, perform the reconstruc- matter ﬁeld, and no smoothing was applied, the Zel’dovich tion and compare the power spectrum ﬁtting results. displacements would take the data back to Lagrangian po- We reduce the number of randoms in each catalogue to sitions, and the displacements would be transferred to the 10, 25 and 50 times the number of data points. As a precau- random catalogue. This process would be equivalent to per- tion to prevent spurious correlations between mocks caused forming no reconstruction. However, working with a discrete by using the same set of randoms, we randomly subsample non-linear galaxy distribution, the density ﬁeld smoothed these for each mock from the initial random catalogue of 100 on small scales will be dominated by incoherent highly non- times the number of data points. To prevent correlations be- linear ﬂuctuations and shot noise and we will not be correct- tween the displacements induced and δ used to calculate the ing for the damping where the BAO signal is the strongest. s 2-point statistics, we use a diﬀerent base of randoms with For a large smoothing scale, the algorithm will only pick up 100 times the number of data points for each. Two sets of modes of the density ﬁeld that are well in the in the linear reconstructed catalogues are created; one using the smaller regime and density perturbations in the quasi-linear regime number of randoms for both ﬁelds which we name R where get washed out making the algorithm less eﬀective. In this i,i i is the ratio of randoms to data points in both; and one that section we empirically measure the optimal smoothing scale. maintains 100 times the number of randoms to calculate the In previous work (Eisenstein et al. 2007; Padmanabhan overdensity but uses the smaller number of randoms in the et al. 2012; Anderson et al. 2014, 2012), a Gaussian smooth- − − shifted catalogue, we name these R where j is the ratio ing kernel of R = 10 20 h 1 Mpc has been used and mildly 100,j of randoms to data points in the shifted ﬁeld. deviating from this has been shown not to alter the results (cid:104) (cid:105) (cid:104) (cid:105) Fig. 11 shows α and σ as a function of the (see Appendix B of Anderson et al. 2012 and Padmanabhan post α,post number of randoms for both cases. Both data sets have a et al. 2012). Here we provide a more extensive test on how (cid:104) (cid:105) (cid:62) (cid:104) (cid:105) α consistent with one for i, j 25. The σ val- the smoothing length alters the measurements and their er- post α,post − ues are consistent implying that the precision of the result rors. A wide range of smoothing lengths between 5 h Mpc 1 − is only sensitive to the number of randoms in the shifted and 40 h Mpc 1 on the CMASS and LOWZ mocks are con- ﬁeld and increasing the number of randoms in the initial sidered. (cid:104) (cid:105) overdensity ﬁeld is inconsequential as this ﬁeld is smoothed. Fig. 10 shows how the smoothing scale aﬀects α and (cid:104) (cid:105) Note that the galaxy ﬁeld is also smoothed, but its shot- σ recovered from the mocks. The bias in the measurement α − − noise is dominant and, unlike the randoms it is strongly of α is reduced most using the 5 h 1 Mpc and 15 h 1 Mpc − − clustered, changing the importance of the smoothing on the for CMASS and 8 h 1 Mpc and 10 h 1 Mpc for LOWZ. For (cid:104) (cid:105) ﬁeld. In the R and R catalogues, the α val- a larger smoothing scale the bias is reduced from the pre 100,10 10,10 post ues are no longer consistent, suggesting for either random reconstruction value but the samples tend to become biased (cid:104) (cid:105) catalogue there needs to be more than 10 times the number low. In the CMASS mocks the σ value is reduced the most − − of randoms compared to data points. with a smoothing scale of 10 h 1 Mpc and 15 h 1 Mpc. In (cid:104) (cid:105) 7.3 Finite diﬀerence method the LOWZ measurements the σ value is reduced the most − with a smoothing scale of 10 h 1 Mpc. When the scale is There are a number of options for ﬁnding solutions to smaller than this, the algorithm quickly breaks down due Eq. (8), including methods based in Fourier space or in to the increased non-linear and shot noise contribution to conﬁguration space as used by Padmanabhan et al. (2012). the estimate of the displacements and the error increases To check that the approximations used in the conﬁguration sharply. Conversely when the smoothing scale is increased, space method of Padmanabhan et al. (2012) give the same the result is a steady decline in the error reduction. solution as our Fourier based method, we have implemented Below the optimal smoothing length, the reconstructed both. The conﬁguration space method solves for the poten- catalogues still perform better than the pre-reconstruction tial as deﬁned in Eq. (5) and the equation we want to solve dat−a. For the CMAS−S sample all smoothing lengths between is Eq. 8 rewritten in terms of the potential. 8 h 1 Mpc and 40 h 1 Mpc give an improvement on every − mock and the 5 h 1 Mpc smoothing kernel gives an improved − ∇ f ∇ · ∇ δ result in 595 out of the 600 mocks. For the LOWZ sample, all 2φ + ( φ )ˆr = g . (25) smoothing lengths give an improvement in over 96% of the b r b mocks. The average values of the best ﬁt α and σ values are α shown for each smoothing scale for both samples are shown We solve this on a grid using ﬁnite diﬀerences to ap- in Table 4. From these results we deduce that the optimal proximate the derivatives. The potential at each grid point (cid:13) c 0000 RAS, MNRAS 000, 000–000 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A.Burden

text: et

text: al.

text: smoothing

text: scale

text: for

text: CMASS

text: is

text: h

text: Mpc

text: and

text: h

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text: Table

text: BAO

text: scale

text: errors

text: recovered

text: varying

text: the

text: percentage

text: of

text: the

text: survey

text: that

text: lies

text: along

text: an

text: edge.

text: for

text: LOWZ.

label: section

text: Number

text: of

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text: Sample

text: edge

text: galaxies

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text: random

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text: results

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text: et

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text: α

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text: post

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text: number

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text: shifted

text: and

text: h

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text: the

text: CMASS

text: and

text: LOWZ

text: mocks

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text: ﬁeld

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text: increasing

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text: number

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text: initial

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text: the

text: mocks.

text: The

text: bias

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text: measurement

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text: R

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text: pre

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text: algorithm

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text: including

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text: Fourier

text: space

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text: increased

text: non-linear

text: and

text: shot

text: noise

text: contribution

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text: used

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text: et

text: al.

text: the

text: estimate

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text: the

text: displacements

text: and

text: the

text: error

text: increases

text: To

text: check

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text: the

text: approximations

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text: conﬁguration

text: sharply.

text: Conversely

text: when

text: the

text: smoothing

text: scale

text: is

text: increased,

text: space

text: method

text: of

text: Padmanabhan

text: et

text: al.

text: give

text: the

text: same

text: the

text: result

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text: steady

text: decline

text: in

text: the

text: error

text: reduction.

text: solution

text: as

text: our

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text: based

text: method,

text: we

text: have

text: implemented

text: Below

text: the

text: optimal

text: smoothing

text: length,

text: the

text: reconstructed

text: both.

text: The

text: conﬁguration

text: space

text: method

text: solves

text: for

text: the

text: poten-

text: catalogues

text: still

text: perform

text: better

text: than

text: the

text: pre-reconstruction

text: tial

text: as

text: deﬁned

text: in

text: Eq.

text: and

text: the

text: equation

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text: want

text: to

text: solve

text: dat−a.

text: For

text: the

text: CMAS−S

text: sample

text: all

text: smoothing

text: lengths

text: between

text: is

text: Eq.

text: rewritten

text: in

text: terms

text: of

text: the

text: potential.

text: h

text: Mpc

text: and

text: h

text: Mpc

text: give

text: an

text: improvement

text: on

text: every

text: mock

text: and

text: the

text: h

text: Mpc

text: smoothing

text: kernel

text: gives

text: an

text: improved

text: f

text: δ

text: result

text: in

text: out

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text: the

text: mocks.

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text: the

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text: mocks.

text: The

text: average

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text: the

text: best

text: ﬁt

text: α

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text: RAS,

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text: Eﬃcient Reconstruction of Linear Baryon Acoustic Oscillations in Galaxy Surveys. 19 200 pre recon pre recon 50 100 y c n y e c u n eq 0 ue 0 Fr post recon 20% density eq post recon 20% density 100 60% density Fr 200 60% density 100% density 100% density 50 100 00. 9 0.95 1 1.05 1.1 0 0.01 0.02 0.03 0.04 0.05 0.06 (cid:95) (cid:109) (cid:95) Figure B1. Distribution in α and σα for the CMASS mocks at diﬀerent densities pre and post reconstruction. 200 pre recon 100 pre recon 100 50 y y c c n n e que 0 20% density equ 0 post recon 20% density Fre 150 post recRo=n5h(cid:239)1Mpc 6100%0% d ednesnistyity Fr 6100%0% d ednesnistyity 100 R=40h(cid:239)1Mpc 200 100 50 00.8 5 0.9 0.95 1 (cid:95)1.05 1.1 0 0.02 0.04 (cid:109) 0.06 0.08 0.1 (cid:95) Figure B2. Distribution in α and σα for the LOWZ mocks at diﬀerent densities pre and post reconstruction. 250 250 no recon no recon 200 edges 200 edges survey standard ncy150 ncy150 Freque100 Freque100 50 50 00.9 4 0.96 0.98 1 1.02 1.04 1.06 0 0.01 0.015 0.02 0.025 0.028 (cid:109) (cid:95) (cid:95) Figure B3. Distribution in α and σα for the CMASS edges sample compared to the standard reconstructed and non-reconstructed samples. (cid:13) c 0000 RAS, MNRAS 000, 000–000 ##LTFigure## ##LTFigure## ##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##

text: Eﬃcient

label: title

text: Reconstruction

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text: Linear

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text: CMASS

text: edges

text: sample

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text: reconstructed

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text: c

text: RAS,

text: MNRAS

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text: Urbanowicz˙Richmond˙Holyst printed on May 26, 2020 9 100 Unimodal s e c ri p f o Bimodal Gaussian e c n a ri a V 10 1 2 3 4 5 6 D Fig. 7. Variance of prices (Risk) as a function of system dimension. ##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##

text: Urbanowicz˙Richmond˙Holyst

text: printed

text: on

text: May

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text: s

text: e

text: c

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text: f

text: o

text: Bimodal

text: Gaussian

text: e

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text: a

text: ri

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text: V

text: D

text: Fig.

text: Variance

label: caption

text: of

text: prices

text: (Risk)

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text: function

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text: 4 E. PALMGREN AND O. WILANDER 4. From families to categories of setoids It is well-known that the E-category of setoids in Martin-Lo¨f type theory forms a locally cartesian closed (LCC) category; see [7]. It can moreover be shown to be a pretopos with further properties [13]. In fact, one can straightforwardly verify in Coq (see for instance [17]) that the E-category of setoids forms an LCC pretopos. For categories of setoids with equality on objects the constructions are more delicate and this is the subject of this and the next section. Categories can be presented in an essentially algebraic way; cf. [10]. This is a useful formulation especial for doing category theory internally to a category, but also in set C C C C theory. A (small) category is a triple of classes (sets) , , consisting of objects, 0 1 2 C C (cid:47)(cid:47) arrows and composable arrows, equipped with class functions (functions) id : , C C C C 0 1 (cid:47)(cid:47) (cid:47)(cid:47) dom, cod : and cmp, fst, snd : that satisfy the axioms 1 0 2 1 1. dom(id(x)) = x, 2. cod(id(x)) = x, 3. dom(cmp(u)) = dom(fst(u)), 4. cod(cmp(u)) = cod(snd(u)), 41. cod(fst(u)) = dom(snd(u)), 2 and ⇒ 5. fst(u) = fst(v), snd(u) = snd(v) = u = v, ⇒ ∃ ∈ C ∧ 6. dom(f) = cod(g) = u (snd(u) = f fst(u) = g), ⇒ 2 7. fst(u) = id(y) = cmp(u) = snd(u), ⇒ 8. snd(u) = id(x) = cmp(u) = fst(u), ⇒ 9. fst(w) = fst(v), snd(v) = fst(u), snd(u) = snd(z), snd(w) = cmp(u), cmp(v) = fst(z) = cmp(w) = cmp(z). The category Set of sets and functions can be constructed in the standard way in ZF or CZF using these operations, and its categorical properties developed, following e.g. [4]. We can simply obtain a type-theoretic deﬁnition by replacing sets and classes with setoids (with respective size restrictions) and functions by extensional functions. B C B C (cid:47)(cid:47) (cid:47)(cid:47) A functor F : is a triple of extensional functions F : , k = 0, 1, 2, such k k k that all operations of the categories are preserved, that is ◦ ◦ F id = id F , 1 ◦ 0 ◦ F dom = dom F , 0 ◦ ◦ 1 F cod = cod F , 0 ◦ ◦ 1 F fst = fst F , 1 ◦ ◦2 F snd = snd F , 1 ◦ ◦ 2 F cmp = cmp F . 1 2 The axioms 1 – 9 take a more familiar form if we rewrite them using the composition ◦ ≡ predicate Comp(f, g, h) (or f g h) deﬁned by ∃ ∈ C ∧ ∧ ( u )(fst(u) = g snd(u) = f cmp(u) = h). 2 C C Remark 4.1. Any category may be viewed as an E-category by ignoring the equality on objects and deﬁning HomC(a, b) to be the setoid ∈ C ∧ ∼ ((Σf )[dom(f) = a cod(f) = b], ) 1 ∼ (cid:48) (cid:48) where (f, p) (f , p) iﬀ f =C f . Composition and identity are then deﬁned in the obvious 1 way using the axioms above. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: E.

text: PALMGREN

text: AND

text: O.

text: WILANDER

text: From

text: families

text: to

text: categories

text: of

text: setoids

text: It

text: is

text: well-known

text: that

text: the

text: E-category

text: of

text: setoids

text: in

text: Martin-Lo¨f

text: type

text: theory

text: forms

text: a

text: locally

text: cartesian

text: closed

text: (LCC)

text: category;

text: see

text: It

text: can

text: moreover

text: be

text: shown

text: to

text: be

text: a

text: pretopos

text: with

text: further

text: properties

text: In

text: fact,

text: one

text: can

text: straightforwardly

text: verify

text: in

text: Coq

text: (see

text: for

text: instance

text: that

text: the

text: E-category

text: of

text: setoids

text: forms

text: an

text: LCC

text: pretopos.

text: For

text: categories

text: of

text: setoids

text: with

text: equality

text: on

text: objects

text: the

text: constructions

text: are

text: more

text: delicate

text: and

text: this

text: is

text: the

text: subject

text: of

text: this

text: and

text: the

text: next

text: section.

text: Categories

text: can

text: be

text: presented

text: in

text: an

text: essentially

text: algebraic

text: way;

text: cf.

text: This

text: is

text: a

text: useful

text: formulation

text: especial

text: for

text: doing

text: category

text: theory

text: internally

text: to

text: a

text: category,

text: but

text: also

text: in

text: set

text: C

text: C

text: C

text: C

text: theory.

text: A

text: (small)

text: category

text: is

text: a

text: triple

text: of

text: classes

text: (sets)

text: consisting

text: of

text: objects,

text: C

text: C

text: arrows

text: and

text: composable

text: arrows,

text: equipped

text: with

text: class

text: functions

text: (functions)

text: id

text: C

text: C

text: C

text: C

text: dom,

text: cod

text: and

text: cmp,

text: fst,

text: snd

text: that

text: satisfy

text: the

text: axioms

text: dom(id(x))

text: x,

text: cod(id(x))

text: x,

text: dom(cmp(u))

text: dom(fst(u)),

text: cod(cmp(u))

text: cod(snd(u)),

text: cod(fst(u))

text: dom(snd(u)),

text: and

text: fst(u)

text: fst(v),

text: snd(u)

text: snd(v)

text: u

text: v,

text: C

text: dom(f)

text: cod(g)

text: u

text: (snd(u)

text: f

text: fst(u)

text: g),

text: fst(u)

text: id(y)

text: cmp(u)

text: snd(u),

text: snd(u)

text: id(x)

text: cmp(u)

text: fst(u),

text: fst(w)

text: fst(v),

text: snd(v)

text: fst(u),

text: snd(u)

text: snd(z),

text: snd(w)

text: cmp(u),

text: cmp(v)

text: fst(z)

text: cmp(w)

text: cmp(z).

text: The

text: category

text: Set

text: of

text: sets

text: and

text: functions

text: can

text: be

text: constructed

text: in

text: the

text: standard

text: way

text: in

text: ZF

text: or

text: CZF

text: using

text: these

text: operations,

text: and

text: its

text: categorical

text: properties

text: developed,

text: following

text: e.g.

text: We

text: can

text: simply

text: obtain

text: a

text: type-theoretic

text: deﬁnition

text: by

text: replacing

text: sets

text: and

text: classes

text: with

text: setoids

text: (with

text: respective

text: size

text: restrictions)

text: and

text: functions

text: by

text: extensional

text: functions.

text: B

text: C

text: B

text: C

text: A

text: functor

text: F

text: is

text: a

text: triple

text: of

text: extensional

text: functions

text: F

text: k

text: such

text: k

text: k

text: k

text: that

text: all

text: operations

text: of

text: the

text: categories

text: are

text: preserved,

text: that

text: is

text: F

text: id

text: id

text: F

text: F

text: dom

text: dom

text: F

text: F

text: cod

text: cod

text: F

text: F

text: fst

text: fst

text: F

text: ◦2

text: F

text: snd

text: snd

text: F

text: F

text: cmp

text: cmp

text: F

text: The

text: axioms

text: take

text: a

text: more

text: familiar

text: form

text: if

text: we

text: rewrite

text: them

text: using

text: the

text: composition

text: predicate

text: Comp(f,

text: g,

text: h)

text: (or

text: f

text: g

text: h)

text: deﬁned

text: by

text: C

text: u

text: )(fst(u)

text: g

text: snd(u)

text: f

text: cmp(u)

text: h).

text: C

text: C

text: Remark

text: Any

text: category

text: may

text: be

text: viewed

text: as

text: an

text: E-category

text: by

text: ignoring

text: the

text: equality

text: on

text: objects

text: and

text: deﬁning

text: HomC(a,

text: b)

text: to

text: be

text: the

text: setoid

text: C

text: ((Σf

text: )[dom(f)

text: a

text: cod(f)

text: b],

text: where

text: (f,

text: p)

text: (f

text: p)

text: iﬀ

text: f

text: =C

text: f

text: Composition

text: and

text: identity

text: are

text: then

text: deﬁned

text: in

text: the

text: obvious

text: way

text: using

text: the

text: axioms

text: above.

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text: CONSTRUCTING CATEGORIES AND SETOIDS 9 5. Aczel’s iterative sets and setoids It is known that the category of sets inside Constructive Zermelo-Fraenkel set theory (CZF) has good category-theoretic properties [4]. These can preferably be established on basis of the essentially algebraic formulation of categories given in Section 4. Aczel [1] presented a model of CZF in MLTT. This suggests that we may use such models of CZF to build useful categories for type theory. The model builds on the iterative conception of set, which is to say, a set is a, possibly inﬁnite, well-founded tree, and where equality of sets is deﬁned in terms of bisimulation. We consider here a modiﬁcation of Aczel’s standard model of CZF, to be able to add · | | urelements or atoms. For a universe U, T ( ), and a setoid M = ( M , = ) (of urelements), M the set-theoretic universe V (M) = V is inductively deﬁned by the rules | | (cid:47)(cid:47) a : U f : T (a) V b : M . sup(a, f) : V atom(b) : V The equality = is the smallest relation satisfying the two rules V ∀ ∃ ∀ ∃ x : T (a). y : T (b).f(x) = g(y) y : T (b). x : T (a).f(x) = g(y) V V sup(a, f) = sup(b, g) V a = b M . atom(a) = atom(b) V The membership relation is deﬁned by ∈ ⇐⇒ ∃ u sup(a, f) x : T (a).u = f(x) V V ∈ and declaring u atom(b) to be false. We have a = b iﬀ atom(a) = atom(b), so that V M V equality of atoms is exactly that of the setoid. The standard model is the special case when M is the empty setoid (no atoms). | | We say that a setoid M = ( M , = ) belongs to the universe U if there is some m : U | | | | M| | | | (cid:47)(cid:47) (cid:47)(cid:47) with M = T (m), and some e : M M U such that for all x, y : M , ⇐⇒ x = y T (e(x, y)). M For such setoids we have: ∈ Lemma 5.1. If M is a setoid which belongs to U, then the relations x = y and x y V V are propositions in U. ∈ It is crucial that the basic relations and = are interpreted as propositions in the universe U in order to be able to verify that all bounded formulas (∆ -formulas) may be 0 used in the separation scheme of CZF. We will thus consider V (M) where the setoid M belongs to U. There is no principal diﬃculty in extending the construction to ﬁnitely many setoids of atoms. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CONSTRUCTING

text: CATEGORIES

text: AND

text: SETOIDS

text: Aczel’s

text: iterative

text: sets

text: and

text: setoids

text: It

text: is

text: known

text: that

text: the

text: category

text: of

text: sets

text: inside

text: Constructive

text: Zermelo-Fraenkel

text: set

text: theory

text: (CZF)

text: has

text: good

text: category-theoretic

text: properties

text: These

text: can

text: preferably

text: be

text: established

text: on

text: basis

text: of

text: the

text: essentially

text: algebraic

text: formulation

text: of

text: categories

text: given

text: in

text: Section

text: Aczel

text: presented

text: a

text: model

text: of

text: CZF

text: in

text: MLTT.

text: This

text: suggests

text: that

text: we

text: may

text: use

text: such

text: models

text: of

text: CZF

text: to

text: build

text: useful

text: categories

text: for

text: type

text: theory.

text: The

text: model

text: builds

text: on

text: the

text: iterative

text: conception

text: of

text: set,

text: which

text: is

text: to

text: say,

text: a

text: set

text: is

text: a,

text: possibly

text: inﬁnite,

text: well-founded

text: tree,

text: and

text: where

text: equality

text: of

text: sets

text: is

text: deﬁned

text: in

text: terms

text: of

text: bisimulation.

text: We

text: consider

text: here

text: a

text: modiﬁcation

text: of

text: Aczel’s

text: standard

text: model

text: of

text: CZF,

text: to

text: be

text: able

text: to

text: add

text: urelements

text: or

text: atoms.

text: For

text: a

text: universe

text: U,

text: T

text: and

text: a

text: setoid

text: M

text: M

text: (of

text: urelements),

text: M

text: the

text: set-theoretic

text: universe

text: V

text: (M)

text: V

text: is

text: inductively

text: deﬁned

text: by

text: the

text: rules

text: a

text: U

text: f

text: T

text: (a)

text: V

text: b

text: M

text: sup(a,

text: f)

text: V

text: atom(b)

text: V

text: The

text: equality

text: is

text: the

text: smallest

text: relation

text: satisfying

text: the

text: two

text: rules

text: V

text: x

text: T

text: (a).

text: y

text: T

text: (b).f(x)

text: g(y)

text: y

text: T

text: (b).

text: x

text: T

text: (a).f(x)

text: g(y)

text: V

text: V

text: sup(a,

text: f)

text: sup(b,

text: g)

text: V

text: a

text: b

text: M

text: atom(a)

text: atom(b)

text: V

text: The

text: membership

text: relation

text: is

text: deﬁned

text: by

text: u

text: sup(a,

text: f)

text: x

text: T

text: (a).u

text: f(x)

text: V

text: V

text: and

text: declaring

text: u

text: atom(b)

text: to

text: be

text: false.

text: We

text: have

text: a

text: b

text: iﬀ

text: atom(a)

text: atom(b),

text: so

text: that

text: V

text: M

text: V

text: equality

text: of

text: atoms

text: is

text: exactly

text: that

text: of

text: the

text: setoid.

text: The

text: standard

text: model

text: is

text: the

text: special

text: case

text: when

text: M

text: is

text: the

text: empty

text: setoid

text: (no

text: atoms).

text: We

text: say

text: that

text: a

text: setoid

text: M

text: M

text: belongs

text: to

text: the

text: universe

text: U

text: if

text: there

text: is

text: some

text: m

text: U

text: M|

text: with

text: M

text: T

text: (m),

text: and

text: some

text: e

text: M

text: M

text: U

text: such

text: that

text: for

text: all

text: x,

text: y

text: M

text: x

text: y

text: T

text: (e(x,

text: y)).

text: M

text: For

text: such

text: setoids

text: we

text: have:

text: Lemma

text: If

text: M

text: is

text: a

text: setoid

text: which

text: belongs

text: to

text: U,

text: then

text: the

text: relations

text: x

text: y

text: and

text: x

text: y

text: V

text: V

text: are

text: propositions

text: in

text: U.

text: It

text: is

text: crucial

text: that

text: the

text: basic

text: relations

text: and

text: are

text: interpreted

text: as

text: propositions

text: in

text: the

text: universe

text: U

text: in

text: order

text: to

text: be

text: able

text: to

text: verify

text: that

text: all

text: bounded

text: formulas

text: -formulas)

text: may

text: be

text: used

text: in

text: the

text: separation

text: scheme

text: of

text: CZF.

text: We

text: will

text: thus

text: consider

text: V

text: (M)

text: where

text: the

text: setoid

text: M

text: belongs

text: to

text: U.

text: There

text: is

text: no

text: principal

text: diﬃculty

text: in

text: extending

text: the

text: construction

text: to

text: ﬁnitely

text: many

text: setoids

text: of

text: atoms.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: PINNING MODEL WITH HEAVY-TAILED DISORDER 13 → → ∞ ∈ Proposition 3.2. Assume that βˆN βˆ(cid:18)as N . T(cid:19)hen for any ﬁxed δ > 0 and k N there exists Nk such that for any N > Nk − P A(k) = A(k∞) > 1 δ. (3.6) βˆN,N βˆ, Proof. To prove the claim by using the sub-additivity of the probability, it is enough to prove that ∈ { · · · } (cid:18) (cid:19) for any r 1, , k (cid:18) (cid:19) ∈ → → ∞ P r (cid:60) A(k) , r A(k∞) 0, N , (3.7) βˆN,N βˆ, ∈ → → ∞ P r A(k) , r (cid:60) A(k∞) 0, N . (3.8) βˆN,N βˆ, { ∈ We detail the ﬁrst one, the second one follows in an analogous way. On the event r (cid:60) A(k) , r }   βˆN,N A(βˆk,∞) we consider  (cid:88)  ∞ − uˆ(r) := A∈mCka,rx(cid:60)A βˆ i∈A Mi( ) E(YA) < uˆ(βˆk,∞) (3.9)  ∈  becauser (cid:88)A(βˆk,∞) and the set that achieves the maximum is unique. Then A∈mCka,rx(cid:60)A βˆN i∈A Mi(N)−(cid:88)E(YA)  (cid:88) (cid:88) ≤ ∞ − | − | k k | − ∞ | A∈mCka,rx(cid:60)A βˆ i∈A Mi( ) E(YA) + βˆN βˆ i=1 M(cid:88)i(N) + βˆ i=1 (cid:88)Mi(N) Mi( ) | − | k k | − ∞ | = uˆ(r) + βˆN βˆ Mi(N) + βˆ Mi(N) Mi( ) (3.10) i=1 i=1 { ∈ } and in the same wa(cid:88)y, always on the event r (cid:60) A(βˆkN),N, r (cid:88)A(βˆk,∞) , (cid:88) − ≥ − | − | k − k | − ∞ | ∈mCax∈ βˆN Mi(N) E(YA) uˆ(ˆk∞) βˆN βˆ Mi(N) βˆ Mi(N) Mi( ) . (3.11) A k,r A i∈A β, i=1 i=1 (cid:80) Therefore by using the assumption that r (cid:60) A(k) we have that the l.h.s. of (3.10) is larger than (cid:80) βˆN,N − ≤ | − ∞ | | − the l.h.s. of (3.11). Together with (3.9) we obtain 0 < uˆ(ˆk∞) uˆ(r) 2βˆ ki=1 Mi(N) Mi( ) + 2 βˆN | β, βˆ (cid:18)ki=1 Mi(N), and a sim(cid:19)ple inclusion of events give(cid:88)s (cid:88)  ∈ ≤ − ≤ k | − ∞ | | − | k P r (cid:60) A(βˆkN),N, r A(βˆk,∞) P 0 < uˆ(βˆk,∞) uˆ(r) 2βˆ Mi(N) Mi( ) + 2 βˆN βˆ Mi(N) . (3.12) i=1 i=1 → ∞ The proof follows by observing that the r.h.s. converges to 0 as N by Lemma 3.1. (cid:3)

text: PINNING

text: MODEL

text: WITH

text: HEAVY-TAILED

text: DISORDER

font: YQYVXP+txsy

text: Proposition

text: Assume

text: that

text: βˆN

label: equation

text: N

text: for

text: any

text: ﬁxed

text: δ

font: STHIPJ+rtxmi

text: and

text: k

text: N

font: MORZBQ+txsyb

text: there

text: exists

text: Nk

text: such

text: that

text: for

text: any

text: N

text: Nk

text: P

text: A(k)

font: WBMRCJ+rtxr

text: A(k∞)

text: δ.

text: βˆN,N

text: βˆ,

text: Proof.

text: To

text: prove

text: the

text: claim

text: by

text: using

text: the

text: sub-additivity

text: of

text: the

text: probability,

text: it

text: is

text: enough

text: to

text: prove

text: that

font: RZIFXR+txex

text: for

text: any

text: r

text: k

text: P

text: r

text: A(k)

text: r

text: A(k∞)

text: N

text: βˆN,N

text: βˆ,

text: P

text: r

text: A(k)

text: r

text: A(k∞)

text: N

text: βˆN,N

text: βˆ,

text: We

text: detail

text: the

text: ﬁrst

text: one,

text: the

text: second

text: one

text: follows

text: in

text: an

text: analogous

text: way.

text: On

text: the

text: event

text: r

text: A(k)

text: r

text: βˆN,N

text: A(βˆk,∞)

text: we

text: consider

text: uˆ(r)

text: βˆ

text: i∈A

text: Mi(

text: E(YA)

text: uˆ(βˆk,∞)

text: becauser

text: and

text: the

text: set

text: that

text: achieves

text: the

text: maximum

text: is

text: unique.

text: Then

text: βˆN

text: i∈A

text: k

text: k

text: βˆ

text: i∈A

text: Mi(

text: E(YA)

text: βˆN

text: βˆ

text: βˆ

text: Mi(

text: k

text: k

text: uˆ(r)

text: βˆN

text: βˆ

text: Mi(N)

text: βˆ

text: Mi(N)

text: Mi(

text: and

text: in

text: the

text: same

text: always

text: on

text: the

text: event

text: r

text: A(βˆkN),N,

text: r

text: k

text: k

text: ∈mCax∈

text: βˆN

text: Mi(N)

text: E(YA)

text: uˆ(ˆk∞)

text: βˆN

text: βˆ

text: Mi(N)

text: βˆ

text: Mi(N)

text: Mi(

text: A

text: k,r

text: A

text: i∈A

text: β,

text: Therefore

text: by

text: using

text: the

text: assumption

text: that

text: r

text: A(k)

text: we

text: have

text: that

text: the

text: l.h.s.

text: of

text: is

text: larger

text: than

text: βˆN,N

text: the

text: l.h.s.

text: of

text: Together

text: with

text: we

text: obtain

text: uˆ(ˆk∞)

text: uˆ(r)

text: 2βˆ

text: Mi(N)

text: Mi(

text: βˆN

text: β,

text: βˆ

text: Mi(N),

text: and

text: a

text: inclusion

text: of

text: events

text: k

text: k

text: P

text: r

text: A(βˆkN),N,

text: r

text: A(βˆk,∞)

text: P

text: uˆ(βˆk,∞)

text: uˆ(r)

text: 2βˆ

text: Mi(N)

text: Mi(

text: βˆN

text: βˆ

text: Mi(N)

text: The

text: proof

text: follows

text: by

text: observing

text: that

text: the

text: r.h.s.

text: converges

text: to

text: as

text: N

text: by

text: Lemma


text: PINNING MODEL WITH HEAVY-TAILED DISORDER 22 (cid:80) where ρ(Nk) = i>k Mi(N) (cid:18)is deﬁn(cid:18)ed i(cid:18)n (3.19)(cid:19). By(cid:19)usin−g Part(cid:19)(1) of Proposition 3.5 we choose k large enough such that βˆNρ(Nk) < ν/2 with probability 1 ε, uniformly in N. This forces to have ≤ − ≥ − P PˆβˆN,N dH Iˆβ(ˆkN),N, I > δ e Nγν/2 1 2ε. (4.23) proTbhaebilpirtyoolafrfgoelrlothwasn b1y−oεb,suenrvifionrgmtlhyaotnifNk, cisf. lPaorginet e(3n)oPugrohp,otshietinondH3.(5Iˆ.β(ˆkN),N, IˆβˆN,N) < δ/2 wit(cid:3)h In this section we prove Theorem 1.6. The proof is based on the following result S ∈ M S Lemma 4.5. Let ( , dS) be a metric space and let xN be a sequence converging to x¯. Let µN 1( ) be such that for any ε > 0, limN→∞ µN (x : d(xN, x) > ε) = 0. Then µN (cid:42) δx¯. Proof. The proof is a consequence of the Portmanteau’s Lemma [4, Section 2]. (cid:3) Proof of Theorem 1.6. Le(cid:16)t µN = PˆβˆN,N and(cid:17)µ∞ = δIˆβˆ,∞. Note that µN is a random measure o∞n X depending on the discrete disorder w(N), while µ∞ depends on the continuum disorder w( ). Therefore if we couple together these disorders as in Lemma 3.1 we have that by Theorems 1.4, 1.5 (see Remark 3.7) µN I | dH(IˆβˆN,N, I) > δ →P 0 anMd IˆβˆN,N →P Iˆβˆ,∞. To conclude the proof let us observe that the law of µN is a probability measure on 1(X), the space of the probability measures on X, which is a compact space because X is compact. Therefore we can assume that µN has a limit in distribution. We have thus to show t(cid:18)hat this limit is the la(cid:19)w of µ∞. For this purpose it is enough to →(d) show that there exists a subsequence Nk such that µN µ∞. It is not diﬃcult to check that we can ﬁnd a subsequence Nk such that µNk −I | dH(IˆβˆNk,Nk, Ik) > δ P−→a.s. 0 and IˆβˆNk,Nk P−→a.s. Iˆβˆ,∞, therefore by Lemma 4.5 we conclude that µN P(cid:42)a.s. µ∞ and this concludes the proof. (cid:3) k 5. Proof of Theorem 1.7 The goal of this section is to give a proof of Theorem 1.7. ≡ { } As a preliminary fact let us show that if β < βˆ then Iˆ ∞ 0, 1 , while if β > βˆ then { } c β, c Iˆ ∞ (cid:46) 0, 1 . β, To this aim let us consider the maximum of the diﬀerence between the Continuum Energy (2.14) − ≤ − and the entropy (2.20), uˆ ∞ = σˆ ∞(Iˆ ∞) E(Iˆ ∞), deﬁned in (2.26). Then whenever uˆ ∞ 1, − − { β, } ≤ β≤, − β, ≡ { } β, we have that 1 = E( 0, 1 ) uˆ ∞ 1 and this implies that Iˆ ∞ 0, 1 by uniqueness of the β, − β, { } − maximizer. On the other hand, if uˆ ∞ > 1, then there exists I (cid:46) 0, 1 such that U ∞(I) > 1 { } − { β, } (cid:55)→ β, because U ∞( 0, 1 ) = 1, so that 0, 1 (cid:40) Iˆ ∞. In particular, since β uˆ ∞ is non-decreasing, we β, ≡ { } {β, } β, have that Iˆ ∞ 0, 1 if β < βˆ and Iˆ ∞ (cid:46) 0, 1 if β > βˆ . β, c β, c To prove the theorem we proceed in two steps: in the ﬁrst one we show that a.s. for any ε > 0 there ⊂ ∪ − exists β0 = β0(ε) > 0 random for which Iˆβ,∞ [0, ε] [1 ε, 1] for all β < β0. In the second one we

text: PINNING

text: MODEL

text: WITH

text: HEAVY-TAILED

text: DISORDER

text: where

text: ρ(Nk)

text: i>k

text: Mi(N)

text: of

text: Proposition

text: we

text: choose

text: k

text: large

text: enough

text: such

text: that

text: βˆNρ(Nk)

text: with

text: probability

text: ε,

text: uniformly

text: in

text: N.

text: This

text: forces

text: to

text: have

text: P

text: PˆβˆN,N

text: dH

text: Iˆβ(ˆkN),N,

text: I

text: δ

text: e

text: proTbhaebilpirtyoolafrfgoelrlothwasn

text: cisf.

text: lPaorginet

text: IˆβˆN,N)

text: In

text: this

text: section

text: we

text: prove

text: Theorem

text: The

text: proof

text: is

text: based

text: on

text: the

text: following

text: result

text: S

text: M

text: S

text: Lemma

text: Let

text: dS)

text: be

text: a

text: metric

text: space

text: and

text: let

text: xN

text: be

text: a

text: sequence

text: converging

text: to

text: x¯.

text: Let

text: µN

text: be

text: such

text: that

text: for

text: any

text: ε

text: limN→∞

text: µN

text: (x

text: d(xN,

text: x)

text: ε)

text: Then

text: µN

text: δx¯.

text: Proof.

text: The

text: proof

text: is

text: a

text: consequence

text: of

text: the

text: Portmanteau’s

text: Lemma

text: Section

text: Proof

text: of

text: Theorem

text: µN

text: PˆβˆN,N

text: δIˆβˆ,∞.

text: Note

text: that

text: µN

text: is

text: a

text: random

text: measure

text: o∞n

text: X

text: depending

text: on

text: the

text: discrete

text: disorder

text: w(N),

text: while

text: µ∞

text: depends

text: on

text: the

text: continuum

text: disorder

text: w(

text: Therefore

text: if

text: we

text: couple

text: together

text: these

text: disorders

text: as

text: in

text: Lemma

text: we

text: have

text: that

text: by

text: Theorems

text: (see

text: Remark

text: µN

text: I

text: dH(IˆβˆN,N,

text: I)

text: δ

text: →P

text: anMd

text: IˆβˆN,N

text: →P

text: Iˆβˆ,∞.

text: To

text: conclude

text: the

text: proof

text: let

text: us

text: observe

text: that

text: the

text: law

text: of

text: µN

text: is

text: a

text: probability

text: measure

text: on

text: the

text: space

text: of

text: the

text: probability

text: measures

text: on

text: X,

text: which

text: is

text: a

text: compact

text: space

text: because

text: X

text: is

text: compact.

text: Therefore

text: we

text: can

text: assume

text: that

text: µN

text: has

text: a

text: limit

text: in

text: distribution.

text: We

text: have

text: thus

text: to

text: show

text: this

text: limit

text: is

text: the

text: of

text: µ∞.

text: For

text: this

text: purpose

text: it

text: is

text: enough

text: to

text: →(d)

text: show

text: that

text: there

text: exists

text: a

text: subsequence

text: Nk

text: such

text: that

text: µN

text: µ∞.

text: It

text: is

text: not

text: diﬃcult

text: to

text: check

text: that

text: we

text: can

text: ﬁnd

text: a

text: subsequence

text: Nk

text: such

text: that

text: µNk

text: −I

text: dH(IˆβˆNk,Nk,

text: Ik)

text: δ

text: P−→a.s.

text: and

text: IˆβˆNk,Nk

text: P−→a.s.

text: Iˆβˆ,∞,

text: therefore

text: by

text: Lemma

text: we

text: conclude

text: that

text: µN

text: µ∞

text: and

text: this

text: concludes

text: the

text: proof.

text: k

text: Proof

text: of

text: Theorem

text: The

text: goal

text: of

text: this

text: section

text: is

text: to

text: give

text: a

text: proof

text: of

text: Theorem

text: As

text: a

text: preliminary

text: fact

text: let

text: us

text: show

text: that

text: if

text: β

text: βˆ

text: then

text: Iˆ

text: while

text: if

text: β

text: βˆ

text: then

text: c

text: β,

text: c

text: Iˆ

text: β,

text: To

text: this

text: aim

text: let

text: us

text: consider

text: the

text: maximum

text: of

text: the

text: diﬀerence

text: between

text: the

text: Continuum

text: Energy

text: and

text: the

text: entropy

text: uˆ

text: σˆ

text: ∞(Iˆ

text: E(Iˆ

text: deﬁned

text: in

text: Then

text: whenever

text: uˆ

text: β,

text: β≤,

text: β,

text: β,

text: we

text: have

text: that

text: E(

text: uˆ

text: and

text: this

text: implies

text: that

text: Iˆ

text: by

text: uniqueness

text: of

text: the

text: β,

text: β,

text: maximizer.

text: On

text: the

text: other

text: hand,

text: if

text: uˆ

text: then

text: there

text: exists

text: I

text: such

text: that

text: U

text: ∞(I)

text: β,

text: β,

text: because

text: U

text: so

text: that

text: Iˆ

text: In

text: particular,

text: since

text: β

text: uˆ

text: is

text: non-decreasing,

text: we

text: β,

text: {β,

text: β,

text: have

text: that

text: Iˆ

text: if

text: β

text: βˆ

text: and

text: Iˆ

text: if

text: β

text: βˆ

text: β,

text: c

text: β,

text: c

text: To

text: prove

text: the

text: theorem

text: we

text: proceed

text: in

text: two

text: steps:

text: in

text: the

text: ﬁrst

text: one

text: we

text: show

text: that

text: a.s.

text: for

text: any

text: ε

text: there

text: exists

text: β0

text: random

text: for

text: which

text: Iˆβ,∞

text: ε]

text: ε,

text: for

text: all

text: β

text: In

text: the

text: second

text: one

text: we


text: PINNING MODEL WITH HEAVY-TAILED DISORDER 24 ≤ Now let us ﬁx ε > 0 small and β0 = β0(ε) 1 as in Remark 5.3. Let ∩ ε1 = sup Iˆβ,∞ [0, ε], (5.5) ∩ − ε2 = inf Iˆβ,∞ [1 ε, 1]. (5.6) { − } Let εˆ = max ε1, 1 ε2 . If εˆ = 0 we have ﬁnished. Then we may assume that εˆ > 0 and we (cid:32) (cid:33) choose q > 1, C > 0 for which Proposition 5.1 holds for any t < ε, namely ≤ ≤ 1 βσˆ ∞(Iˆβ,∞) β0Xεˆ Cεˆ α1 logαq εˆ , (5.7) By Theorem 2.6 we get a lower bound for the entropy ≥ − E(Iˆ ∞) εˆγ + (1 εˆ)γ. (5.8) β, In particular if ε is small enough, we have that E(Iˆ ∞) − 1 > εˆγ . (5.9) β, 2 ∈ Therefore with a further restrictions on ε and β0, if necessary, by recalling that α, γ (0, 1) we (cid:32) (cid:33) conclude that for all β < β0 E(Iˆβ,∞) − 1 ≤ βσˆ ∞(Iˆβ,∞) ≤ Cεˆ α1 logαq 1εˆ ≤ εˆ2γ < E(Iˆβ,∞) − 1, (5.10) ≡ { } which is a contradiction. Therefore εˆ must be 0 and this implies that Iˆβ,∞ 0, 1 for each β < β0. 6. The Directed polymer in random environment with heavy tails Originally introduced by [12], the directed polymer in random environment is a model to de- scribe an interaction between a polymer chain and a medium with microscopic impurities. From a mathematical point of view we consider the set of all possible paths of a 1 + 1 - dimensional simple random walk starting from 0 and constrained to come back to 0 after N-steps. The impurities — and { } so the medium-polymer interactions — are ideal∈ized×by an i.i.d. sequence ( ωi,j i∈N,j∈Z, P). Each random variable ωi,j is placed on the point (i, j) N Z. For a given path s we deﬁne the Gibbs measure (cid:80) eβσN(s) µβ,N(s) = Q , (6.1) β,N · where σN( ) = i,j ωi,j1(si = j) is the energy and Qβ,N is a normalization constant. In [2] is studied the case in wh∈ich the impurities have heavy tails, namely the distr→ibu∞tion of ω1,1 is regularly varying with index α (0, 2). In this case to have a non-trivial limit as N , we have ∼ − to choose β = βN βˆN1 2/αL(N), with L a slowly varying function, cf. [2, (2.4),(2.5)]. For such a choice of β, cf. [2, Theorem 2.1], one has that the trajectories of the polymer are concentrated in the uniform topology around a favorable curve γˆβN,N. In [2, Theorem 2.2] one shows that there exists a ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PINNING

text: MODEL

text: WITH

text: HEAVY-TAILED

text: DISORDER

text: Now

text: let

text: us

text: ﬁx

text: ε

text: small

text: and

text: β0

text: as

text: in

text: Remark

text: Let

text: ε1

text: sup

text: Iˆβ,∞

text: ε],

text: ε2

text: inf

text: Iˆβ,∞

text: ε,

text: Let

text: εˆ

text: max

text: ε2

text: If

text: εˆ

text: we

text: have

text: ﬁnished.

text: Then

text: we

text: may

text: assume

text: that

text: εˆ

text: and

text: we

text: choose

text: q

text: C

text: for

text: which

text: Proposition

text: holds

text: for

text: any

text: t

text: ε,

text: namely

text: βσˆ

text: ∞(Iˆβ,∞)

text: β0Xεˆ

text: Cεˆ

text: α1

text: logαq

text: εˆ

text: By

text: Theorem

text: we

text: get

text: a

text: lower

text: bound

text: for

text: the

text: entropy

text: E(Iˆ

text: εˆγ

text: εˆ)γ.

text: β,

text: In

text: particular

text: if

text: ε

text: is

text: small

text: enough,

text: we

text: have

text: that

text: E(Iˆ

text: εˆγ

text: β,

text: Therefore

text: with

text: a

text: further

text: restrictions

text: on

text: ε

text: and

text: if

text: necessary,

text: by

text: recalling

text: that

text: α,

text: γ

text: we

text: conclude

text: that

text: for

text: all

text: β

text: β0

text: E(Iˆβ,∞)

text: βσˆ

text: ∞(Iˆβ,∞)

text: Cεˆ

text: α1

text: logαq

text: 1εˆ

text: εˆ2γ

text: E(Iˆβ,∞)

text: which

text: is

text: a

text: contradiction.

text: Therefore

text: εˆ

text: must

text: be

text: and

text: this

text: implies

text: that

text: Iˆβ,∞

text: for

text: each

text: β

text: The

text: Directed

text: polymer

text: in

text: random

text: environment

text: with

text: heavy

text: tails

text: Originally

text: introduced

text: by

text: the

text: directed

text: polymer

text: in

text: random

text: environment

text: is

text: a

text: model

text: to

text: de-

text: scribe

text: an

text: interaction

text: between

text: a

text: polymer

text: chain

text: and

text: a

text: medium

text: with

text: microscopic

text: impurities.

text: From

text: a

text: mathematical

text: point

text: of

text: view

text: we

text: consider

text: the

text: set

text: of

text: all

text: possible

text: paths

text: of

text: a

text: dimensional

text: simple

text: random

text: walk

text: starting

text: from

text: and

text: constrained

text: to

text: come

text: back

text: to

text: after

text: N-steps.

text: The

text: impurities

text: and

text: so

text: the

text: medium-polymer

text: interactions

text: are

text: ideal∈ized×by

text: an

text: i.i.d.

text: sequence

text: ωi,j

text: i∈N,j∈Z,

text: P).

text: Each

text: random

text: variable

text: ωi,j

text: is

text: placed

text: on

text: the

text: point

text: (i,

text: j)

text: N

text: Z.

text: For

text: a

text: given

text: path

text: s

text: we

text: deﬁne

text: the

text: Gibbs

text: measure

text: eβσN(s)

text: µβ,N(s)

text: Q

text: β,N

text: where

text: σN(

text: i,j

text: j)

text: is

text: the

text: energy

text: and

text: Qβ,N

text: is

text: a

text: normalization

text: constant.

text: In

text: is

text: studied

text: the

text: case

text: in

text: wh∈ich

text: the

text: impurities

text: have

text: heavy

text: tails,

text: namely

text: the

text: distr→ibu∞tion

text: of

text: is

text: regularly

text: varying

text: with

text: index

text: α

text: In

text: this

text: case

text: to

text: have

text: a

text: non-trivial

text: limit

text: as

text: N

text: we

text: have

text: to

text: choose

text: β

text: βN

text: βˆN1

text: with

text: L

text: a

text: slowly

text: varying

text: function,

text: cf.

text: For

text: such

text: a

text: choice

text: of

text: β,

text: cf.

text: Theorem

text: one

text: has

text: that

text: the

text: trajectories

text: of

text: the

text: polymer

text: are

text: concentrated

text: in

text: the

text: uniform

text: topology

text: around

text: a

text: favorable

text: curve

text: γˆβN,N.

text: In

text: Theorem

text: one

text: shows

text: that

text: there

text: exists

text: a

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: PINNING MODEL WITH HEAVY-TAILED DISORDER 10 ∈ ∞ Proof. Let I X and let us assume that E(I) < . Note that x (cid:60) I means that there exists δ > 0 ∩ − ∅ such that I (x δ, x + δ) = because I is closed. We consider a, b the left and right closest points to x in I. Then the proof will follow by proving that ∪ { } − ≥ − − − − E(I x ) E(I) (x a)γ + (b x)γ (b a)γ, (2.22) because the r.h.s. is a quantity strictly bigger than 0, since γ < 1. To prove (2.22), we show that the result is true for any ﬁnite set in an ε-neighborhood (in the ∪ { } Hausdorﬀ metric) of I x and then we deduce the result for E(I), by using its deﬁnition (2.20). ∪ { } Let us start to observe that for any ε small enough, if A is a set in an ε-neighborhood of I x , then it can be written as union of two disjoint sets D, C where D is in a ε-neighborhood of I and C in a { } ε-neighborhood of x . In particular this holds when A is a ﬁnite set, and thus ∪ { } ∩ { ∈ ∪ ∈ ∈ { } } BH(I x , ε) X(fin) = A X(fin) : A = D C, D BH(I, ε) and C BH( x , ε) . (cid:48) ∩ (cid:48)(cid:48) Furthermore, we can partition any such D in two disjoint sets D = D [0, x) and D = (x, 1]. ∈ For a ﬁxed set S X , let l be its smallest point bigger than 0 and r its biggest point smaller S S than 1. By using this notation it follows from the deﬁnition of the entropy of a ﬁnite set (2.19) that ∈ ∪ { } ∩ for any such A BH(I x , ε) X(fin) we have ∪ − − ∪{ } − − − − − E(A) = E(D C) = E(D) (lD(cid:48)(cid:48) rD(cid:48))γ+E(C 0, 1 ) lCγ (1 rC)γ+(lC rD(cid:48))γ+(lD(cid:48)(cid:48) rC)γ. (2.23) ∪ { } ≥ − − By Theorem 2.5 we can bound E(C 0, 1 ) lCγ + (1 rC)γ + (rC lC)γ. Putting such expression in (2.22) we obtain ∪ ≥ E(A) = E(D C) − − − − − ≥ (2.24) E(D) (lD(cid:48)(cid:48) rD(cid:48))γ + (lC rD(cid:48))γ + (lD(cid:48)(cid:48) rC)γ + (rC lC)γ E(D) + e(ε), where { − − − − − } e(ε) = inf (lC rD(cid:48))γ + (lD(cid:48)(cid:48) rC)γ + (rC lC)γ (lD(cid:48)(cid:48) rD(cid:48))γ . (cid:48) ∪ (cid:48)(cid:48) ∈ ∩ ∈ { } } ∩ Such inf is taken among all possible D ≥= D D BH(I, ε) X(fin) and C BH( x , ε) ) X(fin). Finally (2.24) implies that inf E(A) inf E(D) + e(ε), where the inf is taken among all possible ∪ ∈ ∪ { } ∩ \{ ∪ { }} → → A −= D C − BH−(I −x , ε) X(fin) I x . By taking the limit for ε 0 we have e(ε) (x a)γ + (b x)γ (b a)γ and the result follows by (2.21), since the r.h.s. of (2.24) is independent of C. (cid:3) ≤ ≤ ∞ Proposition 2.9. For any 0 a < b 1 we have that E([a, b]) = . Proof. Let us consider the case in which a = 0, b = 1, the other cases follow in a similar way. By ≥ { · · · } − ↑ ∞ ↑ ∞ Theorem 2.5 we have that E([0, 1]) E( 0, 1/N, , 1 ) = N1 γ as N because γ < 1. (cid:3)

text: PINNING

text: MODEL

text: WITH

text: HEAVY-TAILED

text: DISORDER

text: Proof.

text: Let

text: I

text: X

text: and

text: let

text: us

text: assume

text: that

text: E(I)

text: Note

text: that

text: x

font: FOEUNW+txsyc

text: I

text: means

text: that

text: there

text: exists

text: δ

text: such

text: that

text: I

text: (x

text: δ,

text: x

text: δ)

text: because

text: I

text: is

text: closed.

text: We

text: consider

text: a,

text: b

text: the

text: left

text: and

text: right

text: closest

text: points

text: to

text: x

text: in

text: I.

text: Then

text: the

text: proof

text: will

text: follow

text: by

text: proving

text: that

text: E(I

text: x

text: E(I)

text: (x

text: a)γ

text: (b

text: x)γ

text: (b

text: a)γ,

text: because

text: the

text: r.h.s.

text: is

text: a

text: quantity

text: strictly

text: bigger

text: than

text: since

text: γ

text: To

text: prove

text: we

text: show

text: that

text: the

text: result

text: is

text: true

text: for

text: any

text: ﬁnite

text: set

text: in

text: an

text: ε-neighborhood

text: (in

text: the

text: Hausdorﬀ

text: metric)

text: of

text: I

text: x

text: and

text: then

text: we

text: deduce

text: the

text: result

text: for

text: E(I),

text: by

text: using

text: its

text: deﬁnition

text: Let

text: us

text: start

text: to

text: observe

text: that

text: for

text: any

text: ε

text: small

text: enough,

text: if

text: A

text: is

text: a

text: set

text: in

text: an

text: ε-neighborhood

text: of

text: I

text: x

text: then

text: it

text: can

text: be

text: written

text: as

text: union

text: of

text: two

text: disjoint

text: sets

text: D,

text: C

text: where

text: D

text: is

text: in

text: a

text: ε-neighborhood

text: of

text: I

text: and

text: C

text: in

text: a

text: ε-neighborhood

text: of

text: x

text: In

text: particular

text: this

text: holds

text: when

text: A

text: is

text: a

text: ﬁnite

text: set,

text: and

text: thus

text: BH(I

text: x

text: ε)

text: X(fin)

text: A

text: X(fin)

text: A

text: D

text: C,

text: D

text: BH(I,

text: ε)

text: and

text: C

text: BH(

text: x

text: ε)

text: Furthermore,

text: we

text: can

text: partition

text: any

text: such

text: D

text: in

text: two

text: disjoint

text: sets

text: D

text: D

text: x)

text: and

text: D

text: (x,

text: For

text: a

text: ﬁxed

text: set

text: S

text: X

text: let

text: l

text: be

text: its

text: smallest

text: point

text: bigger

text: than

text: and

text: r

text: its

text: biggest

text: point

text: smaller

text: S

text: S

text: than

text: By

text: using

text: this

text: notation

text: it

text: follows

text: from

text: the

text: deﬁnition

text: of

text: the

text: entropy

text: of

text: a

text: ﬁnite

text: set

text: that

text: for

text: any

text: such

text: A

text: BH(I

text: x

text: ε)

text: X(fin)

text: we

text: have

text: E(A)

text: E(D

text: C)

text: E(D)

text: lCγ

text: rC)γ+(lC

text: rC)γ.

text: By

text: Theorem

text: we

text: can

text: bound

text: E(C

text: lCγ

text: rC)γ

text: (rC

text: lC)γ.

text: Putting

text: such

text: expression

text: in

text: we

text: obtain

text: E(A)

text: E(D

text: C)

text: E(D)

text: (lC

text: rC)γ

text: (rC

text: lC)γ

text: E(D)

text: e(ε),

text: where

text: e(ε)

text: inf

text: (lC

text: rC)γ

text: (rC

text: lC)γ

text: Such

text: inf

text: is

text: taken

text: among

text: all

text: possible

text: D

text: D

text: D

text: BH(I,

text: ε)

text: X(fin)

text: and

text: C

text: BH(

text: x

text: ε)

text: X(fin).

text: Finally

text: implies

text: that

text: inf

text: E(A)

text: inf

text: E(D)

text: e(ε),

text: where

text: the

text: inf

text: is

text: taken

text: among

text: all

text: possible

text: A

text: D

text: C

text: BH−(I

text: −x

text: ε)

text: X(fin)

text: I

text: x

text: By

text: taking

text: the

text: limit

text: for

text: ε

text: we

text: have

text: e(ε)

text: (x

text: a)γ

text: (b

text: x)γ

text: (b

text: a)γ

text: and

text: the

text: result

text: follows

text: by

text: since

text: the

text: r.h.s.

text: of

text: is

text: independent

text: of

text: C.

text: Proposition

text: For

text: any

text: a

text: b

text: we

text: have

text: that

text: E([a,

text: b])

text: Proof.

text: Let

text: us

text: consider

text: the

text: case

text: in

text: which

text: a

text: b

text: the

text: other

text: cases

text: follow

text: in

text: a

text: similar

text: way.

text: By

text: Theorem

text: we

text: have

text: that

text: E(

text: γ

text: as

text: N

text: because

text: γ


text: 12 Corollary 1. Suppose that X is a forward invariant compact set and that the Koopman operator ◦ ∈ (cid:60){ } Utf = f ϕt admits the eigenfunctions φ C0(X) with the eigenvalues λ < 0, i = λi i (cid:92) 1, . . . , m. Then the intersection of the zero level sets m { ∈ | } M = x X φ (x) = 0 λi i=1 is invariant under ϕt and globally asymptotically stable. (cid:107) − (cid:107) (cid:48) ∀ ∈ { } ⇒ (cid:107) − (cid:107) min z y < (cid:15) i 1, . . . , m min z y < (cid:15) . (11) ∈ ∈∩ y M0i y iM0i (cid:48) Moreover, Theorem 1 implies that, for some (cid:15) > 0, there exists δ > 0 such that (cid:107) − (cid:107) ⇒ (cid:107) − (cid:107) ⇒ (cid:107) − (cid:107) (cid:48) ∀ ≥ min x y < δ min x y < δ min ϕt(x) y < (cid:15) t 0 (12) ∈∩ ∈ ∈ y iM0i y M0i y M0i for i = 1, . . . , m. The proof follows from (11) with z = ϕt(x) and (12). According to Proposition 2, the eigenfunctions considered in Corollary 1 are zero on the ⊆ attractor and it follows that A M. In order to prove the stability of A, several eigenfunctions − are required so that the intersections of their zero level sets satisfy M = A (typically N q eigenfunctions if the attractor is of dimension q). It is noticeable that, despite the fact that the Koopman operator is inﬁnite-dimensional, only a (small) ﬁnite number of eigenfunctions is sufﬁcient to establish global stability of the attractor. Remark 4 (Stable and unstable manifolds). If A is an invariant manifold but not an attractor, it is obviously not possible to obtain M = A, where M is the intersection of the zero level sets of all the eigenfunctions that correspond to an eigenvalue with a strictly negative real part. In this case, the smallest set M is the unstable manifold of A, and the intersection M of the zero level s (cid:60){ } sets of the eigenfunctions φ , with λ > 0, is the stable manifold of A. For the Koopman λi i | operator acting on the space of functions f restricted to M , one can ﬁnd an intersection Ms s | (cid:60){ } of zero level sets of eigenfunctions φ , with λ < 0, that is equal to A. In this case, λi Ms i | Corollary 1 implies that A is globally stable in M (i.e. under the ﬂow ϕt ). s Ms May 26, 2020 DRAFT

text: Corollary

text: Suppose

text: that

text: X

text: is

text: a

text: forward

text: invariant

text: compact

text: set

text: and

text: that

text: the

text: Koopman

text: operator

text: Utf

text: f

text: ϕt

text: admits

text: the

text: eigenfunctions

text: φ

text: with

text: the

text: eigenvalues

text: λ

text: i

text: λi

text: i

text: m.

text: Then

text: the

text: intersection

text: of

text: the

text: zero

text: level

text: sets

text: m

text: M

text: x

text: X

text: φ

text: (x)

text: λi

text: is

text: invariant

text: under

text: ϕt

text: and

text: globally

text: asymptotically

text: stable.

text: min

text: z

text: y

text: i

text: m

text: min

text: z

text: y

text: y

text: y

text: Moreover,

text: Theorem

text: implies

text: that,

text: for

text: some

text: there

text: exists

text: δ

text: such

text: that

text: min

text: x

text: y

text: δ

text: min

text: x

text: y

text: δ

text: min

text: ϕt(x)

text: y

text: t

text: y

text: y

text: y

text: for

text: i

text: m.

text: The

text: proof

text: follows

text: from

text: with

text: z

text: ϕt(x)

text: and

text: According

text: to

text: Proposition

text: the

text: eigenfunctions

text: considered

text: in

text: Corollary

text: are

text: zero

text: on

text: the

text: attractor

text: and

text: it

text: follows

text: that

text: A

text: M.

text: In

text: order

text: to

text: prove

text: the

text: stability

text: of

text: A,

text: several

text: eigenfunctions

text: are

text: required

text: so

text: that

text: the

text: intersections

text: of

text: their

text: zero

text: level

text: sets

text: satisfy

text: M

text: A

text: (typically

text: N

text: q

text: eigenfunctions

text: if

text: the

text: attractor

text: is

text: of

text: dimension

text: q).

text: It

text: is

text: noticeable

text: that,

text: despite

text: the

text: fact

text: that

text: the

text: Koopman

text: operator

text: is

text: inﬁnite-dimensional,

text: only

text: a

text: (small)

text: ﬁnite

text: number

text: of

text: eigenfunctions

text: is

text: sufﬁcient

text: to

text: establish

text: global

text: stability

text: of

text: the

text: attractor.

text: Remark

text: (Stable

text: and

text: unstable

text: manifolds).

text: If

text: A

text: is

text: an

text: invariant

text: manifold

text: but

text: not

text: an

text: attractor,

text: it

text: is

text: obviously

text: not

text: possible

text: to

text: obtain

text: M

text: A,

text: where

text: M

text: is

text: the

text: intersection

text: of

text: the

text: zero

text: level

text: sets

text: of

text: all

text: the

text: eigenfunctions

text: that

text: correspond

text: to

text: an

text: eigenvalue

text: with

text: a

text: strictly

text: negative

text: real

text: part.

text: In

text: this

text: case,

text: the

text: smallest

text: set

text: M

text: is

text: the

text: unstable

text: manifold

text: of

text: A,

text: and

text: the

text: intersection

text: M

text: of

text: the

text: zero

text: level

text: s

text: sets

text: of

text: the

text: eigenfunctions

text: φ

text: with

text: λ

text: is

text: the

text: stable

text: manifold

text: of

text: A.

text: For

text: the

text: Koopman

text: λi

text: i

text: operator

text: acting

text: on

text: the

text: space

text: of

text: functions

text: f

text: restricted

text: to

text: M

text: one

text: can

text: ﬁnd

text: an

text: intersection

text: Ms

text: s

text: of

text: zero

text: level

text: sets

text: of

text: eigenfunctions

text: φ

text: with

text: λ

text: that

text: is

text: equal

text: to

text: A.

text: In

text: this

text: case,

text: λi

text: Ms

text: i

text: Corollary

text: implies

text: that

text: A

text: is

text: globally

text: stable

text: in

text: M

text: (i.e.

text: under

text: the

text: ﬂow

text: ϕt

text: s

text: Ms

text: May

text: DRAFT


text: 14 ∗ (Note that φ (x ) = 0, according to Property 2.) Injecting these two approximations into (7) λi − ∗ (cid:107) − ∗(cid:107) − ∗ and considering x x = x x e , where e is the unit vector in the direction of x x , X X (cid:107) − ∗| → we obtain to ﬁrst order (i.e. by taking x x 0) · ∇ ∗ ∇ ∗ (Je ) φ (x ) = λ φ (x )e . X λi i λi X This equation is valid for every e , i.e. every unit vector, so that we get X ∇ ∗ ∇ ∗ JT φ (x ) = λ φ (x ) . (13) λi i λi ∇ ∗ (cid:54) Since φ (x ) = 0, the Koopman eigenvalue is an eigenvalue of the Jacobian matrix J. λi ∇ ∗ Moreover, φ (x ) is equal (up to a multiplicative constant) to the left eigenvector w of J, λi i (cid:60){∇ ∗ } (cid:61){∇ ∗ } or equivalently φ (x ) and φ (x ) (when λ is complex) are respectively parallel λi λi i (cid:60){ } (cid:61){ } (cid:60){ } (cid:61){ } ∗ to w and w . Thus the zero level sets of φ and φ are tangent at x to a i i λi λi (cid:60){ } (cid:61){ } (cid:60){ } hyperplane whose normal is w and w , respectively. Since the N eigenvectors w i i i (cid:61){ } and w (here we use a shorthand, the imaginary part could be zero) are independent, the i ∗ intersection in a small neighborhood V of x of the zero level sets of the N different functions (cid:60){ } (cid:61){ } ∗ φ and φ can only be x (to prove by contradiction, just take a sequence of points λi λi { } (cid:60){ } (cid:61){ } x in a nested sequence of neighborhoods V such that φ (x ) = 0 and φ (x ) = 0, i i λi i λi i for every i. By taking the limit of along that sequence, it is easy to see that gradients must be dependent.) Next, we show that the zero level sets cannot have another intersection in X. Suppose that there is another intersection. This must be a part of another invariant set which is not connected ∗ to the ﬁxed point. Therefore, the boundary ∂Ω of the basin of attraction Ω of x has a non empty intersection with X, since X is a connected set. Moreover, since X is forward invariant, ∩ ∂Ω X is also forward invariant and contains the limit sets of its trajectories. Consider a point ∈ ∩ x ∂Ω X that belongs to a limit set. By deﬁnition and continuity of the eigenfunctions, ω ∀ we have φ (x ) = 0 i (see (10)). Also, for any arbitrarily small neighborhood V of x and λi ω (cid:15) ω ∈ ∩ ∈ (cid:54) ∗ for all x V Ω, there exist a point x V with x = x and a constant T > 0 such that (cid:15) (cid:15) V V − | | ∗ ϕ T (x ) = x . There is at least one eigenfunction that satisﬁes φ (x ) = C > 0, since x V (cid:15) λi V is the only intersection in V of the zero level sets of the N eigenfunctions. Equivalently we | | −(cid:60){ } ⊂ have φ (x ) = C exp( λ T ) > 0. Therefore, φ is not continuous in V X, which is a λi (cid:15) i λi (cid:15) contradiction. May 26, 2020 DRAFT

text: (Note

text: that

text: φ

text: (x

text: according

text: to

text: Property

text: Injecting

text: these

text: two

text: approximations

text: into

text: λi

text: and

text: considering

text: x

text: x

text: x

text: x

text: e

text: where

text: e

text: is

text: the

text: unit

text: vector

text: in

text: the

text: direction

text: of

text: x

text: x

text: X

text: X

text: we

text: obtain

text: to

text: ﬁrst

text: order

text: (i.e.

text: by

text: taking

text: x

text: x

text: (Je

text: φ

text: (x

text: λ

text: φ

text: (x

text: )e

text: X

text: λi

text: i

text: λi

text: X

text: This

text: equation

text: is

text: valid

text: for

text: every

text: e

text: i.e.

text: every

text: unit

text: vector,

text: so

text: that

text: we

text: get

text: X

text: JT

text: φ

text: (x

text: λ

text: φ

text: (x

text: λi

text: i

text: λi

text: Since

text: φ

text: (x

text: the

text: Koopman

text: eigenvalue

text: is

text: an

text: eigenvalue

text: of

text: the

text: Jacobian

text: matrix

text: J.

text: λi

text: Moreover,

text: φ

text: (x

text: is

text: equal

text: (up

text: to

text: a

text: multiplicative

text: constant)

text: to

text: the

text: left

text: eigenvector

text: w

text: of

text: J,

text: λi

text: i

text: or

text: equivalently

text: φ

text: (x

text: and

text: φ

text: (x

text: (when

text: λ

text: is

text: complex)

text: are

text: respectively

text: parallel

text: λi

text: λi

text: i

text: to

text: w

text: and

text: w

text: Thus

text: the

text: zero

text: level

text: sets

text: of

text: φ

text: and

text: φ

text: are

text: tangent

text: at

text: x

text: to

text: a

text: i

text: i

text: λi

text: λi

text: hyperplane

text: whose

text: normal

text: is

text: w

text: and

text: w

text: respectively.

text: Since

text: the

text: N

text: eigenvectors

text: w

text: i

text: i

text: i

text: and

text: w

text: (here

text: we

text: use

text: a

text: shorthand,

text: the

text: imaginary

text: part

text: could

text: be

text: zero)

text: are

text: independent,

text: the

text: i

text: intersection

text: in

text: a

text: small

text: neighborhood

text: V

text: of

text: x

text: of

text: the

text: zero

text: level

text: sets

text: of

text: the

text: N

text: different

text: functions

text: φ

text: and

text: φ

text: can

text: only

text: be

text: x

text: (to

text: prove

text: by

text: contradiction,

text: just

text: take

text: a

text: sequence

text: of

text: points

text: λi

text: λi

text: x

text: in

text: a

text: nested

text: sequence

text: of

text: neighborhoods

text: V

text: such

text: that

text: φ

text: (x

text: and

text: φ

text: (x

text: i

text: i

text: λi

text: i

text: λi

text: i

text: for

text: every

text: i.

text: By

text: taking

text: the

text: limit

text: of

text: along

text: that

text: sequence,

text: it

text: is

text: easy

text: to

text: see

text: that

text: gradients

text: must

text: be

text: dependent.)

text: Next,

text: we

text: show

text: that

text: the

text: zero

text: level

text: sets

text: cannot

text: have

text: another

text: intersection

text: in

text: X.

text: Suppose

text: that

text: there

text: is

text: another

text: intersection.

text: This

text: must

text: be

text: a

text: part

text: of

text: another

text: invariant

text: set

text: which

text: is

text: not

text: connected

text: to

text: the

text: ﬁxed

text: point.

text: Therefore,

text: the

text: boundary

text: ∂Ω

text: of

text: the

text: basin

text: of

text: attraction

text: Ω

text: of

text: x

text: has

text: a

text: non

text: empty

text: intersection

text: with

text: X,

text: since

text: X

text: is

text: a

text: connected

text: set.

text: Moreover,

text: since

text: X

text: is

text: forward

text: invariant,

text: ∂Ω

text: X

text: is

text: also

text: forward

text: invariant

text: and

text: contains

text: the

text: limit

text: sets

text: of

text: its

text: trajectories.

text: Consider

text: a

text: point

text: x

text: ∂Ω

text: X

text: that

text: belongs

text: to

text: a

text: limit

text: set.

text: By

text: deﬁnition

text: and

text: continuity

text: of

text: the

text: eigenfunctions,

text: ω

text: we

text: have

text: φ

text: (x

text: i

text: (see

text: Also,

text: for

text: any

text: arbitrarily

text: small

text: neighborhood

text: V

text: of

text: x

text: and

text: λi

text: ω

text: ω

text: for

text: all

text: x

text: V

text: Ω,

text: there

text: exist

text: a

text: point

text: x

text: V

text: with

text: x

text: x

text: and

text: a

text: constant

text: T

text: such

text: that

text: V

text: V

text: ϕ

text: T

text: (x

text: x

text: There

text: is

text: at

text: least

text: one

text: eigenfunction

text: that

text: satisﬁes

text: φ

text: (x

text: C

text: since

text: x

text: V

text: λi

text: V

text: is

text: the

text: only

text: intersection

text: in

text: V

text: of

text: the

text: zero

text: level

text: sets

text: of

text: the

text: N

text: eigenfunctions.

text: Equivalently

text: we

text: have

text: φ

text: (x

text: C

text: exp(

text: λ

text: T

text: Therefore,

text: φ

text: is

text: not

text: continuous

text: in

text: V

text: X,

text: which

text: is

text: a

text: λi

text: i

text: λi

text: contradiction.

text: May

text: DRAFT


text: 15 (cid:60){ } { ∗} Finally, since λ < 0, the result follows from Corollary 1 with M = x . i Necessity. (The proof is inspired from results presented in [9].) Since the ﬁxed point is globally stable in X, it follows from Theorem 2.3 in [9] that there exists a C1 diffeomorphism y = h(x) ∗ such that y˙ = J y, h(x ) = 0, and the Jacobian matrix of h at 0 satisﬁes J = I. For this linear h · system, there exist N distinct Koopman eigenfunctions φ˜ (y) = y w that are associated with λi i the eigenvalues of J (see e.g. [14], [16]). It follows that the Koopman operator of (2) has N ◦ C1 eigenfunctions of the form φ = φ˜ h (with the same eigenvalues). Moreover, we have λi λi ∇ ∗ (cid:54) φ (x ) = JT w = w = 0. This concludes the proof. λi h i i Proposition 2 can be interpreted as the global equivalent of the well-known local stability result for a ﬁxed point. While local stability depends on N eigenvalues of the Jacobian matrix, global stability depends on N C1 Koopman eigenfunctions (associated with the same eigenvalues). The support of these particular eigenfunctions corresponds to the basin of attraction of the ∇ ∗ (cid:54) ﬁxed point. Note also that the condition φ (x ) = 0 is necessary to rule out the Koopman λ · · · eigenfunctions of the form φk1 φkN (see Property 1), which are redundant with respect to the λ1 λN basic eigenfunctions φ . λi Remark 6 (Linear systems). In the case of a linear system x˙ = Ax, the eigenfunctions are · ∈ R given by φ (x) = x w where w is a left eigenvector of A. They satisfy φ C1( N) and λi i i λi ∇ ∗ (cid:54) (cid:60){ } φ (x ) = w = 0, so that Proposition 2 only requires λ < 0. We recover the usual λi i i stability criterion for (global) stability of linear systems. Remark 7 (Non hyperbolic case). The result of Proposition 2 cannot be used when the ﬁxed (cid:60){ } point is not hyperbolic (i.e. the Jacobian matrix J has at least one eigenvalue λ = 0). In this i case, the Koopman operator does not possess N eigenfunctions that satisfy the assumptions of (cid:60){ } ∇ ∗ (cid:54) Proposition 2 (i.e. λ < 0 and φ (x ) = 0). However, the operator has a continuous spectrum λ (cid:60){ } with eigenvalues satisfying λ < 0 (see [5]), so that Theorem 1 can be used. For instance, the − − system x˙ = x3 admits a continuous (generalized) eigenfunction φ (x) = exp( 1/(2x2)) with λ − the associated eigenvalue λ = 1, and Theorem 1 implies global stability of the (non hyperbolic) May 26, 2020 DRAFT ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Finally,

text: since

text: λ

text: the

text: result

text: follows

text: from

text: Corollary

text: with

text: M

text: x

text: i

text: Necessity.

text: (The

text: proof

text: is

text: inspired

text: from

text: results

text: presented

text: in

text: Since

text: the

text: ﬁxed

text: point

text: is

text: globally

text: stable

text: in

text: X,

text: it

text: follows

text: from

text: Theorem

text: in

text: that

text: there

text: exists

text: a

text: diffeomorphism

text: y

text: h(x)

text: such

text: that

text: y˙

text: J

text: y,

text: h(x

text: and

text: the

text: Jacobian

text: matrix

text: of

text: h

text: at

text: satisﬁes

text: J

text: I.

text: For

text: this

text: linear

text: h

text: system,

text: there

text: exist

text: N

text: distinct

text: Koopman

text: eigenfunctions

text: φ˜

text: (y)

text: y

text: w

text: that

text: are

text: associated

text: with

text: λi

text: i

text: the

text: eigenvalues

text: of

text: J

text: (see

text: e.g.

text: It

text: follows

text: that

text: the

text: Koopman

text: operator

text: of

text: has

text: N

text: eigenfunctions

text: of

text: the

text: form

text: φ

text: φ˜

text: h

text: (with

text: the

text: same

text: eigenvalues).

text: Moreover,

text: we

text: have

text: λi

text: λi

text: φ

text: (x

text: JT

text: w

text: w

text: This

text: concludes

text: the

text: proof.

text: λi

text: h

text: i

text: i

text: Proposition

text: can

text: be

text: interpreted

text: as

text: the

text: global

text: equivalent

text: of

text: the

text: well-known

text: local

text: stability

text: result

text: for

text: a

text: ﬁxed

text: point.

text: While

text: local

text: stability

text: depends

text: on

text: N

text: eigenvalues

text: of

text: the

text: Jacobian

text: matrix,

text: global

text: stability

text: depends

text: on

text: N

text: Koopman

text: eigenfunctions

text: (associated

text: with

text: the

text: same

text: eigenvalues).

text: The

text: support

text: of

text: these

text: particular

text: eigenfunctions

text: corresponds

text: to

text: the

text: basin

text: of

text: attraction

text: of

text: the

text: ﬁxed

text: point.

text: Note

text: also

text: that

text: the

text: condition

text: φ

text: (x

text: is

text: necessary

text: to

text: rule

text: out

text: the

text: Koopman

text: λ

text: eigenfunctions

text: of

text: the

text: form

text: φkN

text: (see

text: Property

text: which

text: are

text: redundant

text: with

text: respect

text: to

text: the

text: λ1

text: λN

text: basic

text: eigenfunctions

text: φ

text: λi

text: Remark

text: (Linear

text: systems).

text: In

text: the

text: case

text: of

text: a

text: linear

text: system

text: x˙

text: Ax,

text: the

text: eigenfunctions

text: are

text: R

text: given

text: by

text: φ

text: (x)

text: x

text: w

text: where

text: w

text: is

text: a

text: left

text: eigenvector

text: of

text: A.

text: They

text: satisfy

text: φ

text: N)

text: and

text: λi

text: i

text: i

text: λi

text: φ

text: (x

text: w

text: so

text: that

text: Proposition

text: only

text: requires

text: λ

text: We

text: recover

text: the

text: usual

text: λi

text: i

text: i

text: stability

text: criterion

text: for

text: (global)

text: stability

text: of

text: linear

text: systems.

text: Remark

text: (Non

text: hyperbolic

text: case).

text: The

text: result

text: of

text: Proposition

text: cannot

text: be

text: used

text: when

text: the

text: ﬁxed

text: point

text: is

text: not

text: hyperbolic

text: (i.e.

text: the

text: Jacobian

text: matrix

text: J

text: has

text: at

text: least

text: one

text: eigenvalue

text: λ

text: In

text: this

text: i

text: case,

text: the

text: Koopman

text: operator

text: does

text: not

text: possess

text: N

text: eigenfunctions

text: that

text: satisfy

text: the

text: assumptions

text: of

text: Proposition

text: (i.e.

text: λ

text: and

text: φ

text: (x

text: However,

text: the

text: operator

text: has

text: a

text: continuous

text: spectrum

text: λ

text: with

text: eigenvalues

text: satisfying

text: λ

text: (see

text: so

text: that

text: Theorem

text: can

text: be

text: used.

text: For

text: instance,

text: the

text: system

text: x˙

text: admits

text: a

text: continuous

text: (generalized)

text: eigenfunction

text: φ

text: (x)

text: exp(

text: with

text: λ

text: the

text: associated

text: eigenvalue

text: λ

text: and

text: Theorem

text: implies

text: global

text: stability

text: of

text: the

text: (non

text: hyperbolic)

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 ∈ N ﬁxed point at the origin. Note that dnφ /dxn(0) = 0 for all n so that the eigenfunction is λ not analytic. For the sufﬁciency part of Proposition 2, the assumptions on the eigenvalues of the Jacobian matrix are actually not required. We have the following Corollary. Corollary 2. Let X ⊂ RN be a connected, forward invariant, compact set and consider that ∗ ∈ ∈ (2) admits a ﬁxed point x X. If there exist N Koopman eigenfunctions φ C1(X), with λ (cid:60){ } ∇ ∗ (cid:54) ∗ λ < 0 and such that the N vectors φ (x ) = 0 are linearly independent, then x is globally λ asymptotically stable in X. Proof. The result follows directly from Corollary 1 and from the proof of Proposition 2. The Koopman eigenfunctions considered in Proposition 2 and Corollary 2 yield the Lyapunov functions [13], [14] (cid:32) (cid:33) (cid:88) 1/p V N | | (x) = φ (x) p (14) λi i=1 ≥ V ≤ with the integer p 1. According to (6), these Lyapunov functions satisfy (ϕt(x)) (cid:60){ } V ∈ exp( λ t) (x) for x X, where λ is the eigenvalue closest to the imaginary axis. (Note 1 1 that the Lyapunov functions are not necessarily smooth.) This result can be used to deﬁne a (compact) set { ∈ | } Ω = x X V (x) < α α ∩ ∅ which is forward invariant if Ω ∂X = , where ∂X is the boundary of X. This is useful α when it cannot be shown that X is forward invariant. In numerical simulations, the Lyapunov function (14) can also be used to estimate the basin of attraction or to check the accuracy of the computations. In addition, it is shown in [13], [14] that the eigenfunctions are related to (non-quadratic) metrics that are exponentially contracting on X. These metrics could be considered through the differential framework recently developed in [4]. May 26, 2020 DRAFT ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: ﬁxed

text: point

text: at

text: the

text: origin.

text: Note

text: that

text: dnφ

text: for

text: all

text: n

text: so

text: that

text: the

text: eigenfunction

text: is

text: λ

text: not

text: analytic.

text: For

text: the

text: sufﬁciency

text: part

text: of

text: Proposition

text: the

text: assumptions

text: on

text: the

text: eigenvalues

text: of

text: the

text: Jacobian

text: matrix

text: are

text: actually

text: not

text: required.

text: We

text: have

text: the

text: following

text: Corollary.

text: Corollary

text: Let

text: X

text: RN

text: be

text: a

text: connected,

text: forward

text: invariant,

text: compact

text: set

text: and

text: consider

text: that

text: admits

text: a

text: ﬁxed

text: point

text: x

text: X.

text: If

text: there

text: exist

text: N

text: Koopman

text: eigenfunctions

text: φ

text: with

text: λ

text: λ

text: and

text: such

text: that

text: the

text: N

text: vectors

text: φ

text: (x

text: are

text: linearly

text: independent,

text: then

text: x

text: is

text: globally

text: λ

text: asymptotically

text: stable

text: in

text: X.

text: Proof.

text: The

text: result

text: follows

text: directly

text: from

text: Corollary

text: and

text: from

text: the

text: proof

text: of

text: Proposition

text: The

text: Koopman

text: eigenfunctions

text: considered

text: in

text: Proposition

text: and

text: Corollary

text: yield

text: the

text: Lyapunov

text: functions

text: V

text: N

text: (x)

text: φ

text: (x)

text: p

text: λi

text: V

text: with

text: the

text: integer

text: p

text: According

text: to

text: these

text: Lyapunov

text: functions

text: satisfy

text: (ϕt(x))

text: V

text: exp(

text: λ

text: t)

text: (x)

text: for

text: x

text: X,

text: where

text: λ

text: is

text: the

text: eigenvalue

text: closest

text: to

text: the

text: imaginary

text: axis.

text: (Note

text: that

text: the

text: Lyapunov

text: functions

text: are

text: not

text: necessarily

text: smooth.)

text: This

text: result

text: can

text: be

text: used

text: to

text: deﬁne

text: a

text: (compact)

text: set

text: Ω

text: x

text: X

text: V

text: (x)

text: α

text: α

text: which

text: is

text: forward

text: invariant

text: if

text: Ω

text: ∂X

text: where

text: ∂X

text: is

text: the

text: boundary

text: of

text: X.

text: This

text: is

text: useful

text: α

text: when

text: it

text: cannot

text: be

text: shown

text: that

text: X

text: is

text: forward

text: invariant.

text: In

text: numerical

text: simulations,

text: the

text: Lyapunov

text: function

text: can

text: also

text: be

text: used

text: to

text: estimate

text: the

text: basin

text: of

text: attraction

text: or

text: to

text: check

text: the

text: accuracy

text: of

text: the

text: computations.

text: In

text: addition,

text: it

text: is

text: shown

text: in

text: that

text: the

text: eigenfunctions

text: are

text: related

text: to

text: (non-quadratic)

text: metrics

text: that

text: are

text: exponentially

text: contracting

text: on

text: X.

text: These

text: metrics

text: could

text: be

text: considered

text: through

text: the

text: differential

text: framework

text: recently

text: developed

text: in

text: May

text: DRAFT

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