[PPT] - Loop Correlations in Random Wire Models Costanza Benassi 23rd PowerPoint Presentation

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Loop Correlations in Random Wire Models

Costanza Benassi 23rd August 2019 Based on a joint work with Daniel Ueltschi (University of Warwick) (CMP 2019)

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Introduction

Loops as one-dimensional objects living in higher dimensional spaces. Great variety of models for interacting loops on a lattice (random interchange model, lattice permutations...). Expected to share the same qualitative behaviour. We focus on a specific one, the random wire model.

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Plan of the talk

1 Definition of our model. 2 Conjectured behaviour. 3 A partial result. 4 Relationship between our loop model and the XY spin model. Costanza Benassi Random Wire Models 23rd August 2019 3 / 15

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The random wire model

Generalisation of the random current representation of the Ising model

[Griffiths, Hurst, Sherman ’70; Aizenman ’82].

Define GL = (ΛL, EL) with ΛL = (−L, . . . , L)d. Link configurations. m = (me)e∈EL ∈ NEL. Constraint:

e∋x me

even for any x ∈ ΛL.

Pairings. π = (πx)x∈ΛL. We can divide links around each site into

pairs in different ways. πx is such a choice for site x.

2 2 1 1 1 1 1 1 1 1

1 3 4 5 6 7 8 9 2

1 2 1 3 3 2 2 1

1 1 1 1 1 2 1 1 3 2 1 2 3 1 1 2 2 1

w = (m, π) is a wire configuration. WGL is the set of wire configurations.

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The random wire model

Loops naturally appear. They are undirected and have no beginning and no end. The length ℓ of a loop is the number of links belonging to it.

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Notation: For any w ∈ WGL, λ(w) is the number of loops. For any site x ∈ ΛL nx(m) = 1

2

e∋x me.

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The random wire model

We define a probability measure over wire configurations. Let α, β > 0. Pα,β

GL (w) =

1 ZGL(α, β) αλ(w)

e∈EL

βme me!e−

x∈ΛL U(nx(m))

with Partition function: ZGL(α, β) =

w∈WΛL αλ(w) e∈EL βme me! e−

x∈ΛL U(nx),

Potential function: U : N → R.

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Some conjectures in d ≥ 3

Let w ∈ WGL and (ℓ1(w), ℓ2(w), . . . , ℓk(w)) the sequence of the lengths of its loops in decreasing order ℓ1 ≥ ℓ2 ≥ · · · ≥ ℓk. Let Ltot(w) = k

i=1 ℓi(w).

Notice that

ℓ1(w)

Ltot(w), . . . , ℓk(w) Ltot(w)

is a random partition of [0, 1].

A loop of length ℓ can be: macroscopic: ℓ ∼ Ltot(w). microscopic: ℓ ∼ 1. mesoscopic: neither microscopic nor macroscopic.

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Some conjectures in d ≥ 3

A fraction m ∈ [0, 1] of the total loop volume Ltot(w) is occupied by macroscopic loops, and the rest is occupied by microscopic loops. If m = 0, the lengths of the macroscopic loops are distributed according to a PD(ϑ) distribution with ϑ = α

2 .

macroscopic, PD(ϑ) microscopic

m

[Schramm ’05] proved it for the random interachange model on the

complete graph. Conjectured to hold in more generality for spatial loop models [Goldschmidt, Ueltschi, Windridge ’11; Ueltschi ’17]. Numerical results for some specific models [Grosskinsky, Lovisolo, Ueltschi ’12; Nahum, Chalker,

Serna, Ortu˜ no ’13, Barp, Barp, Briol, Ueltschi ’15].

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Set partitions and loop configurations

Let us a fix a wire configuration w. and x = (x1, . . . , xk) be k sites in Zd. We define Yx(w) = {Yi}ℓ

i=1 be the

set partition of {1, 2, . . . , k} such that {1, 2, . . . , k} = ∪ℓ

i=1Yi with Yi ∩ Yj = ∅.

m, n belong to the same subset iff xm, xn belong to the same loop.

x1 x2 x3 x4 x5 x6

x = (x1, x2, x3, x4, x5, x6)

Yx(w) = {{1, 2, 3, 4} , {5, 6}}

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Set partitions and PD distributions

Analogously, for any Y set partition of {1, 2, . . . , k} and given U1, . . . , Uk ∈ [0, 1] i.i.d. unformly, we define PPD(ϑ)[Y ] = PPD(ϑ)[Um, Un are in the same PD(ϑ) partition element iff m, n belong to the same subset]. Y = {{1, 2, 3, 4} , {5, 6}}

U1 U2 U3 U4 U5 U6

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Some conjectures in d ≥ 3

Define a splashing sequence as a sequence of k-ple of sites in Zd x(n) = (x(n)

1 , x(n) 2 , . . . , x(n) k ) such that limn→∞ x(n) i

− x(n)

j

= ∞ ∀i = j. Fix a set partition Y with |Yi| = 1 for all i. We expect lim

n→∞ lim L→∞ Pα,β GL [Yx = Y ] = PPD( α

2 )[Y ] Pα,β

Zd [0 ∈ long loop]k. x1 x2 x3 x4 x5 x6

←

→

U1 U2 U3 U4 U5 U6

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Main result: PD(1) distribution for the random wire model

Our result is in the same spirit. We fix U(n) = log n!, k ∈ N, α = 2 and x(n) = (x(n)

1 , . . . , x(n) 2k ) a splashing sequence.

Theorem (B., Ueltschi (’19))

lim

n→∞ lim L→∞

Y even

E2,β

GL

 1Yx(n)=Y

2k

i=1

1 nx(n)

j

+ 1   = m(d, β)2k

Y even

PPD(1)[Y ]. The sums are over even set partitions of {1, 2, . . . , 2k}, i.e. |Yi| even ∀i. m(d, β) is non-decreasing in both d and β and if d ≥ 3 there exists βc(d) < ∞ such that m(d, β) = 0 if β < βc and m(d, β) > 0 if β > βc.

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What does it actually mean?

Long loops are present when m(d, β) > 0. Multiple long loops occur with positive probability. Since the theorem holds for all k with the same constant m(d, β), this partially proves that the correlations due to long loops are given by Poisson-Dirichlet PD(1).

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The relationship with the XY model

XY spin model ← → Random wire model Spontaneous symmetry breaking ← → Appearance of long loops The random wire expectations above can be written in terms of correlation functions for the XY model (similar to [Brydges, Fr¨

hlich, Spencer ’82]). The

main ingredients of the proof are A major result of [Pfister ’82] about characterisation of extremal Gibbs states for the XY model. Another major result of [Fr¨

hlich, Simon, Spencer ’76] about occurrence
f long-range order at low temperatures.

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To sum up...

General loop model. Expected behaviour: macroscopic loops with PD α

2

.

Partially proved for α = 2. Insight on symmetry breaking for classical spin systems.

THANK YOU!

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Some definitions

We are interested in distributions on the space of ordered partitions of [0,1], i.e. on the space of sequences {Xi}i≥1 such that Xi ∈ [0, 1] for all i ∈ N.

i Xi = 1, i.e. the sequence constitutes a partition of [0, 1].

Xi ≥ Xi+1, i.e. the sequence is ordered.

X1

X2 X3 X4

Poisson-Dirichlet distributions are a family of distributions on this sort
f objects.

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PD(ϑ) distributions and stick breaking construction

Let ν1 be a measure over [0, 1]. For any q ∈ [0, 1] let νq be the same measure rescaled on [0, q] i.e. Pν1(X < s) = Pνq(X < qs) for any s ∈ [0, 1]. Let {Xn}n≥1 be such that

1

X1 is chosen according to ν1;

2

X2 is chosen according to ν1−X1.

3

X3 is chosen according to ν1−X1−X2.

4

. . .

X1
1 − X1

X2

1 − X1 − X2

X3

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PD(ϑ) distributions and stick breaking construction

n Xn = 1 and limn→∞ Xn = 0, i.e. (X1, . . . , Xn) is an unordered

random partition of [0,1]. Suppose ν1 is the measure of a Beta(ϑ) random variable i.e. Pν1(X > s) = (1 − s)ϑ. Rearrange (X1, X2, ...) to be an ordered partition of [0, 1]. This random partition is by definition distributed according to PD(ϑ).

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PD(ϑ) distributions and split merge processes

PD(ϑ) distributions are a family of distributions which are stationary measures for split-merge processes (A. M. Vershik, A. Schmidt (1977), P. Diaconis, E. Mayer-Wolf

et al. (2004)).

Let gs, gm ∈ [0, 1]. Let (Y1, Y2, . . . ) be a random partition at a certain time t ∈ [0, ∞). An element Yi splits at rate Y2

i gs and two elements Yi,

Yj merge at rate 2YiYjgm. This means that during the interval [t,t+dt]:

1 Yj splits with probability Y 2

j gsdt,

2 Yi and Yj (i = j) merge with probability 2YiYjgmdt 3 Nothing happens with probability 1 −

i Y 2 i gsdt − i<j 2YiYjgmdt.

PD(ϑ) is the stationary measure of this process with ϑ = gs

gm .

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An example of split-merge process

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Figure from C. Goldschmidt, D. Ueltschi, P. Windridge (2011). Costanza Benassi Random Wire Models 23rd August 2019 20 / 15

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Why should the conjectures for loop models hold?

Schramm (2005) proved this result for the random interchange model on the complete graph. Macroscopic loops are distributed according to PD(1). Suggested by C. Goldschmidt, D. Ueltschi, and P. Windridge (2011) for a class of loop models related to quantum spin systems. Loops interact in an effective mean-field way in d ≥ 3. It should be possible to define a split-merge process with the measure over loops as stationary measure.

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An effective split-merge process for random currents

Recall the measure Pα,β

GL :

Pα,β

GL (w) =

1 ZGL(α, β) αλ(w)

e∈EL

βme me!e−

x∈ΛL U(nx(m))

Markov process on pairings:

1 Choose uniformly a site. 2 A different pairing at that site is chosen at rate √α if the change

causes a loop to split.

3 A different pairing at that site is chosen with rate

1 √α if the change

merges two loops.

4 A different pairing at that site is chosen with rate 1 if it leaves the

total number of loops unchanged.

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