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

Loop Correlations in Random Wire Models Costanza Benassi 23rd August 2019 Based on a joint work with Daniel Ueltschi (University of Warwick) (CMP 2019) Costanza Benassi Random Wire Models 23rd August 2019 1 / 15

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. 1 1 2 2 3 1 1 1 1 1 2 1 1 2 3 1 1 2 Costanza Benassi Random Wire Models 23rd August 2019 2 / 15

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

The random wire model Generalisation of the random current representation of the Ising model [Griffiths, Hurst, Sherman ’70; Aizenman ’82] . Define G L = (Λ L , E L ) with Λ L = ( − L , . . . , L ) d . Link configurations . m = ( m e ) e ∈E L ∈ N E L . Constraint: � e ∋ x m e even for any x ∈ Λ L . Pairings . π = ( π x ) x ∈ Λ L . We can divide links around each site into pairs in different ways. π x is such a choice for site x . 1 1 1 1 7 8 2 9 2 2 3 2 3 1 1 1 1 4 5 6 1 1 1 1 1 2 1 2 3 1 1 2 1 2 3 1 1 1 1 2 3 1 1 2 2 w = ( m , π ) is a wire configuration . W G L is the set of wire configurations. Costanza Benassi Random Wire Models 23rd August 2019 4 / 15

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. 1 1 1 1 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 3 1 2 3 1 1 1 1 2 2 Notation: For any w ∈ W G L , λ ( w ) is the number of loops. For any site x ∈ Λ L n x ( m ) = 1 � e ∋ x m e . 2 Costanza Benassi Random Wire Models 23rd August 2019 5 / 15

The random wire model We define a probability measure over wire configurations. Let α, β > 0. β m e 1 P α,β m e !e − � x ∈ Λ L U ( n x ( m )) Z G L ( α, β ) α λ ( w ) � G L ( w ) = e ∈E L with Partition function: β me m e ! e − � x ∈ Λ L U ( n x ) , w ∈W Λ L α λ ( w ) � Z G L ( α, β ) = � e ∈E L Potential function: U : N → R . Costanza Benassi Random Wire Models 23rd August 2019 6 / 15

Some conjectures in d ≥ 3 Let w ∈ W G L and ( ℓ 1 ( w ) , ℓ 2 ( w ) , . . . , ℓ k ( w )) the sequence of the lengths of its loops in decreasing order ℓ 1 ≥ ℓ 2 ≥ · · · ≥ ℓ k . Let L tot ( w ) = � k i =1 ℓ i ( w ). � � L tot ( w ) , . . . , ℓ k ( w ) ℓ 1 ( w ) Notice that is a random partition of [0 , 1]. L tot ( w ) A loop of length ℓ can be: macroscopic: ℓ ∼ L tot ( w ). microscopic: ℓ ∼ 1. mesoscopic: neither microscopic nor macroscopic. Costanza Benassi Random Wire Models 23rd August 2019 7 / 15

Some conjectures in d ≥ 3 A fraction m ∈ [0 , 1] of the total loop volume L tot ( 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, no ’13, Barp, Barp, Briol, Ueltschi ’15] . Serna, Ortu˜ Costanza Benassi Random Wire Models 23rd August 2019 8 / 15

� Set partitions and loop configurations Let us a fix a wire configuration w . and x = ( x 1 , . . . , x k ) be k sites in Z d . We define Y x ( w ) = { Y i } ℓ i =1 be the set partition of { 1 , 2 , . . . , k } such that { 1 , 2 , . . . , k } = ∪ ℓ i =1 Y i with Y i ∩ Y j = ∅ . m , n belong to the same subset iff x m , x n belong to the same loop. x 1 x 5 x = ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) x 4 x 2 Y x ( w ) = {{ 1 , 2 , 3 , 4 } , { 5 , 6 }} x 3 x 6 Costanza Benassi Random Wire Models 23rd August 2019 9 / 15

� Set partitions and PD distributions Analogously, for any Y set partition of { 1 , 2 , . . . , k } and given U 1 , . . . , U k ∈ [0 , 1] i.i.d. unformly, we define P PD ( ϑ ) [ Y ] = P PD ( ϑ ) [ U m , U n are in the same PD( ϑ ) partition element iff m , n belong to the same subset] . U 1 U 2 U 3 U 4 U 5 U 6 Y = {{ 1 , 2 , 3 , 4 } , { 5 , 6 }} Costanza Benassi Random Wire Models 23rd August 2019 10 / 15

� � Some conjectures in d ≥ 3 Define a splashing sequence as a sequence of k -ple of sites in Z d x ( n ) = ( x ( n ) 1 , x ( n ) 2 , . . . , x ( n ) k ) such that lim n →∞ � x ( n ) − x ( n ) � = ∞ ∀ i � = j . i j Fix a set partition Y with | Y i | � = 1 for all i . We expect L →∞ P α,β 2 )[ Y ] P α,β Z d [0 ∈ long loop] k . n →∞ lim lim G L [ Y x = Y ] = P PD ( α x 1 x 5 U 1 U 2 U 3 U 4 U 5 U 6 ← → x 4 x 2 x 3 x 6 Costanza Benassi Random Wire Models 23rd August 2019 11 / 15

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 ) 2 k ) a splashing sequence. Theorem (B., Ueltschi (’19))   2 k 1 E 2 ,β � �  = m ( d , β ) 2 k � n →∞ lim lim  1 Y x ( n ) = Y P PD ( 1 ) [ Y ] . G L n x ( n ) + 1 L →∞ Y even i =1 Y even j The sums are over even set partitions of { 1 , 2 , . . . , 2 k } , i.e. | Y i | even ∀ i. m ( d , β ) is non-decreasing in both d and β and if d ≥ 3 there exists β c ( d ) < ∞ such that m ( d , β ) = 0 if β < β c and m ( d , β ) > 0 if β > β c . Costanza Benassi Random Wire Models 23rd August 2019 12 / 15

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). Costanza Benassi Random Wire Models 23rd August 2019 13 / 15

The relationship with the XY model ← → XY spin model Random wire model Spontaneous symmetry ← → Appearance of long loops breaking The random wire expectations above can be written in terms of correlation functions for the XY model (similar to [Brydges, Fr¨ ohlich, 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¨ ohlich, Simon, Spencer ’76] about occurrence of long-range order at low temperatures. Costanza Benassi Random Wire Models 23rd August 2019 14 / 15

To sum up... General loop model. � α � Expected behaviour: macroscopic loops with PD . 2 Partially proved for α = 2. Insight on symmetry breaking for classical spin systems. THANK YOU! Costanza Benassi Random Wire Models 23rd August 2019 15 / 15

� � � � Some definitions We are interested in distributions on the space of ordered partitions of [0,1] , i.e. on the space of sequences { X i } i ≥ 1 such that X i ∈ [0 , 1] for all i ∈ N . � i X i = 1, i.e. the sequence constitutes a partition of [0 , 1]. X i ≥ X i +1 , i.e. the sequence is ordered . X 1 X 2 X 3 X 4 Poisson-Dirichlet distributions are a family of distributions on this sort of objects. Costanza Benassi Random Wire Models 23rd August 2019 16 / 15

� � � � � PD( ϑ ) distributions and stick breaking construction Let ν 1 be a measure over [0 , 1]. For any q ∈ [0 , 1] let ν q be the same measure rescaled on [0 , q ] i.e. P ν 1 ( X < s ) = P ν q ( X < qs ) for any s ∈ [0 , 1]. Let { X n } n ≥ 1 be such that X 1 is chosen according to ν 1 ; 1 X 2 is chosen according to ν 1 − X 1 . 2 X 3 is chosen according to ν 1 − X 1 − X 2 . 3 . . . 4 X 1 X 2 1 − X 1 X 3 1 − X 1 − X 2 Costanza Benassi Random Wire Models 23rd August 2019 17 / 15

PD( ϑ ) distributions and stick breaking construction � n X n = 1 and lim n →∞ X n = 0, i.e. ( X 1 , . . . , X n ) is an unordered random partition of [0,1]. Suppose ν 1 is the measure of a Beta( ϑ ) random variable i.e. P ν 1 ( X > s ) = (1 − s ) ϑ . Rearrange ( X 1 , X 2 , ... ) to be an ordered partition of [0 , 1]. This random partition is by definition distributed according to PD( ϑ ). Costanza Benassi Random Wire Models 23rd August 2019 18 / 15

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 g s , g m ∈ [0 , 1]. Let ( Y 1 , Y 2 , . . . ) be a random partition at a certain time t ∈ [0 , ∞ ). An element Y i splits at rate Y 2 i g s and two elements Y i , Y j merge at rate 2Y i Y j g m . This means that during the interval [t,t+dt]: 1 Y j splits with probability Y 2 j g s dt , 2 Y i and Y j ( i � = j ) merge with probability 2 Y i Y j g m dt 3 Nothing happens with probability 1 − � i Y 2 i g s dt − � i < j 2 Y i Y j g m dt . PD( ϑ ) is the stationary measure of this process with ϑ = g s g m . Costanza Benassi Random Wire Models 23rd August 2019 19 / 15