High temperature regime in spatial random permutations. Lorenzo Taggi, TU Darmstadt (Joint work with Volker Betz, TU Darmstadt)

Transcription

1 High temperature regime in spatial random permutations Lorenzo Taggi, TU Darmstadt (Joint work with Volker Betz, TU Darmstadt)

2 /8 Definition: Nearest neighbor SRP with forced long cycle Set Λ L = [0, L] d Z d and fix two sites z 1, z 2 Λ L Forced open cycle between z 1 and z 2 = set of bijections π : Λ L \ {z 1 } Λ L \ {z 2 } with the constraint that π(x) = x or π(x) x for all x Λ L. S z1 z 2 Λ L Gibbs measure depending on α [0, ) P Λ ({π}) = 1 ( Z(Λ) exp α ) π(x) x, x Λ z 1 z 2 Actual settings 1. forced open cycle between opposite sides. z 1 = centre of the boundary side of the box, 1 S ΛL = z 2 z 2 opp. side to z 1 S z Λ L 2. only cycles, namely nearest-neighbours permutations π : Λ Λ.

3 2/8 Difficulties and results Difficulties and challenges 1. no classical spin system (bijections introduce constraints) 2. energy of the open cycle much smaller than the energy of the system 3. no comparison between the law of π and the law of π conditional on some local features. Results There exists α c < such that for all α > α c, 1. No long cycles exist 2. Exponential decay of correlations 3. Orstein-Zernike behaviour for the forced open cycle in any dimension of Z d.

4 3/8 Orstein-Zernike method r 1 r 2 r 3 r 4

5 /8 Orstein-Zernike method Define a proper function on the sample space f (π) = {(r 1, ξ 1), (r 2, ξ 2),... (r N, ξ N )}, such that Markov process (r i, ξ i ) (r i+1, ξ i+1 ), Λ L P ( r i+1, ξ i+1 (r1, ξ 1),... (r i, ξ i ) ) = Q L (r i+1, ξ i+1, r i, ξ i ) r 1 r 2 r 3 r 4 Regeneration surfaces are defined to be symmetric under a reflection with respect to r i,2. This implies that E[(r i+1 r i ) e 2 ri, ξ i ] = 0. for any r i, ξ i. We prove that, P( r i+1 r i > D log(l) ri, ξ i ) C exp{ D c} with C and c independent on ξ i and positive.

6 /8 Orstein-Zernike method Define a proper function on the sample space f (π) = {(r 1, ξ 1), (r 2, ξ 2),... (r N, ξ N )}, such that Markov process (r i, ξ i ) (r i+1, ξ i+1 ), Λ L ξ 1 ξ ξ ξ P ( r i+1, ξ i+1 (r1, ξ 1),... (r i, ξ i ) ) = Q L (r i+1, ξ i+1, r i, ξ i ) r 1 r 2 r 3 r 4 Regeneration surfaces are defined to be symmetric under a reflection with respect to r i,2. This implies that E[(r i+1 r i ) e 2 ri, ξ i ] = 0. for any r i, ξ i. We prove that, P( r i+1 r i > D log(l) ri, ξ i ) C exp{ D c} with C and c independent on ξ i and positive.

7 5/8 We define a set of indep. r.v. (σ A ) A Z d, where σ A is distributed like P A. Step 1: 1. choose a site x 1 on the vertical line 2. sample σ Λ and keep only the cycle intersecting x 1, γ 1 3. define B 1 = Λ \ γ 1 set N 1 Step 2: 1. choose a site x 2 of the non-matching set N 1 2. sample σ B1 but keep only the cycle intersecting x 2, γ 2 3. define B 2 = B 1 \ γ 2 set N 2

8 5/8 We define a set of indep. r.v. (σ A ) A Z d, where σ A is distributed like P A. Step 1: 1. choose a site x 1 on the vertical line 2. sample σ Λ and keep only the cycle intersecting x 1, γ 1 3. define B 1 = Λ \ γ 1 set N 1 Step 2: 1. choose a site x 2 of the non-matching set N 1 2. sample σ B1 but keep only the cycle intersecting x 2, γ 2 3. define B 2 = B 1 \ γ 2 set N 2

9 5/8 We define a set of indep. r.v. (σ A ) A Z d, where σ A is distributed like P A. Step 1: 1. choose a site x 1 on the vertical line 2. sample σ Λ and keep only the cycle intersecting x 1, γ 1 3. define B 1 = Λ \ γ 1 set N 1 Step 2: 1. choose a site x 2 of the non-matching set N 1 2. sample σ B1 but keep only the cycle intersecting x 2, γ 2 3. define B 2 = B 1 \ γ 2 set N 2

10 5/8 We define a set of indep. r.v. (σ A ) A Z d, where σ A is distributed like P A. Step 1: 1. choose a site x 1 on the vertical line 2. sample σ Λ and keep only the cycle intersecting x 1, γ 1 3. define B 1 = Λ \ γ 1 set N 1 Step 2: 1. choose a site x 2 of the non-matching set N 1 2. sample σ B1 but keep only the cycle intersecting x 2, γ 2 3. define B 2 = B 1 \ γ 2 set N 2

11 5/8 We define a set of indep. r.v. (σ A ) A Z d, where σ A is distributed like P A. Step 1: 1. choose a site x 1 on the vertical line 2. sample σ Λ and keep only the cycle intersecting x 1, γ 1 3. define B 1 = Λ \ γ 1 set N 1 Step 2: 1. choose a site x 2 of the non-matching set N 1 2. sample σ B1 but keep only the cycle intersecting x 2, γ 2 3. define B 2 = B 1 \ γ 2 set N 2

12 5/8 We define a set of indep. r.v. (σ A ) A Z d, where σ A is distributed like P A. Step 1: 1. choose a site x 1 on the vertical line 2. sample σ Λ and keep only the cycle intersecting x 1, γ 1 3. define B 1 = Λ \ γ 1 set N 1 Step 2: 1. choose a site x 2 of the non-matching set N 1 2. sample σ B1 but keep only the cycle intersecting x 2, γ 2 3. define B 2 = B 1 \ γ 2 set N 2

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20 We have the first cluster. We start again from a new site on the vertical line.

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44 Last step: We sample σ Λ\clusters. Lemma If we put together the cycles that we kept at any step, we have a permutation which is distributed like P Λ The procedure defines a stochastic process N 1, N 2, N 3,...

45 7/8 Proposition There exists α c < such that α > α c the following holds. Namely, conditional on any realization of the procedure up to the step i, we have that n N, P( γ i+1 > n x0, σ Λ,... x i, σ Bi ) C exp{ cn}, where c(α), C(α) > 0, γ i+1 cardinality of γ i+1. Theorem Let (W x) x vert line be a sequence of i.i.d. rand. var. distributed like the total population of a Galton-Watson process. Then, n N, P Λ ( max distance from vert. line > n ) P( x vert. line : W x > n )

46 8/8 Open problems 1. convergence to Brownian motion under diffusive scaling? 2. understanding regime of small α 3. monotonicity with respect to α