Where Probability Meets Combinatorics and Statistical Mechanics

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Where Probability Meets Combinatorics and Statistical Mechanics Daniel Ueltschi Department of Mathematics, University of Warwick MASDOC, 15 March 2011 Collaboration with V. Betz, N.M. Ercolani, C. Goldschmidt, S. Tate, Y. Velenik, P. Windridge D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 1 / 13

Random permutations... with cycle weights with spatial structure We are looking for the distribution of long cycles! D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 2 / 13

Random permutations with cycle weights Random object: permutation π S n P n (π) = 1 Z n j 1 θ R j(π) j where R j (π) is the number of j-cycles in the permutation π Properties depend on parameters θ j. Results so far (obtained with Betz, Ercolani, Velenik): Parameters Typical cycle lengths Number of cycles Finite cycles θ n = e nγ, γ > 1 P n(l 1 = n) 1 K 1 R j 0 θ n e nγ, 0 < γ < 1 L 1 /( 1 1 γ log n)1/γ 1 R j Poisson` θ j j θ n n γ 1, γ > 0 L 1 /n γ+1 1 Gamma`γ +1, Γ(γ +1) γ+1 γ E n(k) An γ+1 R j Poisson` θ j j θ n θ L 1 /n Beta(1, θ) E n(k) θ log n R j Poisson` θ j j θ n = n γ, γ > 0, or θ n = e nγ L 1 /n 1 K 1 + Poisson( P θ j j, 0 < γ < 1 ) R j Poisson` θ j j γ 1 γ θ n = e nγ, γ > 1 L 1 /` 1 γ 1 log n 1/γ 1 En(R j ) exp jγ` log n γ 1 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 3 / 13

Methods: Mix of probability & combinatorics Probability distribution of cycle that contains index 1 : P n (L 1 = j) = θ jz n j nz n Use theory of combinatorial structures to get expression for generating function of Z n Saddle point method to get Z n Project 1. More regimes to be studied, e.g. θ n n log n. Aim for more precise results! Theory of combinatorial structures also useful in the study of cluster expansions of statistical mechanics (work in progress with S. Tate) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 4 / 13

Motivation to spatial permutations: quantum Bose gas N bosons in Λ R d State space L 2 sym(λ N ) Hamiltonian operator N H = U(x i x j ) i=1 i + i<j Feynman-Kac formula for Z = Tr e βh : with x = (x 1,..., x N ), Z = dx Λ N π S N e H(x,π) x with Hamiltonian 1 β π (1) H : Λ N S N R (Explicit expression for H in terms of Wiener trajectories) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 5 / 13

Lattice permutations Random objects: permutations of Λ Z d (i.e. bijections Λ Λ ) P(π) = 1 Z(Λ) exp { 1 4β x π(x) 2} x Λ Mean jump length β Expected: critical value β c, with long cycles iff β > β c Project 2. Numerical simulations. Once they work, there are many questions to be investigated! D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 6 / 13

Monte Carlo simulations (dimension d = 2 ) Step 0 (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Monte Carlo simulations (dimension d = 2 ) Step 1 Step 1 Step 1 (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) temperature! = 0.75 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Monte Carlo simulations (dimension d = 2 ) Step 2 Step 2 Step 2 (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) temperature! = 0.75 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Monte Carlo simulations (dimension d = 2 ) Step 3 Step 3 Step 3 (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) temperature! = 0.75 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Monte Carlo simulations (dimension d = 2 ) Step 4 Step 4 Step 4 (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) temperature! = 0.75 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Monte Carlo simulations (dimension d = 2 ) Step 5 Step 5 Step 5 (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) temperature! = 0.75 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Monte Carlo simulations (dimension d = 2 ) Step 10 Step 10 Step 10 (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) temperature! = 0.75 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Monte Carlo simulations (dimension d = 2 ) Step 30 Step 30 Step 30 (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) temperature! = 0.75 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Monte Carlo simulations (dimension d = 2 ) Step 100 Step 100 Step 100 (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) temperature! = 0.75 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Monte Carlo simulations (dimension d = 2 ) Step " Step " Step (Courtesy of Daniel Gandolfo and Jean Ruiz, CNRS, Marseille) temperature! = 0.75 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 7 / 13

Spatial permutations ( annealed ) State space: Ω Λ,N = Λ N S N Gibbs weight e H(x,π) Expectation of random variable Θ : E Λ,N (Θ) = 1 dx Θ(x, π) e H(x,π) Z Λ N π General Hamiltonian: H(x, π) = N ξ(x i x π(i) ) + α j R j (π) i=1 j 1 with α j : parameters, and R j (π) : number of cycles of length j Bose gas: ξ(x) = 1 4β x 2 D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 8 / 13

Main results (with V. Betz) Fraction of points in cycles: ν = lim K lim Λ,N E ( 1 N We suppose that ξ(x) = ξ( x ), e ξ = 1, ê ξ 0 i:l (i) >K Theorem 1. If α j α, there exists ρ c such that ( ) (a) ν = max 0, ρ ρc ρ ( ) (b) When ν > 0 : L (1) νn, L(2) νn,... Poisson-Dirichlet( e α ) L (i)) Theorem 2. If α j = γ log j with γ > 0, there exists ρ c such that ( ) (a) ν = max 0, ρ ρc ρ (b) When ν > 0, L(1) νn 1 Project 3. Extend results to more functions ξ, more weights α j! D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 9 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

Graph (Λ, E). Most relevant: Λ a box in Z 3 Poisson process on edges ρ β (dω) Next: define cycle and loop configurations D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 10 / 13

.. D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles. D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles. D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles. D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles. D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles. D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles. D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles. D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles loops A B A B A B A D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles loops A B A B A B A D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles loops A B A B A B A D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

cycles loops A B A B A B A D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 11 / 13

These are stochastic representations of Heisenberg models of quantum spins Ferromagnetic model: add weight 2 #cycles Antiferromagnetic model: add weight 2 #loops Goal: Show that cycle (loop) lengths are distributed according to Poisson-Dirichlet law Method: Introduce stochastic process such that (i) equilibrium measure is invariant; (ii) effective coagulation-fragmentation process; (iii) Poisson-Dirichlet is invariant measure for the latter Work in progress with C. Goldschmidt & P. Windridge Project 4. Analytical or numerical work in support of these ideas D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 12 / 13

Conclusion The goal is to better understand the distribution of cycle lengths in Random permutations with cycle weights D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 13 / 13

Conclusion The goal is to better understand the distribution of cycle lengths in Random permutations with cycle weights Random permutations with spatial structure D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 13 / 13

Conclusion The goal is to better understand the distribution of cycle lengths in Random permutations with cycle weights Random permutations with spatial structure Cycle and loop models D. Ueltschi (Warwick) Prob., Comb., Stat. Mech. MASDOC 15 Mar 2011 13 / 13

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