Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -- II

2023/05/09 by Clément Cosco, Cosco, Clément, Ofer Zeitouni +1 · 2 citations
Mathematics · #60G50 #60H15 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Primary 82B44 secondary 82D60 #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2305.05758

openalex publication_date 2023/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let W_N(β) = E₀[e^ ∑_n=1^N βω(n,S_n) - Nβ²/2] be the partition function of a two-dimensional directed polymer in a random environment, where ω(i,x), i∈ ℕ, x∈ ℤ² are i.i.d. standard normal and \S_n\ is the path of a random walk. With β=β_N=\widehatβ √(π/log N) and \widehatβ∈ (0,1) (the subcritical window), log W_N(β_N) is known to converge in distribution to a Gaussian law of mean -λ²/2 and variance λ², with λ²=log ((1-\widehatβ²)^-1) (Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper the moments \mathbb E [W_N( β_N)^q] in the subcritical window, and prove a lower bound that matches for q=O(√(log N)) the upper bound derived by us in Cosco, Zeitouni, arXiv:2112.03767 [math.PR]. The analysis is based on appropriate decouplings and a Poisson convergence that uses the method of ''two moments suffice''.

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