2024/12/20 by Coquille, Loren, Dario, Paul, Ny, Arnaud Le
#60K35 #82B20 #82B26 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)
paper · doi:10.48550/arxiv.2412.15782
The goal of this article is to study quantitatively the localisation/delocalisation properties of the discrete Gaussian chain with long-range interactions. Specifically, we consider the discrete Gaussian chain of length N, with Dirichlet boundary condition, range exponent α∈ (1 , ∞) and inverse temperature β∈ (0,∞), and show that: - For α∈ (2 ,3) and β∈ (0 , ∞), the fluctuations of the chain are at least of order N(1)/(2)(α- 2); - For α= 3 and β∈ (0 , ∞), the fluctuations of the chain are of order √(N / ln N) (sharp upper and lower bounds up to multiplicative constants are derived). Combined with the results of Kjaer-Hilhorst, Fröhlich-Zegarlinski and Garban, these estimates provide an (almost) complete picture for the localisation/delocalisation of the discrete Gaussian chain. The proofs are based on graph surgery techniques which have been recently developed by van Engelenburg-Lis and Aizenman-Harel-Peled-Shapiro to study the phase transitions of two dimensional integer-valued height functions (and of their dual spin systems). Additionally, by combining the previous strategy with a technique introduced by Sellke, we are able extend the method to study the q-SOS long-range chain with exponent q ∈ (0 , 2) and show that, for any inverse temperature β∈ (0, ∞) and any range exponent α∈ (1 , ∞): - The fluctuations of the chain are at least of order N(1)/(q)(α-2) \wedge (1)/(2); - The fluctuations of the chain are at most of order N( (1)/(q)α- 1 ) \wedge \frac 12.