Tightness of discrete Gibbsian line ensembles with exponential interaction Hamiltonians

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Abstract

In this paper we introduce a framework to prove the tightness of a sequence of discrete Gibbsian line ensembles LN = {L1k(u),L2k(u),...}, a countable collection of random curves. The sequence of discrete line ensembles LN we consider enjoys a resampling invariance property, which we call (HN,HRW,N)-Gibbs property. We assume that LN satisfies technical Assumptions A1–A4 on (HN,HRW,N) and that the lowest labeled curve with a parabolic shift, LN1 (u)+ u2/2, converges weakly to a stationary process in the topology of uniform convergence on compact sets. Under these assumptions, we prove our main result Theorem 2.18 that LN is tight and the H-Brownian Gibbs property holds for all subsequential limit line ensembles with H(x)= ex. Together with the characterization result in Dimitrov (2021), this proves the convergence to the KPZ line ensemble. As an application of Theorem 2.18, under the weak noise scaling, we show that the scaled log-gamma line ensemble LN converge to the KPZ line ensemble.

Original languageEnglish (US)
Pages (from-to)2106-2150
Number of pages45
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume59
Issue number4
DOIs
StatePublished - 2023
Externally publishedYes

Keywords

  • Gibbs property
  • Line ensemble

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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