Hammersley's harness process: invariant distributions and height fluctuations

2015/04/27 by Timo Seppäläinen, Seppäläinen, Timo, Yun Zhai +1
Mathematics · #60F17 (Secondary) #60K35 (Primary) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1504.07139

openalex publication_date 2015/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the invariant distributions of Hammersley's serial harness process in all dimensions and height fluctuations in one dimension. Subject to mild moment assumptions there is essentially one unique invariant distribution, and all other invariant distributions are obtained by adding harmonic functions of the averaging kernel. We identify one Gaussian case where the invariant distribution is i.i.d. Height fluctuations in one dimension obey the stochastic heat equation with additive noise (Edwards-Wilkinson universality). We prove this for correlated initial data subject to polynomial decay of strong mixing coefficients, including process-level tightness in the Skorohod space of space-time trajectories.

Citations

• Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks

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