Hammersley's harness process: invariant distributions and height fluctuations

摘要

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