Harness processes and harmonic crystals

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Abstract

In the Hammersley harness processes the R-valued height at each site iZd is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Zd. With this representation we compute covariances and show L2 and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L2 at speed t1d/2 (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.

MSC

60K35
82B
82C

Keywords

Harness process
Linear Gaussian processes
Surface dynamics

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