Abstract
Consider the massless free field on thed-dimensional lattice ℤd,d≧3; that is the centered Gaussian field on
with covariances given by the Green function of the simple random walk on ℤd. We show that the probability, that all the spins are positive in a box of volumeN d, decays exponentially at a rate of orderN d−2 logN and compute explicitly the corresponding constant in terms of the capacity of the unit cube. The result is extended to a class of transient random walks with transition functions in the domain of the normal and α-stable law.
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Communicated by J.L. Lebowitz
This research was partially supported by the foundation for promotion of research at the Technion.
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Bolthausen, E., Deuschel, JD. & Zeitouni, O. Entropic repulsion of the lattice free field. Commun.Math. Phys. 170, 417–443 (1995). https://doi.org/10.1007/BF02108336
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DOI: https://doi.org/10.1007/BF02108336