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Probability to be positive for the membrane model in dimensions 2 and 3. (English) Zbl 07088985
Summary: We consider the membrane model on a box \(V_{N}\subset \mathbb{Z} ^{n}\) of size \((2N+1)^{n}\) with zero boundary condition in the subcritical dimensions \(n=2\) and \(n=3\). We show optimal estimates for the probability that the field is positive in a subset \(D_{N}\) of \(V_{N}\). In particular we obtain for \(D_{N}=V_{N}\) that the probability to be positive on the entire domain is exponentially small and the rate is of the order of the surface area \(N^{n-1}\).
MSC:
60G15 Gaussian processes
60G60 Random fields
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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