Entropic repulsion for massless fields.
Random interfaces described in terms of heights \(\{\varphi _x\in \mathbb R;\;x\in \mathbb Z^d\}\) that are zero outside of the sets \(D_N=ND\cap \mathbb Z^d\) for some connected bounded domains \(D\) and large natural numbers \(N\) are investigated. Gibbs probabilities \(P_N\) are given by the massless Hamiltonians of the form \(\sum _{\langle x,y\rangle} V(\varphi _x-\varphi _y)\) with sufficiently smooth convex even \(V\)'s. Various upper and lower bounds are derived for \(\log P_N(\Omega ^+(A_N))\), where \(\Omega ^+(A_N)=\{\varphi ;\;\varphi _x\geq 0\text{ for }x\in A_N\}\) for \(A\subset D\). As corollaries, bounds \[ \limsup _N \sup _{x\in D_N} P_N(\varphi _x>\sqrt {C(\log N)^i}\mid \Omega ^+(D_N))=0 \] and \[ \lim _N \sup _{x\in D_{\delta N}} P_N(\varphi _x<\sqrt {K(\log N)^i}\mid \Omega ^+(D_N))=0 \] describing the entropic repulsion effect are obtained, where \(i=2\) for \(d=2\) and \(i=1\) for \(d>2\).
scientific article; zbMATH DE number 2098357