Random walk representations and entropic repulsion for gradient models (Q2759734)

This is a survey of the mathematical work on the effect of entropic repulsion for gradient interface models. Proofs are (at some length) sketched only, but ideas are made transparent, and heuristics are explained informally. The model is as follows. For a finite set \(A\subset \mathbb Z^d\), define a probability measure \(P_A^U\) on \(\mathbb R^A\) by putting NEWLINE\[NEWLINEP_A^U(d\varphi)=\frac 1{Z_A}\exp \biggl\{-\frac 12 \sum_{i,j\in \overline A: i\sim j}U(\varphi_i-\varphi_j)\biggr\} \prod_{i\in A}d\varphi_i,NEWLINE\]NEWLINE where \(U: \mathbb R\to[0,\infty)\) is symmetric and continuous and not too small at \(\infty\). Here \(\overline A\) is the union of \(A\) and \(\partial A\) is the outer boundary of \(A\). The best studied and best understood special case is the case \(U(x)=x^2\) of a harmonic crystal. In this case, there is a certain random walk representation for the covariances. This representation has been extended by \textit{B. Helffer} and \textit{J. Sjöstrand} [J. Stat. Phys. 74, No. 1/2, 349-409 (1994)] to the case of a uniformly convex \(U\). Section 1 outlines the construction of the underlying random walk. NEWLINENEWLINENEWLINEThe main question is the behavior of the random field \(\varphi\) under the conditional measure \(P_A^U(\cdot\mid\Omega_A^+)\) as \(A\uparrow\mathbb Z^d\), where \(\Omega_A^+=\{\varphi:\varphi_i \geq 0, \forall i\in A\}\). This is a model for a random interface, which is put on one side of a hard wall, which in this (simple) case is just the configuration identical to zero on \(A\). The philosophy is that the combined effect of local fluctuations and global stiffness (coming from long-range correlations under \(P_A^U\)) push the interface away from the wall. The ideas and some background are discussed at the beginning of Section 2. In dimensions \(d\geq 3\), the measure \(P_\infty^U= \lim_{A\uparrow\mathbb Z^d}P_A^U\) may be defined without difficulties. In Section 2, the asymptotics of \(P_\infty^U(\Omega_{\Lambda_n}^+)\) as \(n\to\infty\) are formulated and explained (where \(\Lambda_n\) is the box \([-n,n)^d\cap \mathbb Z^d\)) for the case of a harmonic crystal. Section 3 handles the two-dimensional case, more precisely the asymptotics of \(P_{\Lambda_n}^U(\Omega_{D_n}^+)\) as \(n\to\infty\), for \(D_n=nD\cap\mathbb Z^2\) and any set \(D\subset [-1,1]^2\) having a nice boundary. Finally, in Section 4, the effects of pinning and wetting transition (which appear if a local attractive force is added) are discussed on an informal level.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00044].