Localization of a two-dimensional random walk with an attractive path interaction
Let \((X_ t)_{t \geq 0}\) be the time-continuous, symmetric random walk on \(\mathbb{Z}^ d\), starting from \(X_ 0 = 0\) with generator \[ Af(x) = {1 \over 2} \sum_{y : | y - x | = 1} \bigl( f(y) - f(x) \bigr). \] Let \(S_ T\) be the set of points in \(\mathbb{Z}^ d\) visited by \(X_ t\) up to time \(T > 0\) and set \(N_ T = \) the cardinality of the set \(S_ T\). Define a new distribution \(d \widehat P_ T\) by \(d \widehat P_ T = \exp [- N_ T] dP/E \exp [- N_ T]\). For \(x \in \mathbb{Z}^ d\) and \(r > 0\), put \(B_ x(r) = \{y \in \mathbb{Z}^ d : [y - x | \leq r\}\). The main result is a description of the shape of \(S_ T\) in the two-dimensional case. That is, when \(d = 2\), for any \(\varepsilon > 0\), one has \[ \lim_{T \to \infty} \widehat P_ T \left(\bigcup_{x \in B_ 0 (\rho T^{1/4})} \biggl \{B_ x \bigl( \rho (1 - \varepsilon) T^{1/4} \bigr) \subset S_ T \subset B_ x \bigl( \rho (1 + \varepsilon) T^{1/4} \bigr) \biggr \} \right) = 1, \] where \(\rho = (\lambda/ \pi)^{1/4}\) and \(\lambda\) is the principal eigenvalue of \(-\Delta/2\) in the ball of radius 1. The proof of the result requires a refinement of the analysis of Donsker and Varadhan. Partial discussion is also given to higher dimensions.
scientific article; zbMATH DE number 681387