## Localization of a two-dimensional random walk with an attractive path interaction.(English)Zbl 0819.60028

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.

### MSC:

 60F10 Large deviations 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J25 Continuous-time Markov processes on general state spaces

### Keywords:

large deviation; asymptotic behavior; random walk
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