A Limit Formula for a Class of Gibbs Measures with Long Range Pair Interactions

Let $X_i,\ i=1,2,\dots$ be real-valued {\sl i.i.d.} variables with a compactly supported density. Under certain assumptions on $V,$ we give an asymptotic evaluation of E[\exp(-\half\sum_{i,j=1}^n V(X_i,X_j))] up to the factor $(1+o(1)).$ As an application of this result, we prove a limit formula for a class of Gibbs measures with long range pair interactions

Large Deviation Principle for the Pinned Motion of Random Walks

The large deviation principle is proved for the long time asymptotic of empirical measures associated with the pinned motions of random walks on the square lattice. Random walks are not reversible Markov chains in general, and thus nice property such as the Gaussian bounds on the transition probabilities, which was one of the key tools for proving the large deviations for periodic and reversible Markov chains in [1], are no longer available. For this reason the spectral radius of transition probabilities of random walk comes into play. With the help of Salvatori’s theorem, a sufficient condition is given so that the spectral radius is held to be equal to 1 by certain gauge transform of the transition probabilities, and then the large deviation will be proved under the condition