Large deviation rate functions for the partition function in a log-gamma distributed random potential

We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the $1+1$-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions.Comment: Published in at http://dx.doi.org/10.1214/12-AOP768 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Current fluctuations of a system of one-dimensional random walks in random environment

We study the current of particles that move independently in a common static random environment on the one-dimensional integer lattice. A two-level fluctuation picture appears. On the central limit scale the quenched mean of the current process converges to a Brownian motion. On a smaller scale the current process centered at its quenched mean converges to a mixture of Gaussian processes. These Gaussian processes are similar to those arising from classical random walks, but the environment makes itself felt through an additional Brownian random shift in the spatial argument of the limiting current process.Comment: Published in at http://dx.doi.org/10.1214/10-AOP537 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction

We consider a ballistic random walk in an i.i.d. random environment that does not allow retreating in a certain fixed direction. We prove an invariance principle (functional central limit theorem) under almost every fixed environment. The assumptions are nonnestling, at least two spatial dimensions, and a $2+\epsilon$ moment for the step of the walk uniformly in the environment. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.Comment: Published at http://dx.doi.org/10.1214/009117906000000610 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quenched Point-to-Point Free Energy for Random Walks in Random Potentials

We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and some steps of the walk. The potential can be unbounded, but it is subject to a moment assumption whose strictness is tied to the mixing of the environment, the best case being the i.i.d. environment. We prove that the infinite volume quenched point-to-point free energy exists and has a variational formula in terms of an entropy. We establish regularity properties of the point-to-point free energy, as a function of the potential and as a function on the convex hull of the admissible steps of the walk, and link it to the infinite volume free energy and quenched large deviations of the endpoint of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function.Comment: 39 pages, 3 figures, minor typos fixe