Abstract
We consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero. Our proofs rely on concentration of measure techniques and some combinatorial bounds on families of paths.
Highlights
Introduction and main resultsA number of well-known probabilistic models derive their underlying complexity from a variant of the following simple setup
For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero
Our proofs rely on concentration of measure techniques and some combinatorial bounds on families of paths
Summary
A number of well-known probabilistic models derive their underlying complexity from a variant of the following simple setup. For example in the corner growth model, or directed nearest neighbor last-passage percolation on the 2d lattice, the fundamental issue is to understand the distribution of the maximum-energy path to a given point. This maximal energy represents the last passage time to the point, or time at which it joins the growing corner shape. For almost every environment, the energy of an up-right path selected uniformly at random is asymptotically normally distributed as the mesh gets small This is the content of our main theorem, which is proved for generally distributed weights with nonzero variance and under further moment assumptions (in our setting the weights need not even be identically distributed). We introduce some notation, describe our results, and discuss connections to the corner growth and directed random polymer models
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