Soft edge results for longest increasing paths on the planar lattice

For two-dimensional last-passage time models of weakly increasing paths, interesting scaling limits have been proved for points close the axis (the hard edge). For strictly increasing paths of Bernoulli($p$) marked sites, the relevant boundary is the line $y=px$. We call this the soft edge to contrast it with the hard edge. We prove laws of large numbers for the maximal cardinality of a strictly increasing path in the rectangle [\fl{p^{-1}n -xn^a}]\times[n] as the parameters $a$ and $x$ vary. The results change qualitatively as $a$ passes through the value 1/2.Comment: 14 pages, 2 figure

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

Order of the variance in the discrete Hammersley process with boundaries

We discuss the order of the variance on a lattice analogue of the Hammersley process with boundaries, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope. We show that along characteristic directions the order of the variance of the last passage time is of order N^{2/3} in the model with boundary. These characteristic directions are restricted in a cone starting at the origin, and along any direction outside the cone, the order of the variance changes to O(N) in the boundary model and to O(1) for the non-boundary model. This behaviour is the result of the two flat edges of the shape function

Ratios of partition functions for the log-gamma polymer

We introduce a random walk in random environment associated to an underlying directed polymer model in 1 + 1 dimensions. This walk is the positive temperature counterpart of the competition in- terface of percolation and arises as the limit of quenched polymer measures. We prove this limit for the exactly solvable log-gamma polymer, as a consequence of almost sure limits of ratios of parti- tion functions. These limits of ratios give the Busemann functions of the log-gamma polymer, and furnish centered cocycles that solve a variational formula for the limiting free energy. Limits of ratios of point-to-point and point-to-line partition functions manifest a duality between tilt and velocity that comes from quenched large deviations under polymer measures. In the log-gamma case, we identify a fam- ily of ergodic invariant distributions for the random walk in random environment

The dimension of the range of a transient random walk

We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991).Comment: 37 pages, 5 figure