Timo SEPPÄLÄINEN (University of Wisconsin-Madison)
Features of the log-gamma polymer model: limiting random walk in random environment, stationary polymer, fluctuation exponents
 

After a brief introduction to the general d+1 dimensional lattice model of directed polymer in a random environment, we study properties of the explicitly solvable 1+1 dimensional log-gamma polymer model. This model has an explicit stationary version that can be used to derive various properties. We show that ratios of point-to-point and point-to-line partition functions converge to gamma-distributed limits. One consequence of this is that the quenched polymer measure converges to a random walk in a correlated random environment. This RWRE can then be thought of as an infinitely long polymer, and it is also a positive temperature analogue of the competition interface of last-passage percolation. Time permitting, we may also prove some KPZ fluctuation exponents for the stationary log-gamma polymer.