Scaling for a one-dimensional directed polymer with boundary conditions

Details Scaling for a one-dimensional directed polymer with boundary conditions Scaling for a one-dimensional directed polymer with boundary conditions

2009-11-12

arxiv.org/abs/0911.2446v2

Mathematics

Probability

Mistakes corrected, additional results and references, 46 pages

Scientific paper

We study a 1+1-dimensional directed polymer in a random environment on the integer lattice with log-gamma distributed weights and both endpoints of the path fixed. Among directed polymers this model is special in the same way as the last-passage percolation model with exponential or geometric weights is special among growth models, namely, both permit explicit calculations. With appropriate boundary conditions the polymer with log-gamma weights satisfies an analogue of Burke's theorem for queues. Building on this we prove that the fluctuation exponents for the free energy and the polymer path have their conjectured values. For the model without boundary conditions we get upper bounds on the exponents.

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