Abstract
We study the partition functions associated with non-intersecting polymers in a random environment. By considering paths in series and in parallel, the partition functions carry natural notions of subadditivity, allowing the effective study of their asymptotics. For a certain choice of random environment, the geometric RSK correspondence provides an explicit representation of the partition functions in terms of a stochastic interface. Formally this leads to a variational description of the macroscopic behaviour of the interface and hence the free energy of the associated non-intersecting polymer model. At zero temperature we relate this variational description to the Marčenko–Pastur distribution, and give a new derivation of the surface tension of the bead model.
Highlights
Introduction and SummaryWe study the partition functions associated with a natural model for non-intersecting polymers in a random environment
We turn our discussion to the random polymer model with log-gamma weights, where the geometric RSK correspondence provides an explicit representation of the partition functions in terms of a stochastic interface [14,19]
In the second part we introduce a collection of random functions on the N × N square known as stochastic interfaces, and express the non-intersecting partition functions associated with a particular random environment in terms of a stochastic interface, developing connections with random matrix theory in the process
Summary
We study the partition functions associated with a natural model for non-intersecting polymers in a random environment. By considering paths in series and in parallel, the partition functions carry natural notions of subadditivity allowing the effective study of their asymptotics We use this subadditivity to show that for a small number of paths, the free energy has a linear dependence on the number of paths (Theorem 2.2). We turn our discussion to the random polymer model with log-gamma weights, where the geometric RSK correspondence provides an explicit representation of the partition functions in terms of a stochastic interface [14,19]. This leads to a variational description of the macroscopic behaviour of the interface and the free energy of the associated non-intersecting polymer model.
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