The periodic Schur process and free fermions at finite temperature

Abstract

We revisit the periodic Schur process introduced by Borodin in 2007. Our contribution is threefold. First, we provide a new simpler derivation of its correlation functions via the free fermion formalism. In particular, we shall see that the process becomes determinantal by passing to the grand canonical ensemble, which gives a physical explanation to Borodin's "shift-mixing" trick. Second, we consider the edge scaling limit in the simplest nontrivial case, corresponding to a deformation of the poissonized Plancherel measure on partitions. We show that the edge behavior is described, in a certain crossover regime different from that for the bulk, by the universal finite-temperature Airy kernel, which was previously encountered by Johansson and Le Doussal et al. in other models, and whose extreme value statistics interpolates between the Tracy-Widom GUE and the Gumbel distributions. We also define and prove convergence for a stationary extension of our model. Finally, we compute the correlation functions for a variant of the periodic Schur process involving strict partitions, Schur's P and Q functions, and neutral fermions.

36 pages, 3 figures. v2: final version with several additions: two figures in introduction; some previously missing references; results on the edge behavior of the cylindric Plancherel measure outside the crossover regime, in particular in the nontrivial high temperature regime (see new Theorem 1.2, Proposition 5.2 and Appendix D)

References in corpus (4)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Shifted Schur process and asymptotics of large random strict plane partitions
• Enumeration of Cylindric Plane Partitions - Part II

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• Skew RSK dynamics: Greene invariants, affine crystals and applications to $q$-Whittaker polynomials
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