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.

Free fermions at the edge of interacting systems

“…In that case the mode occupation follows the Fermi-Dirac distribution, ψ † (λ)ψ(µ) β = δ(λ−µ) 1+e −βµ , which leads to a generalization (see e.g. [13][14][15]) that interpolates between the Airy kernel (zero temperature, β → ∞) and the Gumbel kernel (infinite temperature, β → 0). In the following we stick to the Airy kernel (8), namely equal imaginary time and zero temperature.…”

The Free Boundary Schur Process and Applications I

“…A consequence of this is that the free boundary Schur process is neither determinantal nor pfaffian in general, but becomes pfaffian after we perform a certain random vertical shift of the point configuration, that translates in the point process language the "charge mixing" occurring in extended free boundary states. This phenomenon has some similarities with Borodin's shift-mixing for the periodic Schur process [Bor07], but the fermionic picture is rather different: as explained in [BB18], for periodic boundary conditions, Borodin's shift-mixing can be interpreted as the passage to the grand canonical ensemble, needed to apply Wick's theorem at finite temperature. In the case of a single free boundary, the shift goes away, and our approach yields a new derivation of the correlations functions of the pfaffian Schur process, alternative to that by Borodin and Rains [BR05] and the very recent one by Ghosal [Gho17] using Macdonald difference operators.…”

“…that is to say we move all points of S( λ) vertically by a same shift 2D t . Note that, in contrast with the periodic Schur process [Bor07,BB18], we have to shift the point configuration by an even integer. As we shall see, the origin of this shift in the free fermion formalism is rather different.…”

Peaks of cylindric plane partitions

“…When β → ∞ it becomes the usual Airy kernel (by the same argument as above, the Fermi factor in the integrand becoming 1 [0,∞) ) with Fredholm determinant on (s, ∞) the Tracy-Widom GUE distribution [22] (the "universal" asymptotic distribution of the largest eigenvalue of Hermitian random matrices with iid entries). A more subtle β → 0 limit recovers the Gumbel distribution-see [12,3].…”

“…It has been known for a while that cylindric boundary conditions for combinatorial objects correspond to finite temperature systems in the sense of quantum mechanics, with inverse temperature the same as the cylinder's circumference. An example to illustrate both the combinatorics and the mathematical physics is the cylindric Plancherel measure discussed in [3] (and introduced in [6]). The motivation for this paper stems from there: by studying simple distributions on cylindric plane partitions, we obtain interesting asymptotic behavior for their peaks that "crosses over" between a large (infinite) temperature regime governed by the Gumbel distribution (universally the asymptotic maximum of iid random variables that have Gaussian-like tails) and a small (zero) temperature regime governed by the Tracy-Widom [22] distribution (universally the asymptotic maximum of correlated spectra of Hermitian matrices with iid entries).…”