Multicritical Schur measures and higher-order analogues of the Tracy-Widom distribution

Abstract

We introduce multicritical Schur measures, which are probability laws on integer partitions which give rise to non-generic fluctuations at their edge. They are in the same universality classes as one-dimensional momentum-space models of free fermions in flat confining potentials, studied by Le Doussal, Majumdar and Schehr. These universality classes involve critical exponents of the form 1/(2m+1), with m a positive integer, and asymptotic distributions given by Fredholm determinants constructed from higher order Airy kernels, extending the generic Tracy-Widom GUE distribution recovered for m=1. We also compute limit shapes for the multicritical Schur measures, discuss the finite temperature setting, and exhibit an exact mapping to the multicritical unitary matrix models previously encountered by Periwal and Shevitz.

Figures (8)

• Figure 1: Partitions sampled according to certain multicritical measures, namely the minimal measures $\mathbb{P}^{\mathrm{a},m}_\theta$ introduced in Section \ref{['sec:explicit']} for $m=2,3,4$ and $\theta=20$ (sampling is done via a Metropolis--Hastings algorithm). Their Young diagrams are drawn in the Russian convention. The profile of each partition is the piecewise linear function shown in black; for comparison we display in blue the (appropriately rescaled) limit shape obtained for $\theta \to \infty$. The fermionic state indexed by each partition, as detailed in Section \ref{['sec:lattice']}, is illustrated by the dots below the diagram: these indicate filled sites, corresponding to points where the profile has slope $-1$.
• Figure 2: Limit curves $\Omega^{\mathrm{a},m}$ for partitions under the minimal multicritical measures $\mathbb{P}^{\mathrm{a},m}_\theta$ as $\theta \to \infty$ (see Corollary \ref{['corr:asymmmin']}). The limiting densities $\varrho^{\mathrm{a},m}$ in the corresponding fermion models (discussed in Section \ref{['sec:fermions-asymptotics']}) are shown below; they are related to the limit curves by $\Omega'(x) = 1 - 2\varrho(x)$.
• Figure 3: Limit curves $\Omega^{\mathrm{s},m}$ of partitions under the minimal multicritical measures $\mathbb{P}^{\mathrm{s},m}_\theta$ as $\theta \to \infty$ (see Corollary \ref{['corr:symminls']}), and corresponding limiting fermion densities $\varrho^{\mathrm{s},m}$. Note the symmetry under $x \mapsto -x$.
• Figure 4: Saddle points (shown as black dots) of the action $S(z;x)$ for the $m=2$ minimal measure $\mathbb{P}^{\mathrm{a},2}_\theta$, at $x = 1.6$ in the empty region (top left), at $x = 1$ in the bulk (top right) and at $x = -2.8$ in the frozen region (bottom). The contours $c_+, c_+'$ (shown in blue) pass through regions where $\mathrm{Re}(S(z;x))<0$, whereas the contours $c_-,c_-'$ (shown in red) pass through regions where $\mathrm{Re}(S(z;x))>0$.
• Figure 5: Saddle points of the action and contours for the $m=3$ symmetric minimal measure $\mathbb{P}^{\mathrm{s},m}_\theta$, at $x=2$ in the empty region (left), at $x=0$ in the bulk (centre) and at $x=-2$ in the frozen region (right). Here, exchanging $x$ with $-x$ reflects the saddle points about the imaginary axis.

Theorems & Definitions (33)

• Definition 0: Multicritical measures
• Theorem 1: Asymptotic edge fluctuations of multicritical measures
• Theorem 2: Limit shapes of multicritical measures
• Proposition 3: Conjugate partition under a Schur measure
• Definition 4: Minimal multicritical measures
• Corollary 5 of Theorem \ref{thm:mclimitshape}: Limit shapes of minimal multicritical measures
• Definition 5 of Theorem \ref{thm:mclimitshape}: Symmetric minimal multicritical measures
• Corollary 5 of Theorem \ref{thm:mclimitshape} of Theorem \ref{thm:mclimitshape}: Limit shapes of symmetric minimal multicritical measures
• Theorem 5 of Theorem \ref{thm:mclimitshape} of Theorem \ref{thm:mclimitshape}: Edge distributions under Schur measures and unitary matrix integrals
• Definition 5 of Theorem \ref{thm:mclimitshape} of Theorem \ref{thm:mclimitshape}: Multicritical unitary matrix models