Asymptotics of Schur and uniform measures on partitions for general specializations: universality classes, integrable kernels, and error terms

Okounkov's infinite-wedge (free-fermion) formalism expresses the correlation functions of Schur measures on integer partitions, and of the uniform measure, as determinants of an explicit kernel whose generating-function (double-contour) form is governed by a ratio $J(z)=\gamma(z,t)/\gamma(z^{-1},t')$ of two entire functions encoding the two arbitrary specializations. At the time of that work, sharp asymptotics were available only for special families (Poissonized Plancherel, z-measures), and four development directions were left open: asymptotic analysis for general specializations; classification and Riemann-Hilbert treatment of the integrable class where $\partial_z\log J$ is rational; extension of the wedge construction to actions of algebras beyond the Heisenberg algebra; and quantitative refinements of the uniform-measure product formula for local statistics. Later literature resolved substantial parts for special, tuned, symmetric, or dual families — including a complete limit-shape/bulk/edge/cusp/corner theory for the dual (skew-Howe) Schur measure with piecewise-smooth specializations, rigorous multicritical measures, and formal general-specialization edge and cusp analyses that assume symmetric specializations — while the algebraic direction grew into the integrable-probability program (Schur processes, Macdonald processes). The surviving open core is a complete rigorous asymptotic classification for the ordinary two-sided Schur measure with arbitrary independent specializations — phase diagram, degenerate and multicritical transition regimes, and error control — together with quantified convergence rates for the uniform-measure product formula.

Background

A partition $\lambda=(\lambda_1\ge\lambda_2\ge\dots\ge0)$ of an integer is a nonincreasing sequence of nonnegative integers with a fixed sum; partitions label the irreducible representations of the symmetric groups and carry many natural probability measures. The central object of the source paper is the Schur measure: the probability measure on partitions with weight proportional to $s_\lambda(x)\,s_\lambda(y)$, where $s_\lambda$ is the Schur function — the standard basis element of the ring of symmetric functions indexed by $\lambda$ — evaluated at two (finite or infinite) auxiliary alphabets $x$ and $y$. By the Cauchy identity $\sum_\lambda s_\lambda(x)s_\lambda(y)=\prod_{i,j}(1-x_iy_j)^{-1}$ the total mass is explicit; choosing the alphabets, equivalently choosing the power-sum (Miwa) variables $t_k=\frac{1}{k}\sum_i x_i^k$ and $t'_k=\frac{1}{k}\sum_i y_i^k$, is called a specialization. Two special choices recover classical objects: the Poissonized Plancherel measure on partitions (which governs longest increasing subsequences of random permutations) and the z-measures, a two-parameter family (parameters $z$ and $z'$) of measures on partitions arising in the harmonic analysis of the infinite symmetric group $S(\infty)$.

The paper's machinery is the infinite wedge (semi-infinite wedge) representation: a Fock-type space built on the vector space spanned by charge-zero Maya diagrams, which are in bijection with partitions, on which charged free fermions $\psi_k,\psi^*_k$ act. The fermions generate a Heisenberg algebra with modes $\alpha_n$, and the Schur-measure partition function is a tau-function of the Toda lattice hierarchy — a classical completely integrable system of differential equations in the Miwa variables. The probabilistic payoff is that the resulting measures are determinantal point processes: for a random partition, the correlation functions of a particle configuration $X$ (a finite set of modified Frobenius coordinates, i.e. shifted coordinates encoding the boundary profile of the Young diagram of $\lambda$) equal $\det[K(x_i,x_j)]_{x_i,x_j\in X}$, where the correlation kernel has the explicit double-contour (equivalently, generating-function) representation

$$K(z,w)=\frac{\sqrt{zw}}{z-w}\cdot\frac{J(z)}{J(w)},\qquad J(z)=\frac{\gamma(z,t)}{\gamma(z^{-1},t')},\qquad \gamma(z,t)=\exp\Big(\sum_{n\ge1}t_nz^n\Big),$$

with both specializations entering only through $J$. Asymptotic analysis of such kernels, as the specializations grow in a specified scaling regime, yields the local statistics of a random partition around its limit shape — the deterministic boundary curve to which a uniformly random Young diagram converges, in the sense of Vershik. In the known examples the bulk limit is governed by the discrete sine kernel, edges by the Airy kernel and the Tracy-Widom distribution (the universal edge law of random matrix theory), and cusps of the limit shape by the Pearcey kernel. The paper also identifies an integrability structure: the kernel $K$ is called integrable when $(z-w)^{-1}(z\partial_z-w\partial_w)\log(J(z)J(w))$ is a finite sum $\sum_{i=1}^N f_i(z)g_i(w)$ (equivalently, the discrete correlation kernel satisfies that $(i-j)K(i,j)$ has finite rank, i.e. the generating function $\sum_{i,j}z^iw^{-j}(i-j)K(i,j)$ is a finite sum of products $\sum_{i=1}^N f_i(z)g_i(w)$) — the class amenable to Riemann-Hilbert steepest-descent methods — and the source gives as a sufficient condition for this that $\partial_z\log J(z)$ is a rational function, which in turn holds when $\gamma$ is a finite product of powers of linear factors; the Plancherel and z-measure specializations satisfy it.

For the second central object, the uniform measure $M_q(\lambda)\propto q^{|\lambda|}$ on partitions (in the limit $q\to1$ each partition of $n$ is weighted equally), the paper combines the Bloch-Okounkov n-point functions — determinantal formulas involving genus-one theta functions, obtained from the character of the infinite wedge representation — with limit-shape asymptotics to prove, for the correlation functions of the modified Frobenius coordinates $\xi$ of a typical partition, the product formula

$$\lim_{q\to1}\rho(X,q)=\prod_{\xi\in X}\frac{1}{1+e^{\pi|\xi|/\sqrt{6}}},$$

expressing that, locally, a typical partition looks like the trajectory of a random walk. At the source's publication, sharp asymptotics had been completed only for Plancherel-type and z-measure specializations (by Baik-Deift-Johansson, Borodin-Okounkov-Olshanski, and Borodin-Olshanski); the paper formulates the general framework, states that its kernel representation is particularly suitable for asymptotic investigations, and leaves the general theory, the integrable classification, the algebraic extensions, and the uniform-measure error terms undeveloped.

Problem Statement

Overarching objective: carry out the asymptotic and integrable analysis of the determinantal measures on partitions produced by the infinite-wedge formalism for general parameter values, in place of the special cases available when the framework was introduced. The requested results, phrased through the source's own structures, are:

(1) For the Schur measure with arbitrary (independent, not necessarily symmetric) specializations encoded by $J(z)=\gamma(z,t)/\gamma(z^{-1},t')$: determine the asymptotics of the correlation kernel

$$K(z,w)=\frac{\sqrt{zw}}{z-w}\cdot\frac{J(z)}{J(w)}$$

This component comprises the following independently standable sub-claims:

• (1a) the bulk limit: determine the asymptotics of $K$ in the interior of the limit shape and identify the limiting kernel and its universality class;
• (1b) the edge limit: determine the asymptotics of $K$ at smooth boundary points and identify the limiting kernel and its universality class;
• (1c) the transition limits: determine the asymptotics of $K$ in the cusp, corner, and touching regimes, including degenerate and higher-order cases, and identify the limiting kernels and their universality classes;
• (1d) the phase diagram in specialization space and the universality classes: the partition of specialization space according to which of the limiting kernels of (1a)-(1c) occurs;
• (1e) rigorous proofs of (1a)-(1d) with explicit error control.

(2) Within (1): characterize the specializations for which $\partial_z\log J(z)$ is rational (the source's sufficient condition for integrability in the finite-rank sense, satisfied in particular when $\gamma$ is a finite product of powers of linear factors), and develop Riemann-Hilbert / finite-rank operator techniques for this integrable subclass. This component is an ingredient of (1): its classification characterizes the integrable sub-class of the specializations in (1), and the Riemann-Hilbert / finite-rank techniques are the method for the corresponding analysis.

(3) For the uniform measure $M_q(\lambda)\propto q^{|\lambda|}$ in the limit $q\to1$: quantify the local product formula for correlation functions beyond the leading term — derive the rate of convergence, i.e. the error term in formula (3.15) of the source under the scaling $x_i/\sqrt N\to\xi_i$ of its Theorem 7, where $N$ is the expectation of $|\lambda|$ (the source remarks the error term can be found by the Laplace method but does not derive it) — making precise the statement that a typical partition locally looks like a random-walk trajectory.

(4) Extend the infinite-wedge / Toda identification to actions of algebras other than the Heisenberg algebra on the wedge space (beyond the $\mathrm{sl}_2$ action, which the source shows recovers the z-measures), producing new tractable measures on partitions with explicit kernels. This component is an open-ended extension direction.

Resolution boundary: the task is resolved only by completing the full analytic program of (1) and of (3) for general independent specializations, with (2) supplying the classification that organizes the integrable sub-class of (1). A result covering only a proper subset of the sub-claims (1a)-(1e) of (1), or only special, tuned, symmetric, or dual families of specializations, does not by itself resolve (1).

Components (1)-(3) jointly constitute the analytic program on the two measures the source studies; (4) is the representation-theoretic extension direction. Later literature has substantially developed each component for special, tuned, symmetric, or dual families; the surviving source-faithful core is the completion of (1) for general two-sided specializations including degenerate and transition regimes with error control, of the classification in (2) as the organizing principle for the integrable sub-class of (1), and of the quantitative refinements in (3). Component (4) retains the open-ended character of the source's suggestion, while (1)-(3) form the analytic core of the source's program.

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