The probability that the Airy point process - the determinantal point process governing soft-edge fluctuations in random matrix theory, in growth models of the Kardar-Parisi-Zhang (KPZ) universality class, and in related integrable systems - has no points on a union of g large intervals located in the bulk admits a conjectured large-r expansion combining an explicit cubic leading term, a −3g8log⁡r-\frac{3g}{8}\log r correction, and oscillations of order one described by the Riemann theta function of a genus-g hyperelliptic surface (Conjecture 1.1 of Blackstone, Charlier and Lenells, arXiv:2012.01299, published in Random Matrices: Theory and Applications). Only the genus-one case is proved, together with the g=1 classification of admissible endpoint configurations and a numerical verification for g=1,2,3. Subsequent work has proved the Bessel (hard-edge) analogue for all g and the sine-process analogue for two intervals, as well as the Airy bulk-plus-edge configuration with explicit constant, but the genus-g bulk Airy conjecture for g>=2 remains unresolved; its degenerate commensurate-phase regimes and the evaluation of the r-independent constant C(x⃗)C(\vec{x}) are discussed in the source only as remarks and expectations and are likewise open. The problem is to settle Conjecture 1.1 for arbitrary genus g>=2, including its all-negative admissibility clause.

Background

The Airy point process is a determinantal point process on the real line - a random point configuration whose correlation functions are given by minors of a kernel, here the Airy kernel KAi(u,v)=Ai(u)Ai′(v)−Ai′(u)Ai(v)u−vK^{\mathrm{Ai}}(u,v)=\frac{\mathrm{Ai}(u)\mathrm{Ai}'(v)-\mathrm{Ai}'(u)\mathrm{Ai}(v)}{u-v}, where Ai\mathrm{Ai} is the Airy function. It is a universal object: it describes the largest eigenvalues at the soft spectral edge of large Hermitian random matrices - the edge regime where the limiting eigenvalue density vanishes continuously to zero, in contrast to a hard edge where the spectrum is cut off by a wall (the Tracy-Widom law for the largest eigenvalue) - the edge fluctuations of growth models in the Kardar-Parisi-Zhang (KPZ) universality class of stochastic interface growth, the largest parts of random partitions under the Plancherel measure, and the edge statistics of non-intersecting Brownian paths. It has almost surely infinitely many points, a largest point, and no accumulation points.

A gap probability is the probability that a given region is free of points. For a determinantal point process it is a Fredholm determinant: for a finite union of intervals I\mathcal{I}, the probability of finding no points in I\mathcal{I} equals F(I)=det⁡(I−KAiχI)F(\mathcal{I})=\det(I-\mathcal{K}^{\mathrm{Ai}}\chi_{\mathcal{I}}), the determinant of the Airy-kernel integral operator restricted to I\mathcal{I}. One distinguishes two configurations: Ig(e)=(x2g,x2g−1)∪⋯∪(x2,x1)∪(x~0,+∞)\mathcal{I}_g^{(e)}=(x_{2g},x_{2g-1})\cup\cdots\cup(x_2,x_1)\cup(\tilde x_0,+\infty), which includes a gap at the edge (a neighborhood of the rightmost point), and Ig(b)=(x2g,x2g−1)∪⋯∪(x2,x1)\mathcal{I}_g^{(b)}=(x_{2g},x_{2g-1})\cup\cdots\cup(x_2,x_1) with x1<0x_1<0, a union of gg bounded intervals lying in the bulk, i.e. deep inside the infinite point process away from its edge. The single edge-interval case g=0g=0 is the Tracy-Widom distribution, which is expressed through a Painlevé II equation (a second-order nonlinear ODE whose Hastings-McLeod solution appears in the representation); more generally FF can be expressed via systems of coupled Painlevé II equations (Claeys-Doeraene; Xu-Dai). Large gap asymptotics - the behavior of F(rI)F(r\mathcal{I}) as the scale r→+∞r\to+\infty - is a classical and much harder problem than small-gap asymptotics: the Tracy-Widom and one-interval edge cases were settled by Baik-Buckingham-DiFranco and Deift-Its-Krasovsky, and the Airy determinant on one bulk interval together with the edge was computed by Blackstone-Charlier-Lenells and by Krasovsky-Maroudas.

For the sine process (the bulk process of random matrices) the multi-interval problem has a long history: Widom obtained the leading term of the determinant on a union of intervals, and Deift-Its-Zhou showed that the oscillations of order one are described by Riemann theta functions attached to a hyperelliptic Riemann surface of genus equal to the number of intervals, with error terms controlled by Diophantine properties of certain phase integrals; Fahs-Krasovsky later settled the two-interval sine case completely, including the multiplicative constant. The source paper (Blackstone-Charlier-Lenells, 2020/2022) formulated the analogous conjecture for the Airy point process in the bulk and proved it for a single interval.

The conjecture is formulated as follows. Let g≥1g\ge1 and x⃗=(x1,…,x2g)\vec{x}=(x_1,\ldots,x_{2g}) with x2g<⋯<x2<x1x_{2g}<\cdots<x_2<x_1. Define R(z)=∏j=02gz−xj\sqrt{\mathcal{R}(z)}=\prod_{j=0}^{2g}\sqrt{z-x_j} (principal branches), where x0=x0(x⃗)x_0=x_0(\vec{x}) is an auxiliary point determined, together with coefficients (q0,…,qg−1)(q_0,\ldots,q_{g-1}), by the system (1.3) of the paper: gg linear moment equations ∑kmikqk−1=m~i\sum_k m_{ik}q_{k-1}=\tilde m_i involving integrals of sj−1/R(s)s^{j-1}/\sqrt{\mathcal{R}(s)} over the gap intervals, the algebraic condition ∑j=0g+1qjx0j=0\sum_{j=0}^{g+1}q_j x_0^j=0 (with qg+1=−1q_{g+1}=-1 and qg=12∑j=02gxjq_g=\frac12\sum_{j=0}^{2g}x_j), and the requirement x0>max⁡{x1,0}x_0>\max\{x_1,0\}. The admissible set A\mathcal{A} consists of those x⃗=(x1,…,x2g)\vec{x}=(x_1,\ldots,x_{2g}) with x2g<⋯<x2<x1x_{2g}<\cdots<x_2<x_1 for which this system admits a real solution. The two-sheeted Riemann surface XX of R(z)\sqrt{\mathcal{R}(z)} - the hyperelliptic surface w2=R(z)w^2=\mathcal{R}(z) with gg handles (its genus equals the number of intervals) - carries canonical AA- and BB-cycles, normalized holomorphic differentials ωk\omega_k, a period matrix τ∈iRg×g\tau\in i\mathbb{R}^{g\times g}, and the associated Riemann theta function θ(z⃗;τ)=∑n⃗∈Zgeiπ(n⃗tτn⃗+2n⃗tz⃗)\theta(\vec z;\tau)=\sum_{\vec n\in\mathbb{Z}^g}e^{i\pi(\vec n^t\tau\vec n+2\vec n^t\vec z)}. The phases Ωj=2i∫x2j+1x2j∑kqkξkR(ξ)+dξ>0\Omega_j=2i\int_{x_{2j+1}}^{x_{2j}}\frac{\sum_k q_k\xi^k}{\sqrt{\mathcal{R}(\xi)}_+}d\xi>0 define νj=−Ωj2πr3/2\nu_j=-\frac{\Omega_j}{2\pi}r^{3/2}. Under a Diophantine-type condition (1.8) - a polynomial-in-integers lower bound on nonzero integer combinations ∑jmjΩj−1\sum_j m_j\Omega_{j-1} - and the rational independence of Ω0,…,Ωg−1\Omega_0,\ldots,\Omega_{g-1} (condition (1.9)), the conjecture states that, as r→+∞r\to+\infty, F(rx⃗)=exp⁡(c(x⃗) r3−3g8log⁡r+log⁡θ(ν⃗)+C(x⃗)+O(r−3/2)),F(r\vec x)=\exp\Big(c(\vec x)\,r^3-\tfrac{3g}{8}\log r+\log\theta(\vec\nu)+C(\vec x)+O(r^{-3/2})\Big), where c(x⃗)c(\vec x) is an explicit cubic polynomial expression in the xjx_j and qjq_j (eq. (1.11) of the paper) and C(x⃗)C(\vec x) is independent of rr; it also asserts that all-negative configurations {x⃗:x2g<⋯<x1<0}\{\vec x:x_{2g}<\cdots<x_1<0\} lie in A\mathcal{A}. The method of proof for g=1g=1 is representative of the field: a differential identity of Its-Izergin-Korepin-Slavnov type expresses ∂rlog⁡F\partial_r\log F through a 2×22\times2 matrix Riemann-Hilbert problem (a boundary-value problem reconstructing a piecewise-analytic matrix function from prescribed jumps), which is asymptotically solved by the Deift-Zhou nonlinear steepest descent method - construction of a g\mathfrak g-function, AA/BB-cycle opening of lenses, a global parametrix built from genus-gg theta functions, Airy-type local parametrices, and a small-norm analysis - after which the asymptotics of ∂rlog⁡F\partial_r\log F are integrated in rr and simplified by nontrivial theta-function identities; the simplification of the oscillatory terms was, in the authors' words, the most challenging part of the work even for g=1g=1.

What is known beyond the paper: the hard-edge analogue (for the Bessel point process, the determinantal process with the Bessel kernel that governs statistics at an edge where the spectrum is cut off by a hard wall) of the full genus-gg multi-interval problem was proved by Blackstone-Charlier-Lenells for all gg using Birkhoff's ergodic theorem to average theta-function ratios along linear flows on the gg-dimensional torus, with 'good Diophantine' estimates for the error; the sine-process analogue is proved for two intervals; planar (normal-matrix, Coulomb-gas) and Toeplitz analogues have been developed by Charlier, Ameur, Cronvall, Lenells, Marchal and Byun-Park. None of these covers the Airy bulk multi-interval case, whose Riemann-Hilbert problem additionally contains the auxiliary point x0x_0 and the global sign structure imposed by the Airy operator - the obstruction that makes the genus-gg extension nontrivial.

Problem Statement

Establish, rigorously and in full generality, the large-r asymptotics of the gap probability F(rx⃗)=det⁡(I−KAiχrIg(b))F(r\vec x)=\det(I-\mathcal K^{\mathrm{Ai}}\chi_{r\mathcal I_g^{(b)}}) of the Airy point process on g intervals in the bulk, for arbitrary genus g - completing the program opened by Conjecture 1.1 of Blackstone-Charlier-Lenells. Concretely, settle Conjecture 1.1 for g>=2 in the regime in which the source states it: for x⃗∈A\vec x\in\mathcal A satisfying the Diophantine condition (1.8) and the rational independence of the phases Ω0,…,Ωg−1\Omega_0,\ldots,\Omega_{g-1} (condition (1.9)), show that as r→+∞r\to+\infty, F(rx⃗)=exp⁡(c(x⃗) r3−3g8log⁡r+log⁡θ(ν⃗(r,x⃗);τ)+C(x⃗)+O(r−3/2)),F(r\vec x)=\exp\Big(c(\vec x)\,r^{3}-\tfrac{3g}{8}\log r+\log\theta\big(\vec\nu(r,\vec x);\tau\big)+C(\vec x)+O(r^{-3/2})\Big), with all ingredients exactly as in the source: νj=−Ωj2πr3/2\nu_j=-\frac{\Omega_j}{2\pi}r^{3/2}, the auxiliary point x0x_0 and coefficients (q0,…,qg−1)(q_0,\ldots,q_{g-1}) determined by the system (1.3), the genus-g hyperelliptic surface X:w2=R(z)X:w^2=\mathcal R(z), its normalized holomorphic differentials and period matrix τ\tau, the r-independent constant C(x⃗)C(\vec x), and the explicit cubic coefficient c(x⃗)c(\vec x) of eq. (1.11). This includes the conjectured admissibility of all-negative configurations {x⃗:x2g<⋯<x1<0}⊂A\{\vec x: x_{2g}<\cdots<x_1<0\}\subset\mathcal A.

A proof for any fixed small genus (for instance g=2) is valuable progress but does not by itself resolve the stated problem, whose target is arbitrary genus. By contrast, a counterexample to the conjectured expansion, or to the all-negative admissibility clause, resolves the problem just as a proof does. The degenerate regimes in which (1.8) or (1.9) fails, the evaluation of the constant C(x⃗)C(\vec x) in closed form, and a full characterization of A\mathcal A for g>=2 are discussed in the source only as remarks and expectations (Remark 1; the complete g=1 classification in Theorem 1.2); they are related open directions, not components of the goal stated here.

All objects and conditions referenced above - the admissible set A\mathcal{A}, the auxiliary system (1.3) determining (x0,q0,…,qg−1)(x_0,q_0,\ldots,q_{g-1}), the Diophantine condition (1.8), the rational-independence condition (1.9) on the phases Ωj\Omega_j, the genus-gg surface XX with its period matrix τ\tau and theta function, and the explicit cubic coefficient c(x⃗)c(\vec{x}) of eq. (1.11) - are as defined in the background above, following Conjecture 1.1 of Blackstone-Charlier-Lenells.

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