Moment generating function of the tacnode process

Abstract: The tacnode process is a universal determinantal point process arising in non-intersecting particle systems and random tiling models. In this paper, we study the generating function for the counting functions of the tacnode process on a union of $m$ intervals, $m\in\mathbb{N}^{+}$. Our first result provides an integral representation for the $m$-point generating function in terms of the Hamiltonian governing a system of $8m+4$ coupled differential equations. Combined with several differential identities for this Hamiltonian, the representation yields the large gap asymptotics, up to and including the constant term. As further applications, we obtain asymptotic formulae for the expectations, variances, and covariances of the counting functions, and establish a central limit theorem for their joint fluctuations. These results extend the previously known $1$-point theory for the tacnode process to the multi-interval setting with multiple discontinuities.

• The paper extends the analysis of the moment generating function beyond single intervals to multi-interval settings for the tacnode process.
• It establishes a coupled high-dimensional Hamiltonian system linked to a 4×4 Riemann–Hilbert problem, ensuring precise large gap asymptotics.
• The study derives sharp statistical results, including explicit expressions for moments, covariances, and a central limit theorem for multi-point fluctuations.

Moment Generating Function and Large Gap Asymptotics for the Tacnode Process

Introduction and Problem Setting

The paper "Moment generating function of the tacnode process" (2606.17771) presents a comprehensive investigation of the exponential generating function for counting observables in the tacnode point process, a universal determinantal process emerging at critical tangency points in non-intersecting particle systems and random tiling models. The authors extend the analysis of moment generating functions beyond the previously understood single-interval case to multi-interval settings, addressing the multi-point joint fluctuations of the counting function across several intervals where the process exhibits discontinuities.

Letting $\tilde\chi$ be the random point configuration of the tacnode process and $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$, the central object is the $m$-point generating function

$$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$

for distinct points $0 < x_1 < x_2 < \dots < x_m$. This corresponds, via reparametrization, to the exponential moment of the joint counts in the $m$ disjoint intervals forming a partition of $(-x_m, x_m)$. Fundamental results are established on the integrable structure of $F$, its rigorous connection to a large system of coupled nonlinear ODEs, and its precise asymptotic expansion in the so-called large gap regime.

Integrable Structure and Coupled Hamiltonian System

A key structural insight is a representation of $F$ as a Fredholm determinant,

$$F(\mathbf{x}, \mathbf{u}) = \det\left( I - \sum_{j=1}^m (1 - s_j) \mathcal{K}^{\mathrm{tac}} \chi_{A_j} \right)$$

where $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$0 is the integral operator with the tacnode kernel and $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$1 is the characteristic function of the $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$2th interval.

Employing the theory of isomonodromic deformations, the paper demonstrates that for general $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$3, $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$4 can be written in terms of the Hamiltonian of a highly coupled system of $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$5 ODEs. This goes far beyond the $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$6 case, where the structure reduces to the familiar $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$7-dimensional system related to the Painlevé II hierarchy [YZ2024]. These ODEs arise from a Lax pair corresponding to a $N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$8 Riemann-Hilbert problem that encodes the tacnode kernel, and their construction exploits deep analytic and algebraic properties of the underlying RH data, including multiple interaction "cross-terms" which encode the nontrivial covariance structure at the multi-interval level.

An explicit integral formula is proved:

$N(x):=\#\{\,\xi\in\tilde\chi:\xi\in (-x, x)\}$9

where $m$0 is determined by the Hamiltonian of the $m$1-dimensional system. The authors provide sharp asymptotic expansions for the canonical variables in this system, both as $m$2 and, crucially, $m$3.

Asymptotics and Large Gap Expansion

Through steepest descent analysis of the auxiliary RH problems, the paper rigorously extracts the large $m$4 (large gap) asymptotics of $m$5. For general $m$6, the expansion is

$m$7

with explicit formulas for each term:

• $m$8
• $m$9
• $$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$0

Notably, the error term is strongly controlled ($$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$1) and uniform in all parameters, and the expansion is differentiable any number of times in the $$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$2 variables at the cost of increasing the power of $$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$3 in the error.

The appearance of the Barnes $$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$4-function in the constant term generalizes results for the sine, Airy, and Pearcey processes, emphasizing a universality phenomenon in the large gap (thinned) regime for DPPs with higher-order transitions.

Statistical Implications and Central Limit Theorem

By exploiting the explicit large gap expansion, precise asymptotics are obtained for the moments and covariances:

• The expected number of points in $$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$5 is $$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$6;
• The variance is $$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$7, where $$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$8 is the Euler-Mascheroni constant;
• The covariance for $$F(\mathbf{x}, \mathbf{u}) = \mathbb{E}\left[\prod_{j=1}^m \exp(u_j N(x_j))\right]$$9 is $0 < x_1 < x_2 < \dots < x_m$0.

Furthermore, the joint fluctuations of the vector $0 < x_1 < x_2 < \dots < x_m$1 are shown to converge (after proper normalization and centering) to independent normals, yielding a central limit theorem and establishing asymptotic Gaussianity at the level of large gaps. This result is sharp and, together with the explicit covariance, gives the full scaling limit for the linear statistics in this multi-interval tacnode process.

Theoretical and Practical Implications

The extension of integrable probabilistic analysis to multi-interval thinned tacnode processes opens several new directions:

• The multi-interval Hamiltonian structure provides a new instance of high-dimensional isomonodromic integrable systems, and the methods introduced could be adapted to higher-order phase transitions or models with intricate "multi-cut" behavior in their limiting shapes;
• The explicit covariance structure and central limit result are indispensable for understanding rigidity phenomena, extreme value statistics, and mesoscopic scaling limits in non-intersecting systems and random tilings near cusp singularities;
• The techniques—relying on RH analysis, tau function theory, and singular asymptotics—are robust and port readily to other classes of universal determinantal processes with more complex interaction graphs or combinatorial configurations.

This body of results is expected to have deep impact on integrable probability, especially in the study of local-global transitional behaviors, universality classes for edge and cusp singularities, and precise descriptions of statistical fluctuations in interacting particle systems.

Conclusion

The paper provides a mathematically rigorous, integrable framework and asymptotic analysis for the joint exponential moments of multi-interval counting observables in the tacnode process, generalizing classical results for lower-order DPPs to a critically degenerate setting with multi-point discontinuities. The combination of a high-dimensional coupled Hamiltonian system and RH steepest descent analysis yields precise large gap asymptotics, explicit moment and covariance structure, and a central limit theorem, broadening the understanding of universality, rigidity, and rigidity-breaking phenomena in random matrix and interacting particle ensembles.

Reference:

"Moment generating function of the tacnode process" (2606.17771)