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Tomas Lasic Latimer

Publications and source records attributed to Tomas Lasic Latimer.

8 recordsLinked to original sources
arXiv2026

Explicit enumeration and large-valence asymptotics of even-valent maps

Let $\mathscr{N}_g(2ν,j)$ denote the number of connected labelled $2ν$-valent maps of genus $g$ with $j$ vertices. Explicit bivariate formulae for $\mathscr{N}_g(2ν,j)$ have been available only in the planar and toroidal cases. Using a structural formula of Ercolani et al (2023), we translate the problem of determining an explicit bivariate formula for $\mathscr{N}_g(2ν,j)$, $g \geq 2$, to finding finitely many counts with a fixed number of vertices. For $g=2,3$ and $4$ we determine these counts using the associated orthogonal polynomials, yielding explicit bivariate formulae for $\mathscr{N}_g(2ν,j)$ in these genera. Furthermore, the same method applies for every $g\ge5$ at the cost of additional computation. From these formulae we obtain the leading-order asymptotics of $\mathscr{N}_g(2ν,j)$ as $ν\to\infty$ for $g=2,3,4$, and we conjecture the structure of these formulae in general genus. In addition, we establish an analogous reduction to finitely many counts, derive explicit formulae and large-valence asymptotics, and formulate corresponding conjectures for two-legged even-valent maps.

math.CO↗
arXiv2026

On difference-differential Lax pairs and integrals of Painlevé equations in finite characteristic

We collect rank two difference-differential Lax pairs for classical Painlevé equations in the literature and put each in $2\times 2$ matrix form with the coefficient matrix of the spectral equation a degree two matrix polynomial. We describe and apply a general method to obtain integrals of motion in characteristic $p$ from these Lax pairs. For every relevant Painlevé equation, this leads to a countable list of integrals of motion, with one entry for each prime $p$.

nlin.SI↗
arXiv2023

On a class of elliptic orthogonal polynomials and their integrability

Building upon the recent works of Bertola; Fasondini, Olver and Xu, we define a class of orthogonal polynomials on elliptic curves and establish a corresponding Riemann-Hilbert framework. We then focus on the special case, defined by a constant weight function, and use the Riemann-Hilbert problem to derive recurrence relations and differential equations for the orthogonal polynomials. We further show that the sub-class of even polynomials is associated to the elliptic form of Painlevé VI, with the tau function given by the Hankel determinant of even moments, up to a scaling factor. The first iteration of these even polynomials relates to the special case of Painlevé VI studied by Hitchin in relation to self-dual Einstein metrics.

math.CA↗
arXiv2023

Asymptotic behaviours of q-orthogonal polynomials from a q-Riemann Hilbert Problem

We describe a Riemann-Hilbert problem for a family of $q$-orthogonal polynomials, $\{ P_n(x) \}_{n=0}^\infty$, and use it to deduce their asymptotic behaviours in the limit as the degree, $n$, approaches infinity. We find that the $q$-orthogonal polynomials studied in this paper share certain universal behaviours in the limit $n\to\infty$. In particular, we observe that the asymptotic behaviour near the location of their smallest zeros, $x \sim q^{n/2}$, and norm, $\|P_n\|_2$, are independent of the weight function as $n\to\infty$.

math.CA↗
arXiv2022

Asymptotics of Discrete $q$-Freud $\mathrm{II}$ orthogonal polynomials from the $q$-Riemann Hilbert Problem

We investigate a Riemann-Hilbert problem (RHP), whose solution corresponds to a group of $q$-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the polynomials approach infinity. The RHP formulation also enables us to obtain further properties. In particular, we consider how the class of polynomials and their asymptotic behaviours change under translations of the $q$-discrete lattice and determine the asymptotics of related $q$-Painlevé equations.

math.CA↗
arXiv2021

On a class of q-orthogonal polynomials and the q-Riemann Hilbert Problem

We give an explicit solution of a q-Riemann Hilbert problem which arises in the theory of orthogonal polynomials, prove that it is unique, and deduce several properties. Our new results include the asymptotic behaviour of zeroes in the limit as the degree of the polynomial approaches infinity.

math.CA↗