arXiv subjects

Sampad Lahiry

Publications and source records attributed to Sampad Lahiry.

8 recordsLinked to original sources

arXiv2026

Matrix Szeg\H{o} Function and Matrix Orthogonal Polynomials for Multiple Cuts

Following recent developments on matrix-valued orthogonal polynomials (MVOPs), we construct the matrix Szeg\H{o} factorisation of the matrix weight function when it is supported on multiple intervals. We find that on the gaps the matrix function has unitary multiplicative jumps. In the next part, performing the Deift-Zhou steepest descent analysis we study the associated global parametrix which comes from the Riemann-Hilbert problem of the MVOPs. A solution is constructed visualising the rows of the parametrix as sections of vector bundles on a hyper-elliptic Riemann surface. An alternate view point to the global parametrix is also presented using a vanishing lemma. As an application we obtain strong asymptotics of MVOPs for multiple cuts which extends the works of Dea\~no, Kuijlaars, and Rom\'an.

math.CA↗

arXiv2026

Extremal Functions and Widom Factors on Compact Riemann Surfaces

We study the Chebyshev extremal problem on a compact Riemann surface $X$ of genus $g>0$. As an analog to monic polynomials, we consider admissible meromorphic functions having no poles away from a marked point $P_{\infty}$ (with adequate normalisation). For a nonpolar compact set $E\subset X\setminus\{P_{\infty}\}$, we show that the $n$-th root of the Chebyshev constant $t_n(E)$ converges to the capacity of $E$, and we establish the corresponding Bernstein-Walsh inequality. In the second part of the paper, using a Cauchy kernel adapted to Riemann surfaces, we study the refined Szeg\H{o}--Widom asymptotics for the extremals as well as the Widom factors $$ W_n(E)=\frac{t_n(E)}{\operatorname{cap}(E)^n},$$ assuming that $E$ is a finite union of $p$ closed discs with analytic boundaries. The geometry of the Schottky double, which has genus $2g+p-1$, enters explicitly into these asymptotics. We find that there is no analogue of Faber-type asymptotics even for one single boundary curve. We conclude by constructing explicit examples in genus $1$ using the Weierstrass-$\wp$ function.

math.CA↗

arXiv2025

Birth of a gap: Critical phenomena in 2D Coulomb gas

We investigate a family of radially symmetric Coulomb gas systems at inverse temperature $\beta = 2$. The family is characterised by the property that the density of the equilibrium measure vanishes on a ring at radius $r_*$, which lies strictly inside the droplet. The large-$n$ expansion of the logarithm of the partition function is obtained up to the constant term and features a novel term of order $n^{1/4}$. We perform a double scaling limit of the correlation kernel at the $n^{1/4}$ scale and obtain a new limiting kernel in the bulk, which differs from the well-known Ginibre kernel.

math.PR↗

arXiv2025

Partition function of 2D Coulomb gases with radially symmetric potentials and a hard wall

The large $N$ asymptotic expansion of the partition function for the normal matrix model is predicted to have special features inherited from its interpretation as a two-dimensional Coulomb gas. However for the latter, it is most natural to include a hard wall at the boundary of the droplet. We probe how this affects the asymptotic expansion in the solvable case that the potential is radially symmetric and the droplet is a disk or an annulus. We allow too for the hard wall to be strictly inside the boundary of the droplet. It is observed the term of order $\log N$, has then a different rational number prefactor to that when the hard wall is at the droplet boundary. Also found are certain universal (potential independent) numerical constants given by definite integrals, both at order $\sqrt{N}$, and in the constant term.

math.PR↗

arXiv2025

Recurrence relations and the Christoffel-Darboux formula for elliptic orthogonal polynomials

In recent years, there has been significant progress in the theory of orthogonal polynomials on algebraic curves, particularly on genus 1 surfaces. In this paper, we focus on elliptic orthogonal polynomials and establish several of their fundamental properties. In particular, we derive general five-term and seven-term recurrence relations, which lead to a Christoffel-Darboux formula and the construction of an associated point process on the A-cycle of the torus. Notably, the recurrence coefficients in these relations are intricately linked through the underlying elliptic curve equation. Under additional symmetry assumptions on the weight function, the structure simplifies considerably, recovering known results for orthogonal polynomials on the complex plane.

math-ph↗

arXiv2025

Orthogonal polynomials in the spherical ensemble with two insertions

We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{deg}P_{n,N}=n$) with respect to the weight $$\frac{|z-w|^{2NQ_1}}{(1+|z|^2)^{N(1+Q_0+Q_1)+1}}, \quad(Q_0,Q_1 > 0)$$ in the whole complex plane. With $n, N\rightarrow\infty$ and $N-n$ fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted $L^2$ norm and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift-Zhou steepest descent analysis of the associated $2\times 2$ Riemann-Hilbert problem.

math.CA↗

arXiv2025

Properties of the one-component Coulomb gas on a sphere with two macroscopic external charges

The one-component Coulomb gas on the sphere, consisting on $N$ unit charges interacting via a logarithmic potential, and in the presence of two external charges each of strength proportional to $N$, is considered. There are two spherical caps naturally associated with the external charges, giving rise to two distinct phases depending on them not overlapping (post-critical) or overlapping (pre-critical). The equilibrium measure in the post-critical phase is known from earlier work. We determine the equilibrium measure in the pre-critical phase using a particular conformal map, with the parameters therein specified in terms of a root of a certain fourth order polynomial. This is used to determine the exact form of the electrostatic energy for the pre-critical phase. Using a duality relation from random matrix theory, the partition function for the Coulomb gas at the inverse temperature $\beta = 2$ can be expanded for large $N$ in the post-critical phase, and in a scaling region of the post and pre-critical boundary. For the pre-critical phase, the duality identity implies a relation between two electrostatic energies, one for the present sphere system, and the other for a certain constrained log-gas relating to the Jacobi unitary ensemble.

math-ph↗

arXiv2024

Orthogonal polynomials in the normal matrix model with two insertions

We consider orthogonal polynomials with respect to the weight $|z^2+a^2|^{cN}e^{-N|z|^2}$ in the whole complex plane. We obtain strong asymptotics and the limiting normalized zero counting measure (mother body) of the orthogonal polynomials of degree $n$ in the scaling limit $n,N\to \infty$ such that $\frac{n}{N}\to t$. We restrict ourselves to the case $a^2\geq 2c$, $cN$ integer, and $t<t^{*}$ where $t^{*}$ is a constant depending only on $a,c$. Due to this restriction, the mother body is supported on an interval. We also find the two dimensional equilibrium measure (droplet) associated with the eigenvalues in the corresponding normal matrix model. Our method relies on the recent result that the planar orthogonal polynomials are a part of a vector of type I multiple orthogonal polynomials, and this enables us to apply the steepest descent method to the associated Riemann-Hilbert problem.

math.CA↗