arXiv subjects
Publications and source records attributed to Christophe Charlier.
We consider the ``good" Boussinesq equation on the half-line. Assuming existence of the solution, we prove that it can be recovered from the solution of a $3\times 3$ Riemann-Hilbert problem that depends only on the initial and boundary values, and whose jump contour consists of twelve half-lines.
We study $n\times n$ random Hermitian matrix ensembles that are invariant under unitary conjugation. Let $I$ be a finite union of intervals lying in the bulk, and let $m_{k}^{(n)}$ be the $k$-th largest gap between consecutive eigenvalues lying in $I$. We prove that the rescaled gap $\smash{\tau_{k}^{(n)}}$, which is defined by \begin{align*} m_{k}^{(n)} = \frac{1}{2\pi \inf_{I}\rho} \bigg( \frac{\sqrt{32 \log n}}{n} + \frac{3q-8}{2q} \frac{ \log(2\log n)}{n \sqrt{2\log n}} + \frac{4\tau_{k}^{(n)}}{n \sqrt{2\log n}} \bigg), \end{align*} converges in distribution as $n\to +\infty$ to a gamma-Gumbel random variable that is shifted by an explicit constant $c_{V,I}$ depending only on $I$ and on the potential $V$. Here $\rho$ is the density of the equilibrium measure and $q\in \mathbb{N}_{>0}$ is the highest order at which $\rho(x)$ approaches $\inf_{I}\rho$ with $x\in I$; for example, if $\rho(x)=1/(\pi\sqrt{x(1-x)})$, then $q=2$ if $\frac{1}{2}\in \overline{I}$ and $q=1$ otherwise. This work extends a result of Feng and Wei beyond the Gaussian potential.
We obtain precise asymptotics for the weighted number of domino tilings of an L-shaped subset of the Aztec diamond, obtained by removing an approximate rectangle in a corner of the Aztec diamond. By tuning the size of the removed corner, we observe different types of asymptotics. For a small removed corner, the number of tilings is close to that of the full Aztec diamond. Enlarging the removed corner to a critical size, a phase transition described in terms of the Tracy-Widom distribution occurs. Further increasing the size of the removed region, we observe a sharp decrease of the number of tilings, until it is finally approximated by the number of tilings of two smaller disjoint Aztec diamonds. We obtain uniform asymptotics for the number of domino tilings which fully describe these transitions.
We propose an analytical method for the construction of Hartree-Fock phase diagrams for the (fermion) Hubbard model and various generalizations thereof. Such phase diagrams are traditionally constructed numerically, but we argue that, by using asymptotic techniques, it is possible to obtain analytic formulas approximating the curves separating the different phases to very high accuracy. To illustrate the new method, we apply it to the two-dimensional Hubbard model on the square lattice at zero temperature. This yields formulas for the Hartree-Fock phase boundaries that agree with, but also improve on, earlier numerical results. In particular, our results provide the first rigorous proof of the existence of mixed phases in this model.
We study the last passage time in geometric last passage percolation (LPP). As the system size increases, we derive precise large deviation probabilities -- up to and including the constant terms -- for both the lower and upper tails. A key step in proving these results is to establish a duality formula that reformulates the LPP problem in terms of the largest eigenvalue in the Jacobi unitary ensemble (JUE). In addition, we establish a second duality formula, which relates the LPP problem to the truncated unitary ensemble (TUE). Using this, we also derive asymptotics for the moments of the absolute value of characteristic polynomials of the TUE, which may be of independent interest.
We consider the two-dimensional Coulomb gas with a general potential at the determinantal temperature, or equivalently, the eigenvalues of random normal matrices. We prove that the smallest gaps between particles are typically of order $n^{-3/4}$, and that the associated joint point process of gap locations and gap sizes, after rescaling the gaps by $n^{3/4}$, converges to a Poisson point process. As a consequence, we show that the $k$-th smallest rescaled gap has a limiting density proportional to $x^{4k-1}e^{-\frac{\mathcal{J}}{4}x^{4}}$, where $\mathcal{J}=\pi^{2}\int \rho(z)^{3}d^{2}z$ and $\rho$ is the density of the equilibrium measure. This generalizes a result of Shi and Jiang beyond the quadratic potential.
We study the initial-boundary value problem for the Boussinesq equation on the half-line. Assuming that the solution exists, we prove that it can be recovered from its initial-boundary values via the solution of a $3\times 3$ Riemann-Hilbert problem. The contour consists of $18$ arcs on the unit circle, $18$ segments and $18$ half-lines, and the associated jump matrices involve $9$ reflection coefficients.
We introduce a new method for studying gap probabilities in a class of discrete determinantal point processes with double contour integral kernels. This class of point processes includes uniform measures of domino and lozenge tilings as well as their doubly periodic generalizations. We use a Fourier series approach to simplify the form of the kernels and to characterize gap probabilities in terms of Riemann-Hilbert problems. As a first illustration of our approach, we obtain an explicit expression for the number of domino tilings of reduced Aztec diamonds in terms of Pad\'e approximants, by solving the associated Riemann-Hilbert problem explicitly. As a second application, we obtain an explicit expression for the number of lozenge tilings of (simply connected) reduced hexagons in terms of Hermite-Pad\'e approximants. For more complicated domains, such as hexagons with holes, the number of tilings involves a generalization of Hermite-Pad\'e approximants.
Let $\mu$ be a positive measure supported on a domain $\Omega$. We consider the behavior of the balayage measure $\nu:=\mathrm{Bal}(\mu,\partial \Omega)$ near a point $z_{0}\in \partial \Omega$ at which $\Omega$ has an outward-pointing cusp. Assuming that the order and coefficient of tangency of the cusp are $d>0$ and $a>0$, respectively, and that $d\mu(z) \asymp |z-z_{0}|^{2b-2}d^{2}z$ as $z\to z_0$ for some $b > 0$, we obtain the leading order term of $\nu$ near $z_{0}$. This leading term is universal in the sense that it only depends on $d$, $a$, and $b$. We also treat the case when the domain has multiple corners and cusps at the same point. Finally, we obtain an explicit expression for the balayage of the uniform measure on the tacnodal region between two osculating circles, and we give an application of this result to two-dimensional Coulomb gases.
We present a numerical scheme for the solution of the initial-value problem for the ``bad'' Boussinesq equation. The accuracy of the scheme is tested by comparison with exact soliton solutions as well as with recently obtained asymptotic formulas for the solution.
We study blow-up solutions of the ``bad" Boussinesq equation, and prove that a wide range of asymptotic scenarios can happen. For example, for each $T>0$, $x_{0}\in \mathbb{R}$ and $\delta \in (0,1)$, we prove that there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R} \times [0,T)$ such that $|u(x,t)| \leq C \frac{1+x^{2}}{(x-x_{0})^{2}}$ and $u(x_{0},t)\asymp (T-t)^{-\delta}$ as $t\to T$. We also prove that for any $q\in \mathbb{N}$, $T>0$, $x_{0}\in \mathbb{R}$, $\delta \in (0,\frac{1}{2})$, there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R} \times [0,T)$ such that (i) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x,t)|\leq C$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}\leq q$, (ii) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x,t)| \leq C \frac{1+|x|}{|x-x_{0}|}$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}= q+1$, (iii) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x_{0},t)| \asymp (T-t)^{-\delta}$ as $t\to T$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}= q+1$. In particular, when $q=0$, this result establishes the existence of wave-breaking solutions, i.e. solutions that remain bounded but whose $x$-derivative blows up in finite time.
We consider the balayage of a measure $\mu$ defined on a domain $\Omega$ onto its boundary $\partial \Omega$. Assuming that $\Omega$ has a corner of opening $\pi \alpha$ at a point $z_0 \in \partial \Omega$ for some $0 < \alpha \leq 2$ and that $d\mu(z) \asymp |z-z_{0}|^{2b-2}d^{2}z$ as $z\to z_0$ for some $b > 0$, we obtain the precise rate of vanishing of the balayage of $\mu$ near $z_{0}$. The rate of vanishing is universal in the sense that it only depends on $\alpha$ and $b$. We also treat the case when the domain has multiple corners at the same point. Moreover, when $2b\leq \frac{1}{\alpha}$, we provide explicit constants for the upper and lower bounds.
We study large $n$ expansions for the partition function of a Coulomb gas $$Z_n=\frac 1 {\pi^n}\int_{\mathbb{C}^n}\prod_{1\le i<j\le n}|z_i-z_j|^2\prod_{i=1}^n e^{-nQ(z_i)}\, d^2 z_i,$$ where $Q$ is a radially symmetric confining potential on the complex plane $\mathbb{C}$. The droplet is not assumed to be connected, but may consist of a number of disjoint connected annuli and possibly a central disk. The boundary condition is ``soft edge'', i.e., $Q$ is smooth in a $\mathbb{C}$-neighbourhood of the droplet. We include the following possibilities: (i) existence of ``outposts'', i.e., components of the coincidence set which falls outside of the droplet, (ii) a conical (or Fisher-Hartwig) singularity at the origin, (iii) perturbations $Q-\frac h n$ where $h$ is a smooth radially symmetric test-function. In each case, the free energy $\log Z_n$ admits a large $n$ expansion of the form \begin{equation*}\log Z_n=C_1n^2+C_2n\log n+C_3 n+C_4\log n+C_5+\mathcal{G}_{n}+o(1)\end{equation*} where $C_1,\ldots,C_5$ are certain geometric functionals. The $n$-dependent term $\mathcal{G}_n$ is bounded as $n\to\infty$; it arises in the presence of spectral gaps. We use the free energy expansions to study the distribution of fluctuations of linear statistics. We prove that the fluctuations are well approximated by the sum of a Gaussian and certain independent terms which provide the displacement of particles from one component to another. This displacement depends on $n$ and is expressed in terms of the Heine distribution. We also prove (under suitable assumptions) that the number of particles which fall near a spectral outpost converges to a Heine distribution.
We study hole probabilities of two-dimensional Coulomb gases with a general potential and arbitrary temperature. The hole region $U$ is assumed to satisfy $\partial U\subset S$, where $S$ is the support of the equilibrium measure $\mu$. Let $n$ be the number of points. As $n \to \infty$, we prove that the probability that no points lie in $U$ behaves like $\exp(-Cn^{2}+o(n^{2}))$. We determine $C$ in terms of $\mu$ and the balayage measure $\nu = \mathrm{Bal}(\mu|_{U},\partial U)$. If $U$ is unbounded, then $C$ also involves the Green function of $\Omega$ with pole at $\infty$, where $\Omega$ is the unbounded component of $U$. We also provide several examples where $\nu$ and $C$ admit explicit expressions: we consider several point processes, such as the elliptic Ginibre, Mittag-Leffler, and spherical point processes, and various hole regions, such as circular sectors, ellipses, rectangles, and the complement of an ellipse. This work generalizes previous results of Adhikari and Reddy in several directions.
It is well-known that each solution of the mKdV equation gives rise, via the Miura transformation, to a solution of the KdV equation. In this work, we show that a similar Miura-type transformation exists also for the ``good'' Boussinesq equation. This transformation maps solutions of a second-order equation to solutions of the fourth-order Boussinesq equation. Just like in the case of mKdV and KdV, the correspondence exists also at the level of the underlying Riemann--Hilbert problems and this is in fact how we construct the new transformation.
We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel--Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus $0$, then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.
We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let $n$ be the number of points. We focus on two regimes: (a) the ``hard edge regime" where all disk boundaries are at a distance of order $\frac{1}{n}$ from the hard wall, and (b) the ``semi-hard edge regime" where all disk boundaries are at a distance of order $\frac{1}{\sqrt{n}}$ from the hard wall. As $n \to + \infty$, we prove that the moment generating function enjoys asymptotics of the form \begin{align*} & \exp \bigg(C_{1}n + C_{2}\ln n + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})\bigg), & & \mbox{for the hard edge}, \\ & \exp \bigg(C_{1}n + C_{2}\sqrt{n} \hspace{0.12cm} + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}\bigg(\frac{(\ln n)^{4}}{n}\bigg)\bigg), & & \mbox{for the semi-hard edge}. \end{align*} In both cases, we determine the constants $C_{1},\dots,C_{4}$ explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the ``bulk", ``soft edge" and ``semi-hard edge" regimes, the second and higher order cumulants of the disk counting function in the ``hard edge" regime are proportional to $n$ and not to $\sqrt{n}$.
We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant $\Gamma=2$ and that the number $n$ of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order $1/n$ from the hard edge. At distances much larger than $1/\sqrt{n}$, the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the ``semi-hard edge''. More precisely, we provide asymptotics for the correlation kernel $K_{n}(z,w)$ as $n\to\infty$ in two microscopic regimes (with either $|z-w| = \mathcal{O} (1/\sqrt{n})$ or $|z-w| = \mathcal{O} (1/n)$), as well as in three macroscopic regimes (with $|z-w| \asymp 1$). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szeg\H{o} kernels.