Quantum Coulomb Gases

Coulomb气体与Riesz气体讲义

Lectures on Coulomb and Riesz gases

本书是关于Coulomb和Riesz气体的初步版本，源自多所高校及2024年Saint-Flour概率暑期学校的研究生课程讲义。

中等温度区库仑气体的泊松统计特性

Poisson Statistics for Coulomb Gases at Intermediate Temperature Regimes

针对二维库仑气体中等温度区的微观统计问题，通过建立关联函数的定量渐近描述，证明了在特定温度条件下其微观点过程收敛于齐次泊松点过程，扩展了现有结果。

带点电荷的球形库仑气体的自由能

Free energy of spherical Coulomb gases with point charges

本研究针对黎曼球面上带南北极点电荷、具行列式/Pfaffian结构的旋转不变二维库仑气体（对应复平面全平面液滴的弱约束势模型），推导了含常数项的自由能精确渐近展开式。

Local microscopic behavior for 2D Coulomb gases

The study of two-dimensional Coulomb gases lies at the interface of statistical physics and non-Hermitian random matrix theory. In this paper we give a large deviation principle (LDP) for the empirical fields obtained, under the canonical Gibbs measure, by zooming around a point in the bulk of the equilibrium measure, up to the finest averaging scale. The rate function is given by the sum of the "renormalized energy" of Serfaty et al. weighted by the inverse temperature, and of the specific relative entropy. We deduce a local law which quantifies the convergence of the empirical measures of th…

平均场库仑气体的谱隙

Spectral gaps for mean-field Coulomb gases

该研究针对d≥3维排斥单组分库仑气体，在弱耦合条件下证明了粒子数一致的庞加莱不等式，得到过阻尼朗之万动力学的指数弛豫，通过单点不等式、移动极点估计等方法给出谱隙的匹配上下界。

中间温度区间下 Coulomb 气体局部经验测度的大偏差原理

Large Deviation Principle for local empirical measure of Coulomb gases at intermediate temperature regime

本文研究了中间温度区间下Coulomb气体的局部经验场，通过确定临界温度标度，证明了在超临界和亚临界状态下分别满足基于熵和基于能量的LDP，并在临界状态下推导出熵与能量竞争的LDP型结果。

Microscopic description of Log and Coulomb gases

These are the lecture notes of a course taught at the Park City Mathematics Institute in June 2017. They are intended to review some recent results, obtained in large part with Thomas Leblé, on the statistical mechanics of systems of points with logarithmic or Coulomb interactions. After listing some motivations, we describe the "electric approach" which allows to get concentration results, Central Limit Theorems for fluctuations, and a Large Deviations Principle expressed in terms of the microscopic state of the system.

Graphs, local zeta functions, Log-Coulomb Gases, and phase transitions at finite temperature

We study a log-gas on a network (a finite, simple graph) confined in a bounded subset of a local field (i.e. R, C, Q_{p} the field of p-adic numbers). In this gas, a log-Coulomb interaction between two charged particles occurs only when the sites of the particles are connected by an edge of the network. The partition functions of such gases turn out to be a particular class of multivariate local zeta functions attached to the network and a positive test function which is determined by the confining potential. The methods and results of the theory of local zeta functions allow us to establish t…

Non-Archimedean Coulomb Gases

This article aims to study the Coulomb gas model over the $d$-dimensional $p$-adic space. We establish the existence of equilibria measures and the $Γ$-limit for the Coulomb energy functional when the number of configurations tends to infinity. For a cloud of charged particles confined into the unit ball, we compute the equilibrium measure and the minimum of its Coulomb energy functional. In the $p$-adic setting the Coulomb energy is the continuum limit of the minus a hierarchical Hamiltonian attached to a spin glass model with a $p$-adic coupling.

A note on the second order universality at the edge of Coulomb gases on the plane

We consider in this note a class of two-dimensional determinantal Coulomb gases confined by a radial external field. As the number of particles tends to infinity, their empirical distribution tends to a probability measure supported in a centered ring of the complex plane. A quadratic confinement corresponds to the complex Ginibre Ensemble. In this case, it is also already known that the asymptotic fluctuation of the radial edge follows a Gumbel law. We establish in this note the universality of this edge behavior, beyond the quadratic case. The approach, inspired by earlier works of Kostlan a…

Log-Coulomb gases in the projective line of a $p$-field

This article extends recent results on log-Coulomb gases in a $p$-field $K$ (i.e., a nonarchimedean local field) to those in its projective line $\mathbb{P}^1(K)$, where the latter is endowed with the $PGL_2$-invariant Borel probability measure and spherical metric. Our first main result is an explicit combinatorial formula for the canonical partition function of log-Coulomb gases in $\mathbb{P}^1(K)$ with arbitrary charge values. Our second main result is called the "$(q+1)$th Power Law", which relates the grand canonical partition functions for one-component gases in $\mathbb{P}^1(K)$ (where…

Extremal particles of two-dimensional Coulomb gases and random polynomials on a positive background

We study the outliers for two models which have an interesting connection. On the one hand, we study a specific class of planar Coulomb gases which are determinantal. It corresponds to the case where the confining potential is the logarithmic potential of a radial probability measure. On the other hand, we study the zeros of random polynomials that appear to be closely related to the first model. Their behavior far from the origin is shown to depend only on the decaying properties of the probability measure generating the potential. A similar feature is observed for their behavior near the ori…

约束在半空间中的库仑气体的极小化子

Minimizers for Coulomb gases constrained to a halfspace

研究二次陷阱中库仑相互作用粒子在约束于半空间时的分布变化，证明存在相变，解决Byun等人的猜想。

Statistical mechanics of Coulomb gases as quantum theory on Riemann surfaces

Statistical mechanics of 1D multivalent Coulomb gas may be mapped onto non-Hermitian quantum mechanics. We use this example to develop instanton calculus on Riemann surfaces. Borrowing from the formalism developed in the context of Seiberg-Witten duality, we treat momentum and coordinate as complex variables. Constant energy manifolds are given by Riemann surfaces of genus $g\geq 1$. The actions along principal cycles on these surfaces obey ODE in the moduli space of the Riemann surface known as Picard-Fuchs equation. We derive and solve Picard-Fuchs equations for Coulomb gases of various char…

二维库仑气体与多个哨所

Two-dimensional Coulomb gases with multiple outposts

本文研究了二维库仑气体在多个哨所存在时的粒子分布行为，证明了哨所附近粒子数量的联合分布收敛于多维Heine分布，并揭示了哨所间粒子数量的强相关性。

交换子、平均场和超临界平均场极限用于库仑/里兹气体

Commutators, mean-field, and supercritical mean-field limits for Coulomb/Riesz gases

本文通过模调能量方法研究了库仑/里兹气体的交换子估计及其在平均场和超临界平均场极限中的应用。

Distribution of the Wigner-Smith time-delay matrix for chaotic cavities with absorption and coupled Coulomb gases

Within the random matrix theory approach to quantum scattering, we derive the distribution of the Wigner-Smith time delay matrix $\mathcal{Q}$ for a chaotic cavity with uniform absorption, coupled via $N$ perfect channels. In the unitary class $β=2$ we obtain a compact expression for the distribution of the full matrix in terms of a matrix integral. In the other symmetry classes we derive the joint distribution of the eigenvalues. We show how the large $N$ properties of this distribution can be analysed in terms of two interacting Coulomb gases living on two different supports. As an applicati…

Local laws and rigidity for Coulomb gases at any temperature

We study Coulomb gases in any dimension $d \geq 2$ and in a broad temperature regime. We prove local laws on the energy, separation and number of points down to the microscopic scale. These yield the existence of limiting point processes generalizing the Ginibre point process for arbitrary temperature and dimension. The local laws come together with a quantitative expansion of the free energy with a new explicit error rate in the case of a uniform background density. These estimates have explicit temperature dependence, allowing to treat regimes of very large or very small temperature, and exh…

Fluctuations of Two-Dimensional Coulomb Gases

We prove a Central Limit Theorem for the linear statistics of two-dimensional Coulomb gases, with arbitrary inverse temperature and general confining potential, at the macroscopic and mesoscopic scales and possibly near the boundary of the support of the equilibrium measure. This can be stated in terms of convergence of the random electrostatic potential to a Gaussian Free Field. Our result is the first to be valid at arbitrary temperature and at the mesoscopic scales, and we recover previous results of Ameur-Hendenmalm-Makarov and Rider-Virág concerning the determinantal case, with weaker ass…

Equidistribution of jellium energy for Coulomb and Riesz Interactions

For general dimension $d$ we prove the equidistribution of energy at the micro-scale in $\mathbb R^d$, for the optimal point configurations appearing in Coulomb gases at zero temperature. At the microscopic scale, i.e. after blow-up at the scale corresponding to the interparticle distance, in the case of Coulomb gases we show that the energy concentration is precisely determined by the macroscopic density of points, independently of the scale. This uses the "jellium energy" which was previously shown to control the next-order term in the large particle number asymptotics of the minimum energy.…

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For general radially symmetric potentials, we derive the asymptotic expansions of the log-partition functions up to and including the $O(1)$-terms as the number $N$ of particles increases. Notably, our findings stress that the formulas of the $O(\log N)$- and $O(1)$-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions prop…

Gaussian Fluctuations and Free Energy Expansion for Coulomb Gases at Any Temperature

We obtain concentration estimates for the fluctuations of Coulomb gases in any dimension and in a broad temperature regime, including very small and very large temperature regimes which may depend on the number of points. We obtain a full Central Limit Theorem for the fluctuations of linear statistics in dimension 2, valid for the first time down to microscales and for possibly very small or very large temperatures. We show that a similar CLT can also be obtained in any larger dimension conditional on a "no phase-transition" assumption, as soon as one can obtain a precise enough error rate for…

Concentration for Coulomb gases on compact manifolds

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafaï, Hardy and Maïda on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green …

Supersymmetric gauge theories, Coulomb gases and Chern-Simons matrix models

We develop Coulomb gas pictures of strong and weak coupling regimes of supersymmetric Yang-Mills theory in five and four dimensions. By relating them to the matrix models that arise in Chern-Simons theory, we compute their free energies in the large N limit and establish relationships between the respective gauge theories. We use these correspondences to rederive the N^3 behaviour of the perturbative free energy of supersymmetric gauge theory on certain toric Sasaki-Einstein five-manifolds, and the one-loop thermal free energy of N=4 supersymmetric Yang-Mills theory on a spatial three-sphere.

Coulomb and Riesz gases: The known and the unknown

We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in $\mathbb{R}^d$ interacting with the Riesz potential $\pm |x|^{-s}$ (resp. $-\log|x|$ for $s=0$). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g. in random matrix theory). The main question addressed in the article is how to properly define the associated infinite point process and characterize it using some (renormalized) equili…

通过量子吉布斯采样计算量子库仑气体和分子的自由能

Computing the free energy of quantum Coulomb gases and molecules via quantum Gibbs sampling

本文提出量子算法估算二维和三维空间中相互作用量子库仑气体和分子系统在有限温度下的自由能及总吉布斯态，通过有限秩低能截断方法解决奇异相互作用和无限维希尔伯特空间问题，首次给出吉布斯采样在连续变量量子系统中的严格混合时间保证。

The high temperature crossover for general 2D Coulomb gases

We consider $N$ particles in the plane influenced by a general external potential that are subject to the Coulomb interaction in two dimensions at inverse temperature $β$. At large temperature, when scaling $β=2c/N$ with some fixed constant $c>0$, in the large-$N$ limit we observe a crossover from Ginibre's circular law or its generalization to the density of non-interacting particles at $β=0$. Using several different methods we derive a partial differential equation of generalized Liouville type for the crossover density. For radially symmetric potentials we present some asymptotic results an…

库仑气体的若干方面

Aspects of Coulomb gases

本文以说明性笔记形式介绍库仑气体的若干数学方面，重点讨论精确可解性及一阶、二阶全局渐近，并特别关注 Ginibre 等二维随机矩阵模型。

2D Coulomb Gases and the Renormalized Energy

We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case β=\infty corresponds to "weighted Fekete sets" and also falls within our analysis. It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Ben Arous and Zeitouni. By a suitable splitting of the Hamiltonian,…

On the extreme value statistics of normal random matrices and 2D Coulomb gases: Universality and finite N corrections

In this paper we extend the orthogonal polynomials approach for extreme value calculations of Hermitian random matrices, developed by Nadal and Majumdar [1102.0738], to normal random matrices and 2D Coulomb gases in general. Firstly, we show that this approach provides an alternative derivation of results in the literature. More precisely, we show convergence of the rescaled eigenvalue with largest modulus of a normal Gaussian ensemble to a Gumbel distribution, as well as universality for an arbitrary radially symmetric potential. Secondly, it is shown that this approach can be generalised to …

带径向对称势和硬壁的二维库仑气体的配分函数

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

该研究针对带径向对称势的二维库仑气体可解模型，探究液滴处硬壁位置对配分函数大$N$渐近展开的影响，发现不同硬壁位置下$\text{log}\thinspace N$阶项系数有差异，还得到了$\text{sqrt}(N)$阶及常数项的普适数值常数。

Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

We investigate a one-parameter family of Coulomb gases in two dimensions, which are confined to an ellipse, due to a hard wall constraint, and are subject to an additional external potential. At inverse temperature $β=2$ we can use the technique of planar orthogonal polynomials, borrowed from random matrix theory, to explicitly determine all $k$-point correlation functions for a fixed number of particles $N$. These are given by the determinant of the kernel of the corresponding orthogonal polynomials, which in our case are the Gegenbauer polynomials, or a subset of the asymmetric Jacobi polyno…

带圆形根型与跃变型奇点的二维库仑气体的配分函数

Partition functions of two-dimensional Coulomb gases with circular root- and jump-type singularities

本文研究带径向对称势的随机正规矩阵特征值模对应的随机多项式，推导其矩生成函数的大n渐近展开，证实相关猜想，给出带圆形奇点的二维库仑气体首个自由能展开并得到a=0情形的新结果。

Higher Dimensional Coulomb Gases and Renormalized Energy Functionals

We consider a classical system of n charged particles in an external confining potential, in any dimension d larger than 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter scales like the inverse of n (mean-field scaling). By a suitable splitting of the Hamiltonian, we extract the next to leading order term in the ground state energy, beyond the mean-field limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new "renormalized energy" functional providing a way to compute the total Coulomb energy o…

Priors leading to well-behaved Coulomb and Riesz gases versus zeroth-order phase transitions -- a potential-theoretic characterization

We give a potential-theoretic characterization of priors which have the property that the corresponding Coulomb gas is "well-behaved" and similarly for more general Riesz gases. This means that the laws of the empirical measures of the corresponding random point process satisfy a Large Deviation Principle with a rate functional which depends continuously on the temperature, in the sense of Gamma-convergence. Equivalently, there is no zeroth-order phase transition at zero temperature, in the mean field regime. This is shown to be the case for the Hausdorff measure on a compact Lipschitz hypersu…

Point processes, hole events, and large deviations: random complex zeros and Coulomb gases

We consider particle systems (also known as point processes) on the line and in the plane, and are particularly interested in "hole" events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDP-inspired techniques to the study of hole probabilities, and the determination of the most likely configurations of particles that have large holes. As an application of this approach…

Coulomb gases under constraint: some theoretical and numerical results

We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a larg…

Universality of the number variance in rotational invariant two-dimensional Coulomb gases

An exact map was established by Lacroix-A-Chez-Toine, Majumdar, and Schehr in [44] between the $N$ complex eigenvalues of complex non-Hermitian random matrices from the Ginibre ensemble, and the positions of $N$ non-interacting Fermions in a rotating trap in the ground state. An important quantity is the statistics of the number of Fermions $\mathcal{N}_a$ in a disc of radius $a$. Extending the work [44] covering Gaussian and rotationally invariant potentials $Q$, we present a rigorous analysis in planar complex and symplectic ensembles, which both represent 2D Coulomb gases. We show that the …

平面库仑气体的空穴概率与测度的扫除

Hole probabilities and balayage of measures for planar Coulomb gases

研究二维库仑气体空穴概率，当点数\(n\to\infty\)时，证明无点在\(U\)中的概率为\(\exp(-Cn^{2}+o(n^{2}))\)，通过平衡测度等确定\(C\)，给出\(\nu\)和\(C\)显式表达式例子，推广前人结果。

On the Gross-Pitaevskii equation for trapped dipolar quantum gases

We study the time-dependent Gross-Pitaevskii equation describing Bose-Einstein condensation of trapped dipolar quantum gases. Existence and uniqueness as well as the possible blow-up of solutions are studied. Moreover, we discuss the problem of dimension-reduction for this nonlinear and nonlocal Schrodinger equation.

带点电荷的二次场中行列式库仑气体的自由能展开

Free energy expansion of determinantal Coulomb gases in the quadratic fields with a point charge

研究复平面上与外部势Q(z)=1/(1-τ^2)(|z|^2-τ Re z^2)-2c log|z-a|相关的行列式库仑气体，在液滴单连通或双连通区域推导出包含常数项的自由能展开，所有系数显式计算，推广了各向同性情况τ=0的结果，并将常数项与液滴的Liouville作用量关联。

Canonical and grand canonical partition functions of Dyson gases as tau-functions of integrable hierarchies and their fermionic realization

The partition function for a canonical ensemble of 2D Coulomb charges in a background potential (the Dyson gas) is realized as a vacuum expectation value of a group-like element constructed in terms of free fermionic operators. This representation provides an explicit identification of the partition function with a tau-function of the 2D Toda lattice hierarchy. Its dispersionless (quasiclassical) limit yields the tau-function for analytic curves encoding the integrable structure of the inverse potential problem and parametric conformal maps. A similar fermionic realization of partition functio…

Canonical partition functions: ideal quantum gases, interacting classical gases, and interacting quantum gases

In statistical mechanics, for a system with fixed number of particles, e.g., a finite-size system, strictly speaking, the thermodynamic quantity needs to be calculated in the canonical ensemble. Nevertheless, the calculation of the canonical partition function is difficult.\textbf{ }In this paper, based on the mathematical theory of the symmetric function, we suggest a method for the calculation of the canonical partition function of\ ideal quantum gases, including ideal Bose, Fermi, and Gentile gases. Moreover, we express the canonical partition functions of interacting classical and quantum …

Large Deviation Principle for Empirical Fields of Log and Riesz Gases

We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the …

库仑气体的线性统计量：高阶累积量

Linear statistics for Coulomb gases: higher order cumulants

本文研究旋转不变库仑气体中光滑线性统计量的高阶累积量，证明其在大N极限下仅由液滴边界处函数导数值决定，并给出二维Ginibre系综的替代推导及大偏差形式。

The Kramers Problem for Quantum Fermi Gases with Velocity - Dependent Collision Frequency and Specular - Diffusive Boundary Conditions

The classical Kramers problem of the kinetic theory is solved. The Kramers problem about isothermal sliding for quantum Fermi gases is considered. Quantum gases with the velocity - dependent collision frequency are considered. Specular - diffusive boundary conditions are applied. Dependence of isothermal sliding on the resulted chemical potential is found out.

Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases

We study the stability of the standing wave solutions of a Gross-Pitaevskii equation describing Bose-Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions.

Existence of solitary waves in dipolar quantum gases

We study a nonlinear Schroedinger equation arising in the mean-field description of dipolar quantum gases. Under the assumption of sufficiently strong dipolar interactions, the existence of standing waves, and hence solitons, is proved together with some of their properties. This gives a rigorous argument for the possible existence of solitary waves in Bose-Einstein condensates, which originate solely due to the dipolar interaction between the particles.

The Kramers Problem for Quantum Fermi Gases with Velocity--Dependent Collision Frequency and Diffusive Boundary Conditions

The classical Kramers problem of the kinetic theory is analytically solved. The Kramers problem about isothermal sliding for quantum Fermi gases is considered. Quantum gases with the velocity-dependent collision frequency are considered. Diffusive boundary conditions are applied. Dependence of isothermal sliding on the resulted chemical potential is found out.

Supersymmetric soliton solution in (1+1)-dimensional Ultracold Quantum Gases

We obtain by dimensional reduction a $(1+1)$ supersymmetric system introduced in the description of ultracold quantum gases. The correct supercharges are identified and their algebra is constructed. Finally novel static self-dual solutions emerge, satisfying a Liouville type differential equation.