We consider the two matrix model with an even quartic potential W(y)=y^4/4+alpha y^2/2 and an even polynomial potential V(x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M_1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 x 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M_1. Our results generalize earlier results for the case alpha=0, where the external field on the third measure was not present.
Conserved quantities in a bosonic tight-binding chain with non-Hermitian quartic terms
The University of Tokyo（东京大学）；Institute for Physics of Intelligence, The University of Tokyo（东京大学智能物理研究所）；Trans-Scale Quantum Science Institute, The University of Tokyo（东京大学跨尺度量子科学研究所）
A critical phenomenon in the two-matrix model in the quartic/quadratic case
本文研究了二矩阵模型在四次/二次情形下特征值统计量的一个临界行为。对于某些参数，其中一个矩阵的特征值分布有一个极限，该极限在支撑的内部以指数1/2消失。本文的主要结果是一个新的内核，描述了当地的特征值相关性附近的临界点。该核表示为与Painlevé II方程的Hastings-McLeod解相关的4 x 4 Riemann-Hilbert问题。然后，我们比较新的内核与其他两个关键的现象，出现在文献中之前。首先，我们表明，临界内核的情况下出现的二次消失的限制特征值分布可以从新的内核检索通过一个双标度极限。其次，我们简要地讨论了最近分析的非碰撞布朗运动中的结奇异性与结奇异性的关系。虽然该模型中的极限密度也在某个内点以1/2的指数消失，但局部尺度下的过程与我们在双矩阵模型中得到的过程不同。
We study the eigenvalue trajectories of a time dependent matrix $ G_t = H+i t vv^*$ for $t \geq 0$, where $H$ is an $N \times N$ Hermitian random matrix and $v$ is a unit vector. In particular, we establish that with high probability, an outlier can be distinguished at all times $t>1+N^{-1/3+ε}$, for any $ε>0$. The study of this natural process combines elements of Hermitian and non-Hermitian analysis, and illustrates some aspects of the intrinsic instability of (even weakly) non-Hermitian matrices.
The partition functions of Hermitian one-matrix models are known to be tau-functions of the KP hierarchy. In this paper we explicitly compute the elements in Sato grassmannian these tau-functions correspond to, and use them to compute the $n$-point functions of Hermitian one-matrix model.
We consider two different genus expansions of the free energy functions of Hermitian one-matrix models, one using fat graphs, one using ordinary graphs (thin graphs). Some structural results are first proved for the thin version of genus expansion using renormalized coupling constants, and then applied to the fat version.
We use the Riemann-Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers $\mathscr{N}_j(g)$ of $4$-valent connected graphs with $j$ vertices on a compact Riemann surface of genus $g$. We explicitly evaluate these numbers for Riemann surfaces of genus $0,1,2,$ and $3$. Also, for a Riemann surface of an arbitrary genus $g$, we also calculate the leading term in the asymptotics of $\mathscr{N}_j(g)$ as the number of vertices tends to in…
We use genus zero free energy functions of Hermitian matrix models to define spectral curves and their special deformations. They are special plane curves defined by formal power series with integral coefficients generalizing the Catalan numbers. This is done in two different versions, depending on two different genus expansions, and these two versions are in some sense dual to each other.
A Novel Approach to Non-Hermitian Random Matrix Models
本文提出了一种研究非埃尔米特随机矩阵系综谱性质的新方法。类似于复格林函数通过不连续性对厄米特系综的实谱进行编码，这里提出的格林函数的四元数扩展在非厄米特系综的情况下直接导致复谱，并对特征向量的一些谱特性进行编码。四元数的标准二乘二矩阵表示导致了最近在图解方法的背景下提出的所谓矩阵值预解式的推广[1-6]。我们证明了四元数格林函数服从自由变量微积分[7，8]。特别地，矩阵格林函数的四元数函数逆，在[9]之后称为Blue函数，服从简单加法定律，正如不久前观察到的那样[1，3]。利用这一定律，我们推导出新的、通用的、算法性的和有效的方法来寻找所有形式为H+ iH '的非埃尔米特系综的非全纯格林函数，其中系综H和H'是独立的（在Voiculescu [7]的意义上是自由的）埃尔米特系综任意测量。我们证明了一个简单的重新推导的非厄米随机矩阵模型的几个例子的谱特性的方法的力量。
This paper provides the constructive loop vertex expansion for stable matrix models with (single trace) interactions of arbitrarily high even order in the Hermitian and real symmetric cases. It relies on a new and simpler method which can also be applied in the previously treated complex case. We prove analyticity in the coupling constant of the free energy for such models in a domain uniform in the size of the matrix
We investigate the existence and properties of a double asymptotic expansion in $1/N^{2}$ and $1/\sqrt{D}$ in $\mathrm{U}(N)\times\mathrm{O}(D)$ invariant Hermitian multi-matrix models, where the $N\times N$ matrices transform in the vector representation of $\mathrm{O}(D)$. The crucial point is to prove the existence of an upper bound $η(h)$ on the maximum power $D^{1+η(h)}$ of $D$ that can appear for the contribution at a given order $N^{2-2h}$ in the large $N$ expansion. We conjecture that $η(h)=h$ in a large class of models. In the case of traceless Hermitian matrices with the quartic tetr…
This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and $W$-representation. We apply it to prove the formula for the Jack functions averages - appropriate analogue of characters for the $β$-deformed Hermitian Gaussian matrix model. We also sketch the construction of $W$-operators from Calogero-Ruijsenaars H…
We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the c…
Given a diagonalizable $N\times N$ matrix $H$, whose non-degenerate spectrum consists of $p$ pairs of complex conjugate eigenvalues and additional $N-2p$ real eigenvalues, we determine all metrics $M$, of all possible signatures, with respect to which $H$ is pseudo-hermitian. In particular, we show that any compatible $M$ must have $p$ pairs of opposite eigenvalues in its spectrum so that $p$ is the minimal number of both positive and negative eigenvalues of $M$. We provide explicit parametrization of the space of all admissible metrics and show that it is topologically a $p$-dimensional torus…
