We present a method, based on loop equations, to compute recursively, all the terms in the large $N$ topological expansion of the free energy for the 2-hermitian matrix model, in the case where the support of the density of eigenvalues is not connected. We illustrate the method by computing the free energy of a statistical physics model on a discretized torus.
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 give expansions of reproducing kernels of the Christoffel-Darboux type in terms of Schur polynomials. For this, we use evaluations of averages of characteristic polynomials and Schur polynomials in random matrix ensembles. We explicitly compute new Schur averages, such as the Schur average in a $q$-Laguerre ensemble, and the ensuing expansions of random matrix kernels. In addition to classical and $q$-deformed cases on the real line, we use extensions of Dotsenko-Fateev integrals to obtain expressions for kernels on the complex plane. Moreover, a known interplay between Wronskians of Laguer…
In this note we examine a possible extension of the matrix integral representation of knot invariants beyond the class of torus knots. In particular, we study a representation of the SU(2) quantum Racah coefficients by double matrix integrals. We find that the Racah coefficients are mapped to expansion coefficients in some basis of double integrals. The transformed coefficients have a number of interesting algebraic properties.
We present an inversion formula which can be used to obtain resolvent expansions near embedded thresholds. As an application, we prove for a class of quantum waveguides the absence of accumulation of eigenvalues and the continuity of the scattering matrix at all thresholds.
We consider the homogeneous Bose gas on a unit torus in the mean-field regime when the interaction strength is proportional to the inverse of the particle number. In the limit when the number of particles becomes large, we derive a two-term expansion of the one-body density matrix of the ground state. The proof is based on a cubic correction to Bogoliubov's approximation of the ground state energy and the ground state.
We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $τ$-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter $\hbar$ are $τ$-functions of the $\hbar$-KP hierarchy and the expansion in $\hbar$ for the $\hbar$-KP coincides with the genus expansion for these models. Furthermore, …
We present a detailed study of the scattering system given by the Neumann Laplacian on the discrete half-space perturbed by a periodic potential at the boundary. We derive asymptotic resolvent expansions at thresholds and eigenvalues, we prove the continuity of the scattering matrix, and we establish new formulas for the wave operators. Along the way, our analysis puts into evidence a surprising relation between some properties of the potential, like the parity of its period, and the behaviour of the integral kernel of the wave operators.
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 consider the $1/N$-expansion of the moments of the proper delay times for a ballistic chaotic cavity supporting $N$ scattering channels. In the random matrix approach, these moments correspond to traces of negative powers of Wishart matrices. For systems with and without broken time reversal symmetry (Dyson indices $β=1$ and $β=2$) we obtain a recursion relation, which efficiently generates the coefficients of the $1/N$-expansion of the moments. The integrality of these coefficients and their possible diagrammatic interpretation is discussed.
We define a new large $N$ limit for general $\text{O}(N)^{R}$ or $\text{U}(N)^{R}$ invariant tensor models, based on an enhanced large $N$ scaling of the coupling constants. The resulting large $N$ expansion is organized in terms of a half-integer associated with Feynman graphs that we call the index. This index has a natural interpretation in terms of the many matrix models embedded in the tensor model. Our new scaling can be shown to be optimal for a wide class of non-melonic interactions, which includes all the maximally single-trace terms. Our construction allows to define a new large $D$ …
The generating function of the cumulants in random matrix models, as well as the cumulants themselves, can be expanded as asymptotic (divergent) series indexed by maps. While at fixed genus the sums over maps converge, the sums over genera do not. In this paper we obtain alternative expansions both for the generating function and for the cumulants that cure this problem. We provide explicit and convergent expansions for the cumulants, for the remainders of their perturbative expansion (in the size of the maps) and for the remainders of their topological expansion (in the genus of the maps). We…
We present an explicit formula for the expected value of a product of several independent symplectically invariant matrices in which the trace and real part function may be applied, possibly to different subexpressions. This takes the form of a topological expansion; however, each term has two topologies: one for the trace, and another for the real part. The traces and real parts can always be written in terms of index contraction, but in some cases, it is possible to write the expression as a product in which the two functions are applied to bracketed intervals in a legal bracket diagram. We …
Asymptotic expansion of beta matrix models in the one-cut regime
我们证明了在β矩阵模型中存在一个1/N展开到所有的阶数，该模型具有一个封闭的、非临界的势，对应于一个具有连接支撑的平衡测度。因此，展开式的系数可以通过Chekhov和Eynard的“拓扑递归”来递归地获得。我们的方法依赖于结合的先验界上的扩散和Schwinger-Dyson方程的研究，由于使用经典的复分析技术。这些先验界可以根据Boutet de Monvel，Pastur和Shcherbina的方法导出，或者对于严格凸势，可以利用测度的集中性导出。这样做，我们扩展的战略Guionnet和Maurel-Segala，从埃尔米特模型（β = 2）和微扰电位，一般β模型。Johansson以前曾考虑过1/N中第一个修正的存在，最近Kriecherbauer和Shcherbina也考虑过。在这里，通过采取类似的假设，我们将结果推广到1/N中的所有阶。
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…
We consider a class of matrix integrals over the unitary group $U(N)$ with an infinite set of couplings characterized by a series $f(q)=\sum_{n \ge 1} a_n q^n$, with $a_n \in \mathbb{Z}$. Such integrals arise in physics as the partition functions of free four-dimensional gauge theories on $S^3$ and, in particular, as the superconformal index of super Yang-Mills theory. We show that any such model can be expressed in terms of a system of free fermions in an ensemble parameterized by the infinite set of couplings. Integrating out the fermions in a given quantum state leads to a convergent expans…
We present a method to derive asymptotics of eigenvalues for trace-class integral operators $K:L^2(J;dλ)\circlearrowleft$, acting on a single interval $J\subset\mathbb{R}$, which belong to the ring of integrable operators \cite{IIKS}. Our emphasis lies on the behavior of the spectrum $\{λ_i(J)\}_{i=0}^{\infty}$ of $K$ as $|J|\rightarrow\infty$ and $i$ is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant $\det(I-γK)|_{L^2(J)}$ as $|J|\rightarrow\infty$ and $γ\uparrow 1$ in a Stokes type scaling regime. Concrete asymptotic formulæ\, are obtained f…
