Related papers: Random matrices with external source and KP $\tau$…
We construct a family of random matrix models for the q-deformed Gaussian random variables G_\mu=a_\mu+a^\star_\mu where the annihilation operators a_\mu and creation operators a^\star_\nu fulfil the q-deformed commutation relation a_\mu…
We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^{\alpha} e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times…
We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for…
We study Matrix Quantum Mechanics on the Euclidean time orbifold $S_1/\mathbb{Z}_2$. Upon Wick rotation to Lorentzian time and taking the double-scaling limit this theory provides a toy model for a big-bang/big crunch universe in two…
In arXiv:hep-th/0310113 we started a program of creating a reference-book on matrix-model tau-functions, the new generation of special functions, which are going to play an important role in string theory calculations. The main focus of…
We present a review of the spin Hurwitz numbers, which count the ramified coverings with spin structures. They are related to peculiar $Q$ Schur functions, which are actually related to characters of the Sergeev group. This allows one to…
This manuscript describes a technique for computing partial rank-revealing factorizations, such as, e.g, a partial QR factorization or a partial singular value decomposition. The method takes as input a tolerance $\varepsilon$ and an…
In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of $1$'s in the partitions of $n$. A new expansion for Euler's partition function $p(n)$ is derived in this…
Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…
Kendall's tau and conditional Kendall's tau matrices are multivariate (conditional) dependence measures between the components of a random vector. For large dimensions, available estimators are computationally expensive and can be improved…
We consider a Gaussian random matrix theory in the presence of an external matrix source. This matrix model, after duality (a simple version of the closed/open string duality), yields a generalized Kontsevich model through an appropriate…
Adler, Shiota and van Moerbeke observed that a tau function of the Pfaff lattice is a square root of a tau function of the Toda lattice hierarchy of Ueno and Takasaki. In this paper we give a representation theoretical explanation for this…
This is a review of recent results on the integrable structure of the ordinary and modified melting crystal models. When deformed by special external potentials, the partition function of the ordinary melting crystal model is known to…
Partition functions of some two-dimensional statistical models can be represented by means of Grassmann integrals over loops living on two-dimensional torus. It is shown that those Grassmann integrals are topological invariants, which…
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…
We develop a matrix-test dual framework for $C^*$-convex families of completely positive maps $\CP(\mathscr S,\mathscr T)$, where $\mathscr S$ is an operator system and $\mathscr T$ is a unital $C^*$-algebra. Matrix tests $(k,f,s)$ induce…
We find a relationship between the partition function mass zeros and the spectral properties of the QCD Dirac operator in the context of chiral Random Matrix Theory. Introducing the concept of normal modes we see that certain features of…
We present novel equivalences in random matrix and tensor models between complex and self-adjoint theories with nontrivial quadratic terms in the action, established through an intermediate field representation. More precisely, we show that…
A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities…
We study the expansion coefficients of the tau function of the KP hierarchy. If the tau function does not vanish at the origin, it is known that the coefficients are given by Giambelli formula and that it characterizes solutions of the KP…