Related papers: Tau functions as Widom constants
Generic c=1 four-point conformal blocks on the Riemann sphere can be seen as the coefficients of Fourier expansion of the tau function of Painlev\'e VI equation with respect to one of its integration constants. Based on this relation, we…
We study families of quantum field theories of free bosons on a compact Riemann surface of genus g. For the case g > 0 these theories are parameterized by holomorphic line bundles of degree g-1, and for the case g=0 - by smooth closed…
We provide a determinantal formula for tau-functions of the KP hierarchy in terms of rectangular, constant matrices $A$, $B$ and $C$ satisfying a rank one condition. This result is shown to generalize and unify many previous results of…
We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus one mirror curve these are conjectured to be the q-difference Painlev\'e…
A class of Riemann-Hilbert problems corresponding to quasi-permutation monodromy matrices is solved in terms of Szeg\"o kernel on auxiliary Riemann surfaces. The tau-function of Schlesinger system turns out to be closely related to…
In this paper we have two issues coming from the same background. The first one is to describe a certain ratio of Fredholm determinants of integral operators arising from the Riemann zeta function by using the solution of a single integral…
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…
We propose $q$-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlev\'e $\tau$ function. Namely we propose formula for $\tau$ function for $q$-difference Painlev\'e equation corresponding to $A_7^{(1)}{}'$ surface (and $A_1^{(1)}$…
In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with quasi-permutation monodromy groups which correspond to non-singular branched coverings of $\CP1$. The solution is given in terms of Szeg\"o kernel on the…
We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce the matrix integral to the…
We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz…
We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green…
It is well known that tau functions of the KP hierarchy satisfy addition formulas. We consider the general addition formula in the determinant form and take a certain limit of it. It expresses certain shifts of a tau function in terms of…
Given a semisimple Frobenius manifold, we construct a class of integrable deformations of its hierarchy of topological type. We show that these integrable deformations have polynomial tau-structures, and conjecture that for the…
In this paper we find fractional Riemann-Liouville derivatives for the Takagi-Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi-Landsberg functions which have arbitrary bounded coefficients in the…
The paper contains some new results and a review of recent achievements, concerning the multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by…
The probability for the exclusion of eigenvalues from an interval $(-x,x)$ symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter $ a $ (a…
In this paper, we investigate approximate solutions for nonlinear Fredholm integro-differential equations of fractional order. We present an operational Tau method by obtaining the Tau matrix representation. We solve a special class of…
Non-perturbative partition functions of quantum theories constitute a class of $\tau-$functions, which are distinguished satisfying Hirota's bilinear identities(BI). To make this statement general, there must be a proper definition of…
Connections between classical and quantum integrable systems are analyzed from the viewpoint of Slavnov products of Bethe states. It is well known that, modulo model dependent aspects, the functional structure of Slavnov products generally…