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In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight $w_0(x)$ defined on an interval by $w_0(x)e^{-tx}.$ It is a well-known fact that under such a…
We study the full asymptotic expansion of the monodromy data ({\it i.e.}, Stokes multipliers) for the first Painlev\'{e} transcendent (PI) with large initial data or large pole parameters. Our primary approach involves refining the complex…
We find four kinds of six-parameter family of coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$. Each system is the first example which gave higher-order…
We study the parameter dependence of complex geodesics with prescribed boundary value and direction on bounded strongly linearly convex domains. As an important application we establish a quantitative relationship between the regularity of…
Hamiltonian Monodromy is the simplest topological obstruction to the existence of global action-angle coordinates in a completely integrable system. We show that this property can be studied in a neighborhood of a focus-focus singularity by…
We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the…
We establish the relation between two objects: an integrable system related to Painlev\'e II equation, and the symplectic invariants of a certain plane curve S(TW). This curve describes the average eigenvalue density of a random hermitian…
The squared singular values of the product of $M$ complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in…
We study the dependence of the tau function of Painlev\'e I equation on the generalized monodromy of the associated linear problem. In particular, we compute connection constants relating the tau function asymptotics on five canonical rays…
We study the Rayleigh-B{\'e}nard convection in a 2-D rectangular domain with no-slip boundary conditions for the velocity. The main mathematical challenge is due to the no-slip boundary conditions, since the separation of variables for the…
We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax…
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…
In this paper, two-to-one mappings and involutions without any fixed point on finite fields of even characteristic are investigated. First, we characterize a closed relationship between them by implicit functions and develop an AGW-like…
In this paper, we focus on the relationship between the fifth Painlev\'{e} equation and a Jacobi weight perturbed with random singularities, \begin{equation*} w(z)=\left(1-z^2\right)^{\alpha}{\rm…
We elucidate the relation between Painlev\'e equations and four-dimensional rank one ${\cal N= 2}$ theories by identifying the connection associated to Painlev\'e isomonodromic problems with the oper limit of the flat connection of the…
To any positive contraction $Q$ on $\ell^2(W)$, there is associated a determinantal probability measure ${\mathbf P}^Q$ on $2^W$, where $W$ is a denumerable set. Let $\Gamma$ be a countable sofic finitely generated group and $G = (\Gamma,…
We propose multidimensional versions of the Painlev\'e VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of…
We prove that the existence of a dispersionless Lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation (PDE) is equivalent to the canonical conformal structure defined by the symbol being…
For a given spectral curve, the theory of topological recursion generates two different families $\omega_{g,n}$ and $\omega_{g,n}^\vee$ of multi-differentials, which are for algebraic spectral curves related via the universal $x-y$ duality…
This paper offers a comprehensive treatment of the question as to whether a binary relation can be consistent (transitive) without being decisive (complete), or decisive without being consistent, or simultaneously inconsistent or…