Time-Dependent Orthogonal Polynomials and Theory of Soliton –Applications to Matrix Model, Vertex Model and Level Statistics

Supersymmetric vertex models with domain wall boundary conditions

“…In [2,3] it was found that the partition functions of the DW model can be represented as determinants. Taking the homogenous limit of the spectral parameters, Sogo found that the partition function satisfies the Toda differential equations [4]. Then by using the equations Korepin and Zinn-Justin obtained the bulk free energy of the system [5].…”

p‐adic difference‐difference Lotka‐Volterra equation and ultra‐discrete limit

“…By quantization of the orbits, there appears quantum chaos and, as it is very mysterious, its partition function has a very similar structure as zeta functions in number theory [1,22]. (Level statistics in quantum chaos is also connected with integrable systems [21].) Using the resemblance of zeta functions, Connes proposed a kind of unification of number theory and quantum statistical physics in order to solve the Riemannian conjecture of the zeta-function [2,5].…”

On density of state of quantized Willmore surface - a way to a quantized extrinsic string in

“…Recently I exactly quantized the elastica of the Bernoulli-Euler functional (1-4) preserving its local length [25]. Then I found that its moduli is completely represented by the MKdV equation and closely related to the two-dimensional quantum gravity [28][29][30]. The quantized elastica obeys the MKdV equation and at a critical point, the Painlevé equation of the first kind appears [25] while in the quantized two-dimensional gravity which is defined at a critical point of the discrete tiling model, there appears the Painlevé equation of the first kind with the KdV hierarchy [28][29][30].…”

“…Hence my result (3-2) can be extended to such a case and then it means that the algorithm of the calculation of the partition function of the extrinsic string in R 3 is essentially the same as above arguments. In other words, the quantization of the string immersed in R 3 can be partially performed even though only the string in R n n < 3 had been studied as the two dimensional gravity [28][29][30].…”