4 research outputs found
Solitons and Normal Random Matrices
We discuss a general relation between the solitons and statistical mechanics
and show that the partition function of the normal random matrix model can be
obtained from the multi-soliton solutions of the two-dimensional Toda lattice
hierarchy in a special limit
SLE_k: correlation functions in the coefficient problem
We apply the method of correlation functions to the coefficient problem in
stochastic geometry. In particular, we give a proof for some universal patterns
conjectured by M. Zinsmeister for the second moments of the Taylor coefficients
for special values of kappa in the whole-plane Schramm-Loewner evolution
(SLE_kappa). We propose to use multi-point correlation functions for the study
of higher moments in coefficient problem. Generalizations related to the
Levy-type processes are also considered. The exact multifractal spectrum of
considered version of the whole-plane SLE_kappa is discussed
Constrained Reductions of 2D dispersionless Toda Hierarchy, Hamiltonian Structure and Interface Dynamics
Finite-dimensional reductions of the 2D dispersionless Toda hierarchy,
constrained by the ``string equation'' are studied. These include solutions
determined by polynomial, rational or logarithmic functions, which are of
interest in relation to the ``Laplacian growth'' problem governing interface
dynamics. The consistency of such reductions is proved, and the Hamiltonian
structure of the reduced dynamics is derived. The Poisson structure of the
rationally reduced dispersionless Toda hierarchies is also derivedComment: 18 pages LaTex, accepted to J.Math.Phys, Significantly updated
version of the previous submissio