Workshop ‘New Topics on Stochastic and Quantum
Interacting Particle Systems’
10 December 2013
All talks will take place in Room 338, Faculty of Science Building No.1,
University of Tokyo
Organisers: Seiji Miyashita (Tokyo) and Makoto Katori (Chuo)

PROGRAMME (version 2)
10:00-10:10 (10 min.) Seiji MIYASHITA (Univ. of Tokyo)
Opening address
10:10-11:00 (50 min.) Makoto KATORI (Chuo Univ.)
An elliptic extension of Dyson’s Brownian motion model-11:10 (10 min.) discussion
11:10-12:00 (50 min.) Nizar DEMNI (Univ. de Rennes)
I. Dunkl processes with positive multiplicities and reﬂected BM in Weyl chambers-12:10 (10 min.) discussion[12:10-13:30 (80 min.) lunch]
13:30-14:00 (30 min.) Sergio ANDRAUS (Univ. of Tokyo)
Limiting regimes of interacting particle systems using Dunkl operators-14:10 (10 min.) discussion
14:10-15:00 (50 min.) Nizar DEMNI (Univ. de Rennes)
II. Jacobi process: from the univariate process to the free one-15:10 (10 min.) discussion
15:10-16:00 (50 min.) Yutaka MATSUO (Univ. of Tokyo)
Dunkl operator and degenerate double aﬃne Hecke algebra in supersymmetric gauge theories-16:10 (10 min.) discussion[16:10-16:30 (20 min.) coﬀee break]
16:30-17:00 (30 min.) Chihiro MATSUI (Univ. of Tokyo)
Multi-state extension of the asymmetric simple exclusion processes-17:10 (10 min.) discussion
17:10-17:40 (30 min.) Takashi MORI (Univ. of Tokyo)
Thermodynamic limit in long-range interacting systems-17:50 (10 min.) discussion[18:30-20:30 A buﬀet supper will be prepared.]1

Workshop ‘New Topics on Stochastic and Quantum
Interacting Particle Systems’ABSTRACTS
Makoto KATORI (Chuo Univ.)
An elliptic extension of Dyson’s Brownian motion model
We introduce an elliptic extension of Dyson’s Brownian motion model, which is a temporally
inhomogeneous stochastic process of noncolliding diﬀusion particles deﬁned on a circle. Using
elliptic determinant evaluations related to the reduced aﬃne root system of type A (RosengrenSchlosser (2006)), we give a determinantal martingale representation of the process and provethat the process is determinantal. By taking the temporally homogeneous limit of the present
elliptic determinantal process, equilibrium and nonequilibrium dynamics of noncolliding diﬀusion processes are studied.
Nizar DEMNI (Univ. de Rennes)
I. Dunkl processes with positive multiplicities and reﬂected BM in Weyl chambers
I’ll show how one proves that the radial Dunkl process is the unique strong solution of a singular
SDE when all the multiplicity values are positive. While a ﬁrst proof was given by B. Schapira,the one discussed here relies on multivoque SDEs and has the merit to answer a conjectureof Gallardo and Yor about the the ﬁniteness of the total sum of the jumps performed by theDunkl process. In this respect, I’ll also exhibit related unpublished results by D. Lepingle.
When the multiplicity function vanishes, we derive a Tanaka type SDE for the reﬂected BM
in a Weyl chamber. I’ll give a necessary and suﬃcient condition ensuring that the boundary
process is the sum of local times of real BMs on the walls of the chamber.
II. Jacobi process: from the univariate process to the free one
I’ll start with the relation between the Jacobi process and projections of the Brownian motion
on spheres. This aspect allows to deﬁne the matrix-Jacobi process as the radial part of anupper-left corner of a BM on the unitary group. The eigenvalues of this process are shownto evolve like particles in the Weyl alcove of type BC and is a conditioned process in Doob’s
sense. I’ll give another construction of the eigenvalues process as a limit of a markov chain on
the Gelfand-Tsetlin graph. Finally, I’ll describe the asymptotics of the moments of the matrix
process as the size tends to inﬁnity
Sergio ANDRAUS (Univ. of Tokyo)
Limiting regimes of interacting particle systems using Dunkl operators
We examine the extension of Dyson’s Brownian motion and the Wishart-Laguerre processes toreal positive values of the parameter beta, which we refer to as interacting Brownian motions
and interacting Bessel processes. Using analytical and numerical techniques, we investigate theform of these systems’ time-scaled particle distributions in the steady state, showing that theycorrespond to the eigenvalue distribution of the beta-Hermite and beta-Laguerre ensembles ofrandom matrices, and we give estimates of their relaxation time. We also prove that in thefreezing limit (when beta tends to inﬁnity), the particle distribution of these systems becomesa sum of delta functions located at the zeroes of the Hermite and Laguerre polynomials. Our
analysis is carried out using Dunkl operators, and in particular, Dunkl’s intertwining operators.
We obtain our results by deriving previously unknown expressions for the intertwining operators2

of type A and B.
[1] SA, M. Katori, S. Miyashita, J. Phys. A 45 395201 (2012)[2] SA, M. Katori, S. Miyashita, arXiv:1309.2733
Yutaka MATSUO (Univ. of Tokyo)
Dunkl operator and degenerate double aﬃne Hecke algebra in supersymmetric gauge theories
We review the the role of Dunkl operators and the symmetry (degenerate double aﬃne Heckealgebra) in the instanton moduli space of supersymmetric gauge theory. Recently it reveals
an interesting relation between the partition function of 4D gauge theories and 2D integrablemodels.
Chihiro MATSUI (Univ. of Tokyo)
Multi-state extension of the asymmetric simple exclusion processes
There are few far-from-equilibrium systems which are exactly solvable. One of those few examples is the asymmetric simple exclusion process, the one-dimensional stochastic process withvolume exclusion. We consider the multi-state extension of the asymmetric simple exclusion
process based on the fact that the Markov matrix of this process satisﬁes the algebraic relationsof the Temperley-Lieb algebra. Besides the steady states, we derive the exact expressions ofparticle-density proﬁles and particle currents on the steady states under the closed boundary
conditions.
Takashi MORI (Univ. of Tokyo)
Thermodynamic limit in long-range interacting systems
Existence of thermodynamic limit is a fundamental result of statistical mechanics in short-range
interacting systems. Its conventional proof [1] is valid only for short-range interacting systems,
so giving the proof of existence of nontrivial thermodynamic limit for long-range interacting
systems would be important and interesting. I will explain the proof brieﬂy and derive theuseful variational expressions of the entropy and the free energy.
[1] D. Ruelle, ”Statistical Mechanics: Rigorous Results”

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