# Stochastic Integrable Systems Reading Seminar 2013-14

(Covering topics related to random matrices, representation theory, integrable systems and interacting stochastic particle systems)

### Term 1 - 2013/14. KPZ equation and its universality class

• 01/10/2013. Nikos Zygouras, Introduction to KPZ and its universality class, the dicussion of the reading list
• 08/10/2013. Jon Warren, THE KARDAR-PARISI-ZHANG EQUATION AND UNIVERSALITY CLASS (review paper by I. Corwin)
• 21/10/2013. Pierre Le Doussal (CNRS), A solution for the KPZ equation with flat initial conditions
• 29/10/2013. Barnaby Garrod, Random Matrices and Determinantal Processes I
• 05/11/2013. Mihail Poplavskyi, Random Matrices and Determinantal Processes II
• 12/11/2013. Ioanna Nteka, The class of Schur measures
• 19/11/2013. Jere Koskela, Markov dynamics on Gelfan-Tsetlin patterns

### Term 2 - 2013/14. q-TASEP, ASEP, Macdonald processes

• 14/01/2014. Roger Tribe, FROM DUALITY TO DETERMINANTS FOR Q-TASEP AND ASEP (after Borodin, Corwin, Sasamoto)
• 21/01/2014. Nikos Zygouras, On the Macdonald Processes by Borodin-Corwin
• 28/01/2014. Oleg Zaboronski. ODE's for product moments in ASEP - after Borodin, Corwin, Sasamoto; Seminar notes
• 04/02/2014. Cyril Labbe. Continuous limit of ASEP and KPZ, after Bertini, Giacomin
• 11/02/2014. Oleg Zaboronski. Solving ODE's for ASEP product moments - after Borodin, Corwin, Sasamoto
• 18/02/2014. Oleg Zaboronski. Solving ODE's for ASEP product moments (initial conditions) - after Borodin, Corwin, Sasamoto
• 25/02/2014. Chin Lun. Fredholm determinants for tau-exponential moments of ASEP - after Borosdin, Corwin, Sasamoto
• 04/03/2014. Paul Chleboun. Deriving duality relations for Exclusion Processes using a ‘Quantum Hamiltonian’ approach – After G. M. Schuetz
• 11/3/2014. Rostyslav Kozhan. Resonances associated with random matrices.

Abstract: We employ the theory of orthogonal polynomials on the unit
circle to compute the eigenvalue distribution of truncated unitary
random matrices (with one row and column removed). These appear as
scattering resonances of open quantum systems. We also show that zeros
of orthogonal polynomials with decaying random Verblunsky coefficients
asymptotically behave like these eigenvalues. Finally, we will briefly
discuss the distribution of the resonances of Hermitian random matrices
coupled to the discrete Laplacian on the lattice $Z_+$. We reduce this
problem to the resonance problem for Jacobi operators and use the theory
of orthogonal polynomials on the real line. Joint work with Rowan Killip.

### Term 3 - 2013/14. Duality and symmetries of Markov processes

• 29/04/2014. Bruce Westbury. Uq(SL2). Notes
• 06/05/2014. Bruce Westbury. Uq(SL2) and models of statistical physics. Notes
• 13/05/2014. Bruce Westbury, Uq(SL2) and ASEP.

Abstract. The first part of the talk continues on the theme of the representation
theory discussed last week. Last week I showed that for S a two dimensional vector space and L > 0

(large), the L-th tensor power has an action of the quantum group
U_q(sl(2)) and an action of the Temperley-Lieb algebra and that these
actions commute. This week I will describe these actions in an
alternative basis. This is taken from

Frenkel, Igor B., and Mikhail G. Khovanov.
“Canonical Bases in Tensor Products and Graphical Calculus for
$U_q(\germ S\germ l_2)$.”
Duke Mathematical Journal 87, no. 3 (1997): 409–80.

The second part of the talk moves on to ASEP. I will discuss the badly named concept of "duality" following Roger's
talk last term. I will discuss the construction of the model and write the generator in
terms of the Temperley-Lieb operators defined last week. Finally I will discuss the example of a duality for ASEP given in:

Schütz, Gunter M. “Duality Relations for Asymmetric Exclusion Processes.”
Journal of Statistical Physics 86, no. 5–6 (1997): 1265–87.

• 20/05/2014. YACINE BARHOUMI-ANDRÉANI, ON THE COEFFICIENTS OF THE CHARACTERISTIC POLYNOMIAL OF A
RANDOM UNITARY MATRIX.Abstract (with references)
• 10/06/2014. Li-Cheng Tsai (Guest speaker) Dyson's Brownian Motion with Infinitely Many Particles.

Abstract: Dyson's Brownian Motion (DBM) describes the evolution of the
eigenvalues of random matrices driven by i.i.d. Brownian motions. It
forms a system of diffusions with singular long range interactions. In
this talk I will describe a proof the strong existence and pathwise
uniqueness of DBM with infinitely many particles (integer indexed),
corresponding to the bulk limit. This result allows a dynamical
description of this infinite system, and it applies for a general set
of initial configurations

• 17/06/2014. Partha Dey, High temperature scaling limits of directed polymers with heavy tail disorder.

Abstract: The directed polymer model at intermediate disorder regime was introduced by Alberts-Khanin-Quastel. They proved that at inverse temperature $\beta * n^{-c}$ with $c=1/4$ the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was performed under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. In a work with Nikos Zygouras, we show that this conjecture is valid and we further extend it by exhibiting the non-universal limiting behaviour in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of $c$.

1) A survey on KPZ.

http://arxiv.org/pdf/1106.1596.pdf

2) Determinantal processes related to paths and various models in the KPZ universality

http://arxiv.org/pdf/math-ph/0510038.pdf

http://arxiv.org/pdf/1301.7450.pdf

3) Some Fredholm and Pfaffian representations in KPZ

http://arxiv.org/pdf/1208.5669.pdf

http://arxiv.org/pdf/1206.4573.pdf

4) Symmetric functons / Schur & Macdonald functions, ASEP & q-TASEP

http://arxiv.org/pdf/math/0503508.pdf

http://arxiv.org/pdf/1111.4408.pdf

http://arxiv.org/pdf/1207.5035.pdf

5) (geometric) RSK correspondence

http://arxiv.org/pdf/1308.4631.pdf