Columbia / Courant Joint Probability Seminar Series:

Kardar-Parisi-Zhang Universality

Columbia University, October 4, 2013



Herbert Spohn [Interacting diffusions in the KPZ universality class]
Jeremy Quastel [Exact formulas for random growth off a flat interface]
Tea break
Leonid Petrov [Markov Dynamics on Macdonald Processes] (Slides)

Practical Information
The talks will take place at Columbia University Mathematics building in room 520, on October 4, 2013. No registration necessary. However, if you would like to attend the group lunch after the talks, please RSVP.

Directions to Columbia. Event poster.

For further information, please contact the organizers.

Titles and Abstracts

Herbert Spohn (T.U. Munich)
Interacting diffusions in the KPZ universality class

Abstract: We discuss a "chain" of Brownian motions. Nearest neighbors interact through a drift depending only on their relative displacement. The reversible case is well studied with Gaussian fluctuations of order t^{1/4}. For the non-reversible case one expects non-Gaussian fluctuations of order t^{1/3}. We explain recent results pointing towards this conjecture. This is joint work with P.L. Ferrari, T. Sasamoto, and T. Weiss.

Jeremy Quastel (U. Toronto)
Exact formulas for random growth off a flat interface

Abstract: We describe formulas for the asymmetric simple exclusion process starting from half-flat and flat initial data. In the flat case the formula can be written as a Fredholm Pfaffian. This is joint work with Janosch Ortmann and Daniel Remenik.

Leonid Petrov (Northeastern)
Markov Dynamics on Macdonald Processes

Abstract: Since the end of 1990's there has been a significant progress in understanding the long time nonequilibrium behavior of certain integrable (1+1)-dimensional interacting particle systems and random growth models in the KPZ universality class. The miracle of integrability in most cases (with the notable exception of the partially asymmetric simple exclusion process) can be traced to an extension of the Markovian evolution to a suitable (2+1)-dimensional random growth model whose remarkable properties yield the solvability.
So far, there have been two sources of such extensions. The first one originated from a classical combinatorial bijection known as the Robinson-Schensted-Knuth correspondence (RSK, for short) in the works of Johansson, O'Connell and their co-authors. The second approach introduced by Borodin-Ferrari was based on an idea of Diaconis-Fill of extending intertwined ``univariate'' Markov chains to a ``bivariate'' Markov chain that projects to either of the initial ones.
Using a fairly general framework of Macdonald processes of Borodin-Corwin, we present a way to unify these two approaches. This also provides new examples of integrable (2+1)- and (1+1)-dimensional particle systems, including 1) New Robinson--Schensted-type correspondences between words and pairs of Young tableaux; 2) The q-PushASEP (1+1)-dimensional particle system having a nice interpretation as a model of traffic on a one-lane highway in which cars are able to accelerate or slow down; 3) Randomized insertion algorithm associated with large triangular matrices over a finite field.
Based on joint works with Alexei Borodin, Alexey Bufetov, and Ivan Corwin.