MATH 7241 • Probability 1 (graduate)
Integrable Probability: Random Polymers, Random Tilings, and Interacting Particle Systems
One of the versions of my “job talk” in Fall 2013 describing many facets of integrable probability. It is supposed to be accessible. The talk is based on [15] and on the previous two talks. More details can also be found in arXiv:1106.1596 [math.PR] by Corwin and arXiv:1212.3351 [math.PR] by Borodin and Gorin.
Integrable probability: From representation theory to Macdonald processes
These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the $(q,t)$-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer’s partition function.
MATH 3081 • Probability and Statistics (2 sections)
Spectral theory for the q-Boson particle system
We develop spectral theory for the generator of the $q$-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system.
The q-PushASEP: A New Integrable Model for Traffic in 1+1 Dimension
We introduce a new interacting (stochastic) particle system $q$-PushASEP which interpolates between the q-TASEP introduced by Borodin and Corwin (see arXiv:1111.4408, and also arXiv:1207.5035; arXiv:1305.2972; arXiv:1212.6716) and the $q$-PushTASEP introduced recently by Borodin and Petrov (arXiv:1305.5501).
Markov Dynamics on Interlacing Arrays
The talk is devoted to the construction of Markov dynamics on interlacing arrays which act nicely on the Macdonald measures. In a particular case one gets $q$-deformed Robinson-Schensted insertion algorithms. The talk is based on [12], [13], see also [14], [15].
Nearest neighbor Markov dynamics on Macdonald processes
Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin (arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters $(q,t)$, where $0\le q, t < 1$. Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes.