Universality of local statistics for noncolliding random walks
We consider the $N$-particle noncolliding Bernoulli random walk — a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in{0,1}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$.
Lectures on Integrable probability: Stochastic vertex models and symmetric functions
We consider a homogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes.
Higher spin six vertex model and symmetric rational functions
We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes.
MATH 8380 • Random Matrices
MATH 3100 • Introduction to Probability
q-randomized Robinson-Schensted-Knuth correspondences and random polymers
We introduce and study $q$-randomized Robinson-Schensted-Knuth (RSK) correspondences which interpolate between the classical ($q=0$) and geometric ($q\to 1$) RSK correspondences (the latter ones are sometimes also called tropical).
Simulations of stochastic higher spin vertex model
The stochastic higher spin vertex model introduced in my paper with Ivan Corwin ([arXiv:1502.07374 [math.PR]][ivan6v]) generalizes the stochastic six vertex model considered by Borodin, Corwin, and Gorin, arXiv:1407.6729 [math.PR]. Here are some simulations related to this model.
Dear colleagues:
Feel free to use these pictures to illustrate your research in talks and papers, with attribution (CC BY-SA 4.0 (opens in new tab)). Some of the images are available in very high resolution upon request.