### Parameter permutation symmetry in particle systems and random polymers

[2019/12/12]

Many integrable stochastic particle systems in one space dimension (such as TASEP - Totally Asymmetric Simple Exclusion Process - and its various deformations, with a notable exception of ASEP) remain integrable when we equip each particle $x_i$ with its own jump rate parameter $\nu_i$. It is a consequence of integrability that the distribution of each particle $x_n(t)$ in a system started from the step initial configuration depends on the parameters $\nu_j$, $j\le n$, in a symmetric way. A transposition $\nu_n \leftrightarrow \nu_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$. For $q$-Hahn TASEP and its degenerations (namely, $q$-TASEP and beta polymer) we realize the transposition $\nu_n \leftrightarrow \nu_{n+1}$ as an explicit Markov swap operator acting on the single particle $x_n(t)$. For beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are

Markov duality and contour integral formulas for joint moments. In particular, our constructions lead to a continuous time Markov process $\mathsf{Q}^{(\mathsf{t})}$ preserving the time $\mathsf{t}$ distribution of the $q$-TASEP (with step initial configuration, where $\mathsf{t}\in \mathbb{R}_{>0}$ is fixed). The dual system is a certain transient modification of the stochastic $q$-Boson system. We identify asymptotic survival probabilities of this transient process with $q$-moments of the $q$-TASEP, and use this to show convergence of the process $\mathsf{Q}^{(\mathsf{t})}$ with arbitrary initial data to its stationary distribution.

Setting $q=0$, we recover the results about the usual TASEP established recently in this paper by a different approach based on Gibbs ensembles of interlacing particles in two dimensions.

### PushTASEP in inhomogeneous space

[2019/10/20]

We consider the PushTASEP (pushing totally asymmetric simple exclusion process, also sometimes called long-range TASEP) with the step initial configuration evolving in an inhomogeneous space. That is, the rate of each particle’s jump depends on the location of this particle. We match the distribution of the height function of this PushTASEP with Schur processes. Using this matching and determinantal structure of Schur processes, we obtain limit shape and fluctuation results which are typical for stochastic particle systems in the Kardar-Parisi-Zhang universality class. PushTASEP is a close relative of the usual TASEP. In inhomogeneous space the former is integrable, while the integrability of the latter is not known.

### Mapping TASEP back in time

We obtain a new relation between the distributions $\mu_t$ at different times $t ≥ 0$ of the continuous-time TASEP (Totally Asymmetric Simple Exclusion Process) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions μt backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $\mu_t$ which in turn brings new identities for expectations with respect to $\mu_t$. Based on a joint work with Axel Saenz.

Note that the original keynote presentation contained videos which are not included in the PDF download.

### Course notes on random matrices

MATH 8380 Random Matrices (Fall 2019) • PDF course notesCourse page

### MATH 8380 • Random Matrices

[Fall 2019 semester]

### Course notes • Found typo or mistake? Let me know!

• Chapter 1. Some history
• Chapter 2. Gaussian Unitary Ensemble and Dyson Brownian Motion
• Interlude. Markov chains and stochastic differential equations
• Chapter 3. Two ways to derive the GUE eigenvalue distribution
• Chapter 4. Wigner Semicircle Law
• Chapter 5. Orbital measures and free operations
• Chapter 6. Representation-theoretic discrete analogue of the GUE
• Chapter 7. Determinantal point processes
• Chapter 8. Asymptotics via contour integrals
• Chapter 9. Harish-Chandra-Itsykson-Zuber integral and random matrix distributions
• Chapter 10. Universality
• Chapter 11. Markov maps on corners processes and generalizations

### Mapping TASEP back in time

[2019/07/04]

We obtain a new relation between the distributions $\mu_t$ at different times $t\ge 0$ of the continuous-time TASEP (Totally Asymmetric Simple Exclusion Process) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions $\mu_t$ backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to an stationary Markov dynamics preserving $\mu_t$ which in turn brings new identities for expectations with respect to $\mu_t$.

The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang-Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.

### 2020 travel

##### January

15-17 • Denver, CO • AMS Joint Mathematics Meeting

##### February

3-7 • Los Angeles, CA • Workshop on Asymptotic Algebraic Combinatorics at IPAM

27-28 • Storrs, CT • University of Connecticut

##### March

13-15 • Charlottesville, VA • 2020 Spring Southeastern Sectional Meeting at University of Virginia, with a Special Session on Integrable Probability

16-20 • New York, NY • FRG Conference on Integrable Probability at Columbia University

##### April

18-19 • Columbus, OH • Workshop on Stochastic Spatial Processes

22-24 • Madison, WI • Meeting of the Integrable Probability FRG

##### July

27-7 • Oxford, UK • CMI-HIMR Integrable Probability Summer School

##### August

17-21 • Seoul, South Korea • World Congress in Probability and Statistics