A sample of the stationary dynamics on TASEP distribution

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.

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First page of the course notes

Course notes on random matrices

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

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First page of the course notes

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

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Multitime distribution in the backwards Hammersley-type process

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.

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Erratum to "Stochastic higher spin vertex models on the line"

[2019/05/19]

(PDF)

This is an erratum to the paper “Stochastic higher spin vertex models on the line”. The aim of the note is to address two separate errors in the paper: finite vertical spin Plancherel identities, and a false duality claim. The other main statements of the paper (the definition of new stochastic particle systems, duality relations for them, and contour integral observables) are not affected.

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Yang-Baxter equation and its bijectivisation

Yang-Baxter random fields and stochastic vertex models

[2019/05/15]

Bijectivization refines the Yang-Baxter equation into a pair of local Markov moves which randomly update the configuration of the vertex model. Employing this approach, we introduce new Yang-Baxter random fields of Young diagrams based on spin $q$-Whittaker and spin Hall-Littlewood symmetric functions. We match certain scalar Markovian marginals of these fields with (1) the stochastic six vertex model; (2) the stochastic higher spin six vertex model; and (3) a new vertex model with pushing which generalizes the $q$-Hahn PushTASEP introduced recently by Corwin-Matveev-Petrov (2018). Our matchings include models with two-sided stationary initial data, and we obtain Fredholm determinantal expressions for the $q$-Laplace transforms of the height functions of all these models. Moreover, we also discover difference operators acting diagonally on spin $q$-Whittaker or (stable) spin Hall-Littlewood symmetric functions.

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