PushTASEP in inhomogeneous space
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
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
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.
Correction to "Stochastic higher spin vertex models on the line"
(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.
Correction to "Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz"
This is a correction to Theorems 7.3 and 8.12 in our paper. These statements claimed to deduce the spatial Plancherel formula (spatial biorthogonality) of the ASEP and XXZ eigenfunctions from the corresponding statements for the eigenfunctions of the q-Hahn system. Such a reduction is wrong.