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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
  • 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

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Multitime distribution in the backwards Hammersley-type process

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.

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



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


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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A simulation of the square ice, due to Shreyas Balaji (MIT UROP)

From infinite random matrices over finite fields to square ice

Asymptotic representation theory of symmetric groups is a rich and beautiful subject with deep connections with probability, mathematical physics, and algebraic combinatorics. A one-parameter deformation of this theory related to infinite random matrices over a finite field leads to a randomization of the classical Robinson-Schensted correspondence between words and Young tableaux. Exploring such randomizations we find unexpected applications to six vertex (square ice) type models and traffic systems on a 1-dimensional lattice.

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