# Law of Large Numbers for infinite random matrices over a finite field

@article{Bufetov2014LawOL, title={Law of Large Numbers for infinite random matrices over a finite field}, author={Alexey Bufetov and Leonid Petrov}, journal={Selecta Mathematica}, year={2014}, volume={21}, pages={1271-1338} }

Asymptotic representation theory of general linear groups $$\hbox {GL}(n,F_\mathfrak {q})$$GL(n,Fq) over a finite field leads to studying probability measures $$\rho $$ρ on the group $$\mathbb {U}$$U of all infinite uni-uppertriangular matrices over $$F_\mathfrak {q}$$Fq, with the condition that $$\rho $$ρ is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points (in other words…

## 18 Citations

### Infinite 𝑝-adic random matrices and ergodic decomposition of 𝑝-adic Hua measures

- Mathematics
- 2020

Neretin constructed an analogue of the Hua measures on the infinite $p$-adic matrices $Mat\left(\mathbb{N},\mathbb{Q}_p\right)$. Bufetov and Qiu classified the ergodic measures on…

### Limits and fluctuations of p-adic random matrix products

- MathematicsSelecta Mathematica
- 2021

We show that singular numbers (also known as invariant factors or Smith normal forms) of products and corners of random matrices over $\mathbb{Q}_p$ are governed by the Hall-Littlewood polynomials,…

### qRSt: A Probabilistic Robinson–Schensted Correspondence for Macdonald Polynomials

- MathematicsInternational Mathematics Research Notices
- 2020

We present a probabilistic generalization of the Robinson–Schensted correspondence in which a permutation maps to several different pairs of standard Young tableaux with nonzero probability. The…

### Stochastic Monotonicity in Young Graph and Thoma Theorem

- Mathematics
- 2014

We show that the order on probability measures, inherited from the dominance order on the Young diagrams, is preserved under natural maps reducing the number of boxes in a diagram by $1$. As a…

### q-randomized Robinson-Schensted-Knuth correspondences and random polymers

- Mathematics
- 2015

We introduce and study q-randomized Robinson-Schensted-Knuth (RSK) correspondences which interpolate between the classical (q=0) and geometric (q->1) RSK correspondences (the latter ones are…

### HALF-SPACE MACDONALD PROCESSES

- MathematicsForum of Mathematics, Pi
- 2020

Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the…

### Macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture

- MathematicsAnnals of Mathematics
- 2019

We prove the classification of homomorphisms from the algebra of symmetric functions to $\mathbb{R}$ with non-negative values on Macdonald symmetric functions $P_{\lambda}$, that was conjectured by…

### Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class

- Mathematics
- 2014

Integrable probability has emerged as an active area of research at the interface of probability/mathematical physics/statistical mechanics on the one hand, and representation theory/integrable…

### YANG–BAXTER FIELD FOR SPIN HALL–LITTLEWOOD SYMMETRIC FUNCTIONS

- MathematicsForum of Mathematics, Sigma
- 2019

Employing bijectivization of summation identities, we introduce local stochastic moves based on the Yang–Baxter equation for $U_{q}(\widehat{\mathfrak{sl}_{2}})$ . Combining these moves leads to a…

## References

SHOWING 1-10 OF 78 REFERENCES

### Finite traces and representations of the group of infinite matrices over a finite field

- Mathematics
- 2012

### ERGODIC UNITARILY INVARIANT MEASURES ON THE SPACE OF INFINITE HERMITIAN MATRICES

- Mathematics
- 1996

Let $H$ be the space of all Hermitian matrices of infinite order and $U(\infty)$ be the inductive limit of the chain $U(1)\subset U(2)\subset...$ of compact unitary groups. The group $U(\infty)$…

### Macdonald processes

- Mathematics
- 2014

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters $$q,t \in…

### $\mathfrak{sl}(2)$ operators and Markov processes on branching graphs

- Mathematics
- 2011

We present a unified approach to various examples of Markov dynamics on partitions studied by Borodin, Olshanski, Fulman, and the author. Our technique generalizes Kerov’s operators which first…

### On the generating functions of totally positive sequences I

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1951

A real matrix, finite or infinite, is called totally positive if and only if all its minors, of all orders = 1, 2,..., are non-negative. An infinite sequence
$$ {a_0},{a_1},{a_2}, \ldots ,\quad…

### A Probabilistic Approach Toward Conjugacy Classes in the Finite General Linear and Unitary Groups

- Mathematics
- 1999

Abstract The conjugacy classes of the finite general linear and unitary groups are used to define probability measures on the set of all partitions of all natural numbers. Probabilistic algorithms…

### A Probabilistic Approach toward Conjugacy Classes in the Finite General Linear and Unitary Groups

- Mathematics
- 2007

The conjugacy classes of the nite general linear and unitary groups are used to de ne probability measures on the set of all partitions of all natural numbers. Probabilistic algorithms for growing…

### Cohen–Lenstra heuristics and random matrix theory over finite fields

- Mathematics
- 2013

Abstract. Let g be a random element of a finite classical group G, and let λz-1(g) denote the partition corresponding to the polynomial z - 1 in the rational canonical form of g. As the rank of G…

### Markov processes of infinitely many nonintersecting random walks

- Mathematics
- 2011

Consider an N-dimensional Markov chain obtained from N one-dimensional random walks by Doob h-transform with the q-Vandermonde determinant. We prove that as N becomes large, these Markov chains…