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The random walk on upper triangular matrices over $\mathbb{Z}/m \mathbb{Z}$

Evita Nestoridi, Allan Sly

TL;DR

This paper analyzes a natural Markov chain on the group $G_n(m)$ of uni-upper triangular matrices over $\mathbb{Z}/m\mathbb{Z}$, where at each step a random row is added or subtracted to the row above. The authors prove sharp upper bounds on the mixing time, showing $d_n(t_{n,m}) \le e^{-d}$ with an explicit $t_{n,m}$ that scales as $\gamma(m^2 n \log n + n^2 e^{\delta\sqrt{\log m}}) + c n m^2 \log\log n$ when $m$ is prime (and a variant with $e^{\delta (\log m)^{2/3}}$ for non-prime $m$). The core strategy is an induction on the matrix dimension $n$, reducing the problem to the mixing of the first row via spectral analysis of the second row contributions, and employing the Diaconis–Hough lemma for the prime case plus a generalized DH-type control for composites. The results resolve questions of Stong and Arias-Castro–Diaconis–Stanley, and establish mixing at a rate governed by $m$ and $n$ through explicit, scalable bounds, with the continuous-time formulation clarifying the temporal dynamics. Overall, the paper advances the understanding of random walks on non-commutative finite groups and provides precise, dimension-aware mixing-time estimates reliant on spectral and probabilistic techniques.

Abstract

We study a natural random walk on the $n \times n$ upper triangular matrices, with entries in $\mathbb{Z}/m \mathbb{Z}$, generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this random walk is $O(m^2n \log n+ n^2 m^{o(1)})$. This answers a question of Stong and of Arias-Castro, Diaconis, and Stanley.

The random walk on upper triangular matrices over $\mathbb{Z}/m \mathbb{Z}$

TL;DR

This paper analyzes a natural Markov chain on the group of uni-upper triangular matrices over , where at each step a random row is added or subtracted to the row above. The authors prove sharp upper bounds on the mixing time, showing with an explicit that scales as when is prime (and a variant with for non-prime ). The core strategy is an induction on the matrix dimension , reducing the problem to the mixing of the first row via spectral analysis of the second row contributions, and employing the Diaconis–Hough lemma for the prime case plus a generalized DH-type control for composites. The results resolve questions of Stong and Arias-Castro–Diaconis–Stanley, and establish mixing at a rate governed by and through explicit, scalable bounds, with the continuous-time formulation clarifying the temporal dynamics. Overall, the paper advances the understanding of random walks on non-commutative finite groups and provides precise, dimension-aware mixing-time estimates reliant on spectral and probabilistic techniques.

Abstract

We study a natural random walk on the upper triangular matrices, with entries in , generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this random walk is . This answers a question of Stong and of Arias-Castro, Diaconis, and Stanley.

Paper Structure

This paper contains 14 sections, 36 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The induction lemma
  4. The $\ell^2$ bound
  5. Coupling with Exponentials
  6. Reducing the walk to a smaller dimension
  7. The case where $m$ is a prime
  8. The main lemma for $m$ prime
  9. The eigenvalues for $y \in W_I$
  10. The case where $m$ is not a prime
  11. The eigenvalues $y\in W_I$
  12. The eigenvalues $y\in Q_I$
  13. The eigenvalues $y\in P_2$
  14. The proof of Proposition \ref{['q']}

Key Result

Theorem 1

For the random walk $X_t$ on $G_n(m)$, there exist positive constants $\gamma, \delta, c$, so that for all $n,m \geq 2$ with $m$ prime we have where $t_{n,m}=t'_{n,m} + d \frac{t'_{n,m}}{\log (n+m)},$$t'_{n,m}= \gamma (m^2 n \log n + n^2 e^{\delta \sqrt{ \log m}}) + cnm^2 \log \log n$, and $d>0$.

Theorems & Definitions (65)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof : Proof of Theorems \ref{['main']} and \ref{['maing']}
  • Definition 5
  • Lemma 6
  • proof
  • Proposition 7
  • proof : Proof of Theorems \ref{['mainc']} and \ref{['maingc']}.
  • ...and 55 more