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The speed of random walks on semigroups

Guy Blachar, Be'eri Greenfeld

TL;DR

The paper establishes that the spectrum of speed exponents for random walks on finitely generated semigroups is the full interval $[0,1]$ by constructing semigroups $\mathcal{S}_{\vec{m}}$ whose speed $\mathbb{E}|R_n|$ realizes any target growth $n^{\alpha}$ with $\alpha\in[0,1]$. It provides precise speed estimates in terms of the defining sequence $\vec{m}$, and proves that any increasing speed function can be approached arbitrarily slowly by passing to suitable quotients $\overline{\mathcal{S}}_{\vec{m}}$, thereby showing there is no universal gap between constant and non-constant speeds in the semigroup setting. Beyond speed, the authors introduce a lower-bound framework for the distance from the starting point, via rooted ball spreads on directed graphs, and transfer this to semigroups with no finite right ideals. The results highlight a sharp contrast with groups, where speed exponents are restricted, and collectively reveal a rich, highly flexible landscape for random walks in non-group algebraic structures.

Abstract

We construct, for each real number $0\leq α\leq 1$, a random walk on a finitely generated semigroup whose speed exponent is $α$. We further show that the speed function of a random walk on a finitely generated semigroup can be arbitrarily slow, yet tending to infinity. These phenomena demonstrate a sharp contrast from the group-theoretic setting. On the other hand, we show that the distance of a random walk on a finitely generated semigroup from its starting position is infinitely often larger than a non-constant universal lower bound, excluding a certain degenerate case.

The speed of random walks on semigroups

TL;DR

The paper establishes that the spectrum of speed exponents for random walks on finitely generated semigroups is the full interval by constructing semigroups whose speed realizes any target growth with . It provides precise speed estimates in terms of the defining sequence , and proves that any increasing speed function can be approached arbitrarily slowly by passing to suitable quotients , thereby showing there is no universal gap between constant and non-constant speeds in the semigroup setting. Beyond speed, the authors introduce a lower-bound framework for the distance from the starting point, via rooted ball spreads on directed graphs, and transfer this to semigroups with no finite right ideals. The results highlight a sharp contrast with groups, where speed exponents are restricted, and collectively reveal a rich, highly flexible landscape for random walks in non-group algebraic structures.

Abstract

We construct, for each real number , a random walk on a finitely generated semigroup whose speed exponent is . We further show that the speed function of a random walk on a finitely generated semigroup can be arbitrarily slow, yet tending to infinity. These phenomena demonstrate a sharp contrast from the group-theoretic setting. On the other hand, we show that the distance of a random walk on a finitely generated semigroup from its starting position is infinitely often larger than a non-constant universal lower bound, excluding a certain degenerate case.

Paper Structure

This paper contains 13 sections, 21 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on semigroups and random walks
  3. Cayley graphs
  4. Random walks
  5. Presentations of semigroups
  6. Construction
  7. Speed estimates
  8. Realization of speed functions
  9. Arbitrary speed exponents
  10. Arbitrarily slow speed functions
  11. A lower bound on the distance of random walks
  12. Bounded speed
  13. Appearances of subwords

Key Result

Proposition 3.2

Every element of $a\in\mathcal{S}_{\vec{m}}$ can be uniquely written as where $j_0,j_t\ge 0$, $j_1,\dots,j_{t-1}\ge 1$ and $j_1\preceq j_2\preceq\cdots\preceq j_{t-1}$.

Figures (1)

  • Figure 1: A finite part of the Cayley graph of $\mathcal{S}_{\vec{m}}$ (here $j\succeq i$). Red arrows correspond to right multiplication by $x$ and blue arrows correspond to right multiplication by $y$. Here $j\succeq i$.

Theorems & Definitions (49)

  • Proposition 3.2
  • proof
  • Remark 3.3
  • Corollary 3.4
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • ...and 39 more