Step-reinforced random walks and one-half

TL;DR

This work analyzes step-reinforced random walks (SRRW) in $\mathbb{R}^d$, where each step is either a past step chosen with probability $\alpha$ or a fresh draw from the base measure $\mu$. Under mild moment assumptions, the authors prove a universal phase transition at $\alpha=1/2$: SRRW is recurrent in low dimensions ($d=1$ for all $\alpha\le 1/2$, and $d=2$ for $\alpha<1/2$) and transient for $\alpha>1/2$ in all dimensions; in $d\ge 3$ the process is transient for $\alpha\le 1/2$. In the critical 2D case ($\alpha=1/2$) they establish a precise escape rate, $\lim_{n\to\infty} \frac{\log \|\bm{S}_n\|^2}{\log n}=1$, and show an a.s. angular phase transition of the direction. The analysis combines Lyapunov function methods, convergence of quasi-martingales, and a representation via random recursive trees, linking SRRW to Polya urn dynamics and percolation on random trees. The results settle conjectures of Bertoin and extend recurrence/transience criteria across dimensions, providing a detailed picture of radial and angular behaviors and sharp asymptotics in several regimes.

Abstract

Under suitable moment assumptions, we show that a genuinely d-dimensional step-reinforced random walk undergoes a phase transition between recurrence and transience in dimensions $d=1,2$, and that it is transient for all reinforcement parameters in dimensions $d\geq 3$, which solves a conjecture of Bertoin.

Figures (2)

• Figure 1: Recurrence and transience of the SRRW under $2+\delta$-th moment assumption (second moment assumption for $d=1$)
• Figure 2: An illustration of the forest $\mathscr{F}_8$

Theorems & Definitions (63)

• Definition 1: SRRW
• Remark 1.1: Invariance under measurable maps
• Definition 2
• Remark 1.2
• Definition 3
• Theorem 1.1
• Definition 4
• Corollary 1.2
• Proposition 1.3
• Corollary 1.4: Phase transition of the angular component