Dimension gap and phase transition for one-dimensional random walks with reflective boundary

TL;DR

This work develops a unified thermodynamic formalism for $oldsymbol{ extPsi}$-extensions of expanding interval maps to model one-dimensional random walks with and without a reflective boundary. It derives variational formulas linking $oldsymbol{ extP}(g,oldsymbol{Z})$ and $oldsymbol{ extP}(g,oldsymbol{N})$ to base pressures, establishes sharp criteria for a dimension gap via the drift parameter and conformal measures, and identifies a second-order phase transition in the reflective case using asymptotic covariance. The authors validate the theory through explicit examples (Reflective simple random walk, $oldsymbol{ exta}$-Lüroth maps, Gauss-type systems) and connect the pressure framework to the spectral radius of infinite Hessenberg matrices. The results illuminate how boundary effects shape recurrence/transience, dimensional characteristics, and phase behavior, with potential applications to random walks in broader dynamical environments.

Abstract

We study $\mathbb Z$- and $\mathbb N$-extensions of interval maps with at most countably many full branches modelling one-dimensional random walks without and with a reflective boundary. We analyse the associated Gurevich pressure and explore the relations governing these two cases. For such extensions, we obtain variational formulae for the Gurevich pressure that depend only on the base system. As a consequence, we characterise the systems with a dimension gap and, in the presence of a reflective boundary, provide general conditions in terms of asymptotic covariances for a second order phase transition. As a by-product, we derive a variational formula for the spectral radius of infinite Hessenberg matrices.

Figures (5)

• Figure 3.1: To the left the graph of $L_{\lambda,\mathbb{Z}}$ modelling the non-reflective $\mathfrak{a}(\lambda)$-Lüroth system and to the right the graph of $L_{\lambda,{\mathbb{N}}}$ modelling the reflective $\mathfrak{a}(\lambda)$-Lüroth system.
• Figure 3.2: The dimensions of the transient part $\lambda\mapsto\dim_{H}\left(T^{+}\left(L_{\lambda,{\mathbb{N}}}\right)\right)=\dim_{H}\left(\mathbf{T}_{\lambda}\right)$ (dashed line) and recurrent part $\lambda\mapsto\dim_{H}\left(R\left(L_{\lambda,{\mathbb{N}}}\right)\right)=\dim_{H}\left(\mathbf{R}_{\lambda}\right)$ (solid line) as a function of the parameter $\lambda$ for the Lüroth system considered in subsec:vanstrien.
• Figure 3.3: To the left the graph of $G_{\mathbb{Z}}$ modelling the non-reflective Gauss system and to the right $G_{{\mathbb{N}}}$ modelling the reflective Gauss system.
• Figure 3.4: On the left we have the first eight branches of the conjugate system $V_{\lambda}$ to the right reflective $\mathfrak{a}\left(\lambda\right)$-Lüroth map $L_{\lambda,{\mathbb{N}}}$ with $\lambda=1/2$. On the right the corresponding graph for the system conjugate to the Gauss system $G_{{\mathbb{N}}}$ with conjugacy map given by $p:(0,1)\to\mathbb{R}_{>0}$, $x\mapsto1/x-1$.
• Figure 4.1: The graph of $q\mapsto s(q)$ with the values $\delta_{0}$, $\delta_{\mathbb{Z}}$ and $\delta_{{\mathbb{N}}}$ for $\alpha_{\max}>0$ (left) and for $\alpha_{\max}<0$ (right).

Theorems & Definitions (37)

• Proposition 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Remark
• Theorem 2.7: Second-order phase transition
• Lemma 4.1
• proof