Occupation-condensation transition of a sublinearly vertex-reinforced random walk on regular tree

A vertex-reinforced random walk steps to a neighbour with probability proportional to $1+\beta n^{a}$, where $n$ counts previous visits to that neighbour and $a\in(0,1)$ sets the memory strength.

On the rooted $b$-ary tree the exponential growth of the vertex set drives the walk outward while the reinforcement pulls it back.

We report a sharp condensation transition of the occupation measure at a finite $\beta_c(a,b)$: below it the occupation spreads and the range grows linearly; above it a single vertex holds an $O(1)$ fraction of the time, stable in the observation time, while the range keeps growing very slowly, at a rate better described by $\log t$ than by any power.

We do not find the range to be bounded, and keep this condensation distinct from finite-range localization.

Four estimators locate the same threshold, which shows no systematic drift out to $t=3\times10^{7}$.

In a frozen environment the walk is reversible, with edge conductances $c_{uv}=w_{u}w_{v}$, $w_{v}=1+\beta n_{v}^{a}$, and measure $\mu_{v}\propto w_{v}\sum_{u\sim v}w_{u}$ describing the condensed core, whose neighbour coupling we test directly.

Reversibility places the escape at the frontier within the branching-number criterion for biased walks on trees, predicting $\beta_c\propto b-1$; the measured lines for $b=2,3,4$ collapse under division by $b-1$ to a few percent (bootstrap).

The value $a=1/2$ that governs the walk on $\mathbb{Z}$ enters only as the marginal exponent of the condensed profile.

Near $\beta_c$ the occupancy is non-self-averaging and bimodal, a coexistence-type phenomenology.