AI 中文总结
研究正则树上次线性顶点强化随机游走的占据凝聚转变，通过分析概率与参数关系，发现有限\(\beta_c(a,b)\)处的凝聚转变，四个估计器定位相同阈值，还研究了冻结环境下的可逆性及相关性质，揭示了占有率的非自平均和双峰现象。
AI 中文摘要
顶点强化随机游走以与\(1 + \beta n^{a}\)成比例的概率步向邻居，其中\(n\)为之前对该邻居的访问次数，\(a \in (0,1)\)设定记忆强度。在有根\(b\)元树上，顶点集的指数增长驱使游走向外，而强化作用将其拉回。我们报告了在有限的\(\beta_c(a,b)\)处占据测度的尖锐凝聚转变：低于此值时占据扩散且范围线性增长；高于此值时单个顶点在观测时间内保持\(O(1)\)比例的时间，稳定不变，而范围增长非常缓慢，速率用\(\log t\)描述比任何幂次都更好。我们未发现范围有界，并将此凝聚与有限范围局域化区分开。四个估计器定位到相同阈值，该阈值在\(t = 3×10^{7}\)时无系统漂移。在冻结环境中，游走是可逆的，边电导\(c_{uv} = w_{u}w_{v}\)，\(w_{v} = 1 + \beta n_{v}^{a}\)，测度\(\mu_{v} \propto w_{v}\sum_{u \sim v}w_{u}\)描述凝聚核心，我们直接测试其邻居耦合。可逆性将逃逸置于树上次偏置游走的分支数准则内的前沿，预测\(\beta_c \propto b - 1\)；\(b = 2,3,4\)的测量线在除以\(b - 1\)后坍缩到百分之几（自助法）。控制\(\mathbb{Z}\)上随机游走的\(a = 1/2\)仅作为凝聚轮廓的边际指数进入。在\(\beta_c\)附近，占有率是非自平均且双峰的，呈现共存型现象学。
英文摘要
A vertex-reinforced random walk steps to a neighbour with probability proportional to $1+βn^{a}$, 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 $β_c(a,b)$: 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 $c_{uv}=w_{u}w_{v}$, $w_{v}=1+βn_{v}^{a}$, and measure $μ_{v}\propto w_{v}\sum_{u\sim v}w_{u}$ 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 $β_c\propto b-1$; 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 $β_c$ the occupancy is non-self-averaging and bimodal, a coexistence-type phenomenology.
