论文标题
准传递图上无序随机网络的相变
Phase transition of disordered random networks on quasi-transitive graphs
论文作者
论文摘要
给定一个准透射无限图$ g $，其数量增长率$ {\ rm gr}（g），$ thransient偏见的电网$（g，\，c_1）$带有bias $λ_1\ in（0，\，\，\，\，{\ rm gr}（g）（g）（g）（g）（g）（g）（g）（g）（g）$ bias bias $ as $ liase $（giase）$（c_ g， （{\ rm gr}（g），\ infty）。$ write $ g（p）$ $ g（p）$ $ p $ p $债券债券在$ g $上定义为大耦合所定义的$ g $。令$（g，\，c_1，\，\，c_2，\，p）$为以下有偏见的随机网络：$ g（p）$ in $ g（p）$的开放边缘$ e $以$ c_1（e）$和封闭的边缘$ g $ in $ g（p）$ g（p）$ the tuncationance $ c_2（g）$。我们的主要结果如下：（i）在连接的Quasi Transitive无限图$ g $上具有渗透阈值$ p_c \ in（0，\，1），$ $（g，c_1，c_1，c_2，c_2，c_2，\，\，\，p）$具有非trivial retrivial Rexisrience/Transience/Transiention Transition thresshess $ P_ （0，\，1）$是确定性的，几乎可以肯定的是$（g，\，c_1，\，c_2，\，p）$对于$ p <p_c^*$进行了复发，并且$ p> p_c^*。当且仅当相应的组实际上不是$ \ mathbb {z} $时，cayley图。 （ii）在$ \ mathbb {z}^d $上，对于任何$ d \ geq 1，$ $ $ p_c^{*} = p_c $。在$ d $ - regular树上$ \ mathbb {t}^d $ with $ d \ geq 3 $，$ p_c^{*} =（λ_1\ vee 1）p_c $，因此$ p_c^{*}> p_c $ for任何$λ_1\ in（1，\ in c_1， gr}（\ mathbb {t}^d））。$作为对比，我们还考虑具有唯一电流的相过渡，或者以$（\ Mathbb {z}^d，\，\，c_1，c_1，c_2，c_2，\，\，\，p）$的$ d \ d \ egq 2 $ and previve and previve几乎是Mathb^2， C_2，\，p）$带有$λ_1<1 \leqλ_2$具有[0,1] $中任何$ p \的唯一电流。
Given a quasi-transitive infinite graph $G$ with volume growth rate ${\rm gr}(G),$ a transient biased electric network $(G,\, c_1)$ with bias $λ_1\in (0,\,{\rm gr}(G))$ and a recurrent biased one $(G,\, c_2)$ with bias $λ_2\in ({\rm gr}(G),\infty).$ Write $G(p)$ for the Bernoulli-$p$ bond percolation on $G$ defined by the grand coupling. Let $(G,\, c_1,\, c_2,\, p)$ be the following biased disordered random network: Open edges $e$ in $G(p)$ take the conductance $c_1(e)$, and closed edges $g$ in $G(p)$ take the conductance $c_2(g)$. Our main results are as follows: (i) On connected quasi-transitive infinite graph $G$ with percolation threshold $p_c\in (0,\, 1),$ $(G,\, c_1,\, c_2,\, p)$ has a non-trivial recurrence/transience phase transition such that the threshold $p_{c}^{*}\in (0,\, 1)$ is deterministic, and almost surely $(G,\, c_1,\, c_2,\, p)$ is recurrent for $p<p_c^*$ and transient for $p>p_c^*.$ There is a non-trivial recurrence/transience phase transition for $(G,\, c_1,\, c_2,\, p)$ with $G$ being a Cayley graph if and only if the corresponding group is not virtually $\mathbb{Z}$. (ii) On $\mathbb{Z}^d$ for any $d\geq 1,$ $p_c^{*}= p_c$. And on $d$-regular trees $\mathbb{T}^d$ with $d\geq 3$, $p_c^{*}=(λ_1\vee 1) p_c$, and thus $p_c^{*}>p_c$ for any $λ_1\in (1,\,{\rm gr}(\mathbb{T}^d)).$ As a contrast, we also consider phase transition of having unique currents or not for $(\mathbb{Z}^d,\, c_1,\, c_2,\, p)$ with $d\geq 2$ and prove that almost surely $(\mathbb{Z}^2,\, c_1,\, c_2,\, p)$ with $λ_1<1\leqλ_2$ has unique currents for any $p\in [0,1]$.
