ترغب بنشر مسار تعليمي؟ اضغط هنا

A note on some critical thresholds of Bernoulli percolation

78   0   0.0 ( 0 )
 نشر من قبل Pengfei Tang
 تاريخ النشر 2020
  مجال البحث
والبحث باللغة English
 تأليف Pengfei Tang




اسأل ChatGPT حول البحث

Consider Bernoulli bond percolation a locally finite, connected graph $G$ and let $p_{mathrm{cut}}$ be the threshold corresponding to a first-moment method lower bound. Kahn (textit{Electron. Comm. Probab. Volume 8, 184-187.} (2003)) constructed a counter-example to Lyons conjecture of $p_{mathrm{cut}}=p_c$ and proposed a modification. Here we give a positive answer to Kahns modified question. The key observation is that in Kahns modification, the new expectation quantity also appears in the differential inequality of one-arm events. This links the question to a lemma of Duminil-Copin and Tassion (textit{Comm. Math. Phys. Volume 343, 725-745.} (2016)). We also study some applications for Bernoulli percolation on periodic trees.



قيم البحث

اقرأ أيضاً

We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.
Consider an anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $dgeq 2$, with parameter $p$. We show that the two point connectivity function $P_{p}({(0,dots,0)leftrightarrow (n,0,dots,0)})$ is a monotone functi on in $n$ when the parameter $p$ is close enough to 0. Analogously, we show that truncated connectivity function $P_{p}({(0,dots,0)leftrightarrow (n,0,dots,0), (0,dots,0) leftrightarrowinfty})$ is also a monotone function in $n$ when $p$ is close to 1.
Let $mathbb{G}=left(mathbb{V},mathbb{E}right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $mathbb{Z}$. We choose a collection $mathcal{C}$ of finite connected subgraphs o f $G$ and consider a model of Bernoulli bond percolation on $mathbb{G}$ which assigns probability $q$ of being open to each edge whose projection onto $G$ lies in some subgraph of $mathcal{C}$ and probability $p$ to every other edge. We show that the critical percolation threshold $p_{c}left(qright)$ is a continuous function in $left(0,1right)$, provided that the graphs in $mathcal{C}$ are well-spaced in $G$ and their vertex sets have uniformly bounded cardinality. This generalizes a recent result due to Szabo and Valesin.
We prove that for Bernoulli percolation on $mathbb{Z}^d$, $dgeq 2$, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, w e prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that $p_c^{bond} <1/2$ for certain families of triangulations for which Benjamini & Schramm conjectured that $p_c^{site} leq 1/2$.
In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is adapted to sh ow that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as p approaches p_c. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا