اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدClusters in middle-phase percolation on hyperbolic plane
68
0
0.0
(
0
)
تأليف
Jan Czajkowski
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
I consider p-Bernoulli bond percolation on graphs of vertex-transitive tilings of the hyperbolic plane with finite sided faces (or, equivalently, on transitive, nonamenable, planar graphs with one end) and on their duals. It is known (Benjamini and Schramm) that in such a graph G we have three essential phases of percolation, i. e. 0 < p_c(G) < p_u(G) < 1, where p_c is the critical probability and p_u - the unification probability. I prove that in the middle phase a. s. all the ends of all the infinite clusters have one-point boundary in the boundary of H^2. This result is similar to some results of Lalley.
قيم البحث
اقرأ أيضاً
Expected ballisticity of a continuous self avoiding walk on hyperbolic spaces $mathbb{H}^d$ is established.
At each point of a Poisson point process of intensity $lambda$ in the hyperbolic place, center a ball of bounded random radius. Consider the probability $P_r$ that from a fixed point, there is some direction in which one can reach distance $r$ withou
t hitting any ball. It is known cite{BJST} that if $lambda$ is strictly smaller than a critical intensity $lambda_{gv}$ then $P_r$ does not go to $0$ as $rto infty$. The main result in this note shows that in the case $lambda=lambda_{gv}$, the probability of reaching distance larger than $r$ decays essentially polynomial, while if $lambda>lambda_{gv}$, the decay is exponential. We also extend these results to various related models.
In this article we study the sharpness of the phase transition for percolation models defined on top of planar spin systems. The two examples that we treat in detail concern the Glauber dynamics for the Ising model and a Dynamic Bootstrap process. Fo
r both of these models we prove that their phase transition is continuous and sharp, providing also quantitative estimates on the two point connectivity. The techniques that we develop in this work can be applied to a variety of different dependent percolation models and we discuss some of the problems that can be tackled in a similar fashion. In the last section of the paper we present a long list of open problems that would require new ideas to be attacked.
Poisson processes in the space of $(d-1)$-dimensional totally geodesic subspaces (hyperplanes) in a $d$-dimensional hyperbolic space of constant curvature $-1$ are studied. The $k$-dimensional Hausdorff measure of their $k$-skeleton is considered. Ex
plicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the $k$-dimensional Hausdorff measure of the $k$-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $dgeq 2$, it is shown that in case (ii) the central limit theorem holds for $din{2,3}$ and fails if $dgeq 4$ and $k=d-1$ or if $dgeq 7$ and for general $k$. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin-Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.
Consider the graph $mathbb{H}(d)$ whose vertex set is the hyperbolic plane, where two points are connected with an edge when their distance is equal to some $d>0$. Asking for the chromatic number of this graph is the hyperbolic analogue to the famous
Hadwiger-Nelson problem about colouring the points of the Euclidean plane so that points at distance $1$ receive different colours. As in the Euclidean case, one can lower bound the chromatic number of $mathbb{H}(d)$ by $4$ for all $d$. Using spectral methods, we prove that if the colour classes are measurable, then at least $6$ colours are are needed to properly colour $mathbb{H}(d)$ when $d$ is sufficiently large.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
