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Register a new userClusters in middle-phase percolation on hyperbolic plane
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Jan Czajkowski
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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.
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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 $r$ without 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.
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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.
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