No Percolation at Criticality on Certain Groups of Intermediate Growth

We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form $p_n(v,v) \leq \exp \left [-\Omega (n^\gamma )\right ]$ for some $\gamma>1/2$. The result is new in the case that the graph is of intermediate volume growth.

Algorithm for Identification of Infinite Clusters Based on Minimal Finite Automaton

We propose a finite automaton based algorithm for identification of infinite clusters in a 2D rectangular lattice with L=X×Y cells. The algorithm counts infinite clusters and finds one path per infinite cluster in a single pass of the finite automaton. The finite automaton is minimal according to the number of states among all the automata that perform such task. The correctness and efficiency of the algorithm are demonstrated on a planar percolation problem. The algorithm has a computational complexity of O(L) and could be appropriate for efficient data flow implementation.

Peculiarities of Bi doping of Ge–Sb–Te thin films for PCM devices

The influence of Bi doping on the thermal, electrical, and optical properties of Ge2Sb2Te5 thin films was investigated. The existence of two Bi concentration ranges with different influence of dopant on the properties of thin films was established. At low concentrations (0.5–1.0 wt.% of Bi), anomalous deviations of physical properties from monotonous concentration dependences were observed. This effect is explained by the use of percolation theory, where formation of infinite clusters is accompanied by critical phenomena at critical concentrations.

Fixed price of groups and percolation

We prove that for every finitely generated group Γ, at least one of the following holds: (1) Γ has fixed price; (2) each of its Cayley graphs G has infinitely many infinite clusters for some Bernoulli percolation on G.