Spreading from a seed is studied by Monte Carlo simulation on a square lattice with two types of sites affecting the rates of birth and death. These systems exhibit a critical transition between survival and extinction. For time-dependent background, this transition is equivalent to those found in homogeneous systems (i.e., to directed percolation). For frozen backgrounds, the appearance of the...

In this work we suggest a new model for generating random satisfiable 3CNF formulas. To generate such formulas – randomly permute all 8 ( n 3 ) possible clauses over the variables x1, . . . , xn, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if after its addition the formula remains satisfiable. We study the evolution of this proc...

Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-rigorous statistical mechanics ideas have inspired a message passing algorithm called Belief propagation guided decimation for finding satisfying assignments of Φ. This algorithm can be viewed as an attempt at implementing a certain thought experiment that we call the decimation process. In this paper we i...

In this paper we demonstrate that the disintegration process of radioactive atoms in the discrete time model has a natural interpretation in the theory of random graphs. Moreover, we reduce some problems in this area to disintegration processes with anomalies. For instance , we derive exact closed formull for the probabilities and the moments of the number of isolated vertices in Bernoulli grap...

Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to compare the combinatorial structure directly to the independent discrete process, without renormalizing. The quality of approximation can often be convenient...

Let P be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty n-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that P is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extre...

One of the most notable features of nanometer scale CMOS technology is the increasing magnitude of variability of the key parameters affecting performance of integrated circuits [1]. Although scaling made controlling extrinsic variability more complex, nonetheless, the most profound reason for the future increase in parameter variability is that the technology is approaching the regime of funda...

Abstract. By considering a continuous pruning procedure on Aldous’s Brownian tree, we construct a random variable Θ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in a critical Galton-Watson tree. We also prove that this random variable can be obtained as the a.s...

On the one hand, random permutations appear as a natural model for data in various topics, such as analysis of algorithms, statistical mechanics, or genomic statistics. On the other hand, from the mathematical point of view, various parameters of random permutations have been studied in combinatorics and probability, like the distribution of the lengths of cycles, the repartition of the eigenva...