The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results for the random-cluster model with cluster parameter q ≥ 1 are obtained. These include an explicit pointwise dynamical construction of random-cluster measures for arbitrary graphs, and for unimodular transitive graphs, lack of percolation for ...

negation has many complicated forms and can be represented in a variety of ways. some sentences have explicit negation items and some other sentences convey the notion of negation without having any explicit negation item. negative polarity items (npis) are expressions that need to a negative context to be grammatically correct, and their meanings are compatible with negative sentences and are ...

This work provides exponential tail inequalities for sums of random matrices that depend only on intrinsic dimensions rather than explicit matrix dimensions. These tail inequalities are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the explicit dimensions are large...

We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that Condition L of Comets and Popov [3] is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branchi...

Recentry, Viola (CCC’08) showed that the sum of d small-biased distributions fools degree-d polynomial tests; that is, every polynomial expression of degree at most d in the bits of the sum has distribution very close to that induced by this expression evaluated on uniformly selected random bits. We show that this is tight by showing an explicit construction of a small-bias generator (with expo...

A Moderate Deviation Principle is established for random processes arising as small random perturbations of one-dimensional dynamical systems of the form Xn = f(Xn−1). Unlike in the Large Deviations Theory the resulting rate function is independent of the underlying noise distribution, and is always quadratic. This allows one to obtain explicit formulae for the asymptotics of probabilities of t...

We present a geometric approach to the problem of propagating fronts into an unstable state, valid for an arbitrary continuous-time random walk with a Fisher-Kolmogorov-Petrovski-Piskunov growth/reaction rate. We derive an integral Hamilton-Jacobi type equation for the action functional determining the position of reaction front and its speed. Our method does not rely on the explicit derivation...

Extractors are functions which are able to “extract” random bits from arbitrary distributions which “contain” sufficient randomness. Explicit constructions of extractors have many applications in complexity theory and combinatorics. This manuscript is a survey of recent developments in extractors and focuses on explicit constructions of extractors following Trevisan’s breakthrough result [Tre99].

We prove finite speed of propagation for stochastic porous media equations perturbed by linear multiplicative space-time rough signals. Explicit and optimal estimates for the speed of propagation are given. The result applies to any continuous driving signal, thus including fractional Brownian motion for all Hurst parameters. The explicit estimates are then used to prove that the corresponding ...

We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein’s method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not require the classical use of exchangeable pairs, we employ a chaos expansion in order to construct an explicit exchangeable pair vector for any random variable wh...