Numerical and physical experiments on two-dimensional (2D) turbulence show that the differences of transverse components of velocity field are well described by Gaussian statistics and Kolmogorov scaling exponents. In this case the dissipation fluctuations are irrelevant in the limit of small viscosity. In general, one can assume the existence of a critical space dimensionality d=d(c), at which...

How political opinion, product adoption, rumors, and trends diffuse through social networks has been a fundamental question for many decades in a wide variety of research disciplines. The spread of opinion on networks can be thought of as a state dynamics, where each node decides its state based on interactions with its neighbors. In many such settings, the dynamics of the states of nodes are d...

We study the d-dimensional random Ising model using a suitable type of BethePeierls approximation in the framework of the replica method. We take into account the correct interaction only inside replicated clusters of spins. Our ansatz is that the interaction of the borders of the clusters with the external world can be described via an effective interaction among replicas. The Bethe-Peierls mo...

We approximate the Bolker-Pacala model of population dynamics with the logistic Markov chain and analyze the latter. We find the asymptotics of the degenerated hypergeometric function and use these to prove a local CLT and large deviations result. We also state global limit theorems and obtain asymptotics for the first passage time to the boundary of a large interval.

The mean field approximation to the Ising model is a canonical variational tool that is used for analysis and inference in Ising models. We provide a simple and optimal bound for the KL error of the mean field approximation for Ising models on general graphs, and extend it to higher order Markov random fields. Our bound improves on previous bounds obtained in work in the graph limit literature ...

The electrical dc-conductivity of disordered, noninteracting electrons is calculated in the asymptotic limit of high lattice dimensions d→` . To go beyond the lowest-order contribution in the expansion parameter 1/d of the single bubble diagram, vertex corrections are calculated from an asymptotic expression for the two-particle vertex. A mean-field approximation for the dc conductivity contain...