The almost-sure existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walks. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the almost-sure asymptotic variance of the polymer path under strong disorder are also provided.

We study a continuous-time random walk on the d-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we discuss the diffusive behavior in the case of zero drift and weak attraction.

We model diffusion-controlled crystal growth as an interference problem. The crystal layers grow by nucleation (initiation of crystallization centers) followed by attachment of molecules to the nucleus. A forming crystal layer completes by either spreading across the length of the crystal or by colliding with another spreading crystal layer. This model differs from the classical Johnson-Mehl-Kolmogorov model in that nucleation happens only on boundaries of a ‘seed’ crystal as opposed to nucleation from random points in a given region. Our results also differ from the limiting results found for this classical model. We use the invariant measure of an embedded Markov process to find the growth rate of the crystal in terms of the nucleation rates. Ergodic theorems are then used to derive explicit formulae for some stationary probabilities.

In this article, we prove the existence and uniqueness of solution for the Cauchy problem of the Landau-Lifshitz equation of ferromagnetism with external magnetic field. We also show that the solution is globally regular with the exception of at most finitely many blow-up points. An energy identity at blow-up points is presented.

In first-passage percolation (FPP) models, the passage time Tℓ from the origin to the point ℓeℓ satisfies f(ℓ) := ETℓ = μℓ + o(ℓ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously that f(ℓ) is eventually monotonically increasing. Here, for the Poisson-based Euclidean FPP of Howard and Newman (Prob. Theory Relat. Fields108 (1997), 153–170), we prove an explicit formula for f′(ℓ). In all dimensions, for certain values of the model's only parameter we have f′(ℓ) ≥ C > 0 for large ℓ.

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Zd. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

Consider a network whose nodes are the integer lattice points and whose arcs are fuses of 1Ω resistance. Remove a horizontal segment of N adjacent vertical arcs, forming a ‘crack’ of length N. Subject the network to a uniform potential gradient of v volts per arc in the north-south (or vertical) direction and measure the current in one of the two vertical arcs at the ends of the crack. We write this current in the form e(N)v, and call e(N) the current enhancement.

We show that the enhancement grows at a rate that is the order of the square root of the crack length. Our method is to identify the enhancement with the mean time to exit an interval for a certain integer valued random walk, and then to use some of the well-known Fourier methods for studying random walk. Our random walk has no mean or higher moments and is in the domain of attraction of the Cauchy law. We provide a good approximation to the enhancement using the explicitly known mean time to exit an interval for a Cauchy process. Weak convergence arguments together with an estimate of a recurrence probability enable us to show that the current in an intact fuse, that is in the interior of a crack of length N, grows p roportionally with N/logN.