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We study the small mass limit of the equation describing planar motion of a charged particle of a small mass $$\mu $$ in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity $$\epsilon >0$$ . We show that for all small but fixed frictions the small mass limit of $$q_{\mu , \epsilon }$$ gives the solution $$q_\epsilon $$ to a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by Freidlin and Wentzell, we take the limit of the slow component of the motion $$q_\epsilon $$ and we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.

We consider the energy difference restricted to a finite volume for certain pairs of incongruent ground states (if they exist) in the d-dimensional Edwards-Anderson (EA) Ising spin glass at zero temperature. We prove that the variance of this quantity with respect to the couplings grows at least proportionally to the volume in any dimension greater than or equal to two. An essential aspect of our result is the use of the excitation metastate. As an illustration of potential applications, we use this result to restrict the possible structure of spin glass ground states in two dimensions., Comment: 9 pages. arXiv admin note: text overlap with arXiv:1403.1208

We study ground states of Ising models with random ferromagnetic couplings, proving the triviality of all zero-temperature metastates. This result sheds a new light on the properties of these systems, putting strong restrictions on their possible ground state structure. Open problems related to existence of interface-supporting ground states are stated and an interpretation of the main result in terms of first-passage and random surface models in a random environment is presented.

We consider the solution of the equation r(t) = W(r(t)), r(0) = r 0 > 0 where W(⋅) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α∈ (0,1). We show that for almost all realizations of W(⋅) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to the left of r 0, depending on whether W(r 0) is positive or not. After reaching the equilibrium the trajectory stays in it forever. The problem is motivated by studying the separation between two particles in a Gaussian velocity field which satisfies a local self-similarity hypothesis. In contrast to the case when the forcing term is a Brownian motion (then an analogous statement is a consequence of the Markov property of the process) we show our result using as the principal tools the properties of time reversibility of the law of the f.B.m., see Lemma 2.4 below, and the small ball estimate of Molchan, Commun. Math. Phys. 205 (1999) 97–111.

We study waves in convex scalar conservation laws under noisy initial perturbations. It is known that the building blocks of these waves are shock and rarefaction waves, both are invariant under hyperbolic scaling. Noisy perturbations can generate complicated wave patterns, such as diffusion process of shock locations. However we show that under the hyperbolic scaling, the solutions converge in the sense of distribution to the unperturbed waves. In particular, randomly perturbed shock waves move at the unperturbed velocity in the scaling limit. Analysis makes use of the Hopf formula of the related Hamilton-Jacobi equation and regularity estimates of noisy processes.

Let a random variable x0 and a function f:[a, b]k→[a, b] be given. A hierarchical sequence {xn:n=0, 1, 2,...} of random variables is defined inductively by the relation xn=f(xn−1, 1, xn−1, 2..., xn−1, k), where {xn−1, i:i=1, 2,..., k} is a family of independent random variables with the same distribution as xn−1. We prove a central limit theorem for this hierarchical sequence of random variables when a function f satisfies a certain averaging condition. As a corollary under a natural assumption we prove a central limit theorem for a suitably normalized sequence of conductivities of a random resistor network on a hierarchical lattice.

We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known Buckley–Leverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a Cole–Hopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the head or tail of the fronts. Finally, we pass to the inviscid limit, and establish the same results for the inviscid shock fronts.

We study first-passage percolation models and their higher dimensional analogs—models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to +∞. As corollaries we show that in any dimensiond≥2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or +∞. Proofs employ simple large-deviation estimates and ergodic arguments.

We study sequences of random variables obtained by iterative procedures, which can be thought of as nonlinear generalizations of the arithmetic mean. We prove a strong law of large numbers for a class of such iterations. This gives rise to the concept of generalized expected value of a random variable, for which we prove an analog of the classical Jensen inequality. We give several applications to models arising in mathematical physics and other areas.

We study the conductance of random resistor networks ind≥2 dimensions. It is shown (under some technical assumptions) that if a network exhibits a nonzero conductivity in the infinite-volume limit, then the variance of a finite-volume conductance grows at least like the volume.

An extensive quantity is a family of functionsΨv of random parameters, indexed by the finite regionsV (subsets of ℤd) over whichΨv are additive up to corrections satisfying the boundary estimate stated below. It is shown that unless the randomness is nonessential, in the sense that limΨv/|V| has a unique value in the absolute (i.e., not just probabilistic) sense, the variance of such a quantity grows as the volume ofV. Of particular interest is the free energy of a system with random couplings; for suchΨv bounds are derived also for the generating functionE(etΨ). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment.