Motion in a Random Force Field

We consider the motion of a particle in a random isotropic force field. Assuming that the force field arises from a Poisson field in $\mathbb{R}^d$, $d \geq 4$, and the initial velocity of the particle is sufficiently large, we describe the asymptotic behavior of the particle

Damage spreading in random field systems

We investigate how a quenched random field influences the damage spreading transition in kinetic Ising models. To this end we generalize a recent master equation approach and derive an effective field theory for damage spreading in random field systems. This theory is applied to the Glauber Ising model with a bimodal random field distribution. We find that the random field influences the spreading transition by two different mechanisms with opposite effects. First, the random field favors the same particular direction of the spin variable at each site in both systems which reduces the damage. Second, the random field suppresses the magnetization which, in turn, tends to increase the damage. The competition between these two effects leads to a rich behavior.Comment: 4 pages RevTeX, 3 eps figure

Surface criticality in random field magnets

The boundary-induced scaling of three-dimensional random field Ising magnets is investigated close to the bulk critical point by exact combinatorial optimization methods. We measure several exponents describing surface criticality: $\beta_1$ for the surface layer magnetization and the surface excess exponents for the magnetization and the specific heat, $\beta_s$ and $\alpha_s$. The latter ones are related to the bulk phase transition by the same scaling laws as in pure systems, but only with the same violation of hyperscaling exponent $\theta$ as in the bulk. The boundary disorders faster than the bulk, and the experimental and theoretical implications are discussed.Comment: 6 pages, 9 figures, to appear in Phys. Rev.

OPTIMAL RANDOM SAMPLING DESIGNS IN RANDOM FIELD SAMPLING

A Horvitz-Thompson predictor is proposed for spatial sampling when the characteristic of interest is modeled as a random field. Optimal sampling designs are deduced under this context. Fixed and variable sample size are considered.

Mean field conditions for coalescing random walks

The main results in this paper are about the full coalescence time $\mathsf{C}$ of a system of coalescing random walks over a finite graph $G$. Letting $\mathsf{m}(G)$ denote the mean meeting time of two such walkers, we give sufficient conditions under which $\mathbf{E}[\mathsf{C}]\approx 2\mathsf{m}(G)$ and $\mathsf{C}/\mathsf{m}(G)$ has approximately the same law as in the "mean field" setting of a large complete graph. One of our theorems is that mean field behavior occurs over all vertex-transitive graphs whose mixing times are much smaller than $\mathsf{m}(G)$; this nearly solves an open problem of Aldous and Fill and also generalizes results of Cox for discrete tori in $d\geq2$ dimensions. Other results apply to nonreversible walks and also generalize previous theorems of Durrett and Cooper et al. Slight extensions of these results apply to voter model consensus times, which are related to coalescing random walks via duality. Our main proof ideas are a strengthening of the usual approximation of hitting times by exponential random variables, which give results for nonstationary initial states; and a new general set of conditions under which we can prove that the hitting time of a union of sets behaves like a minimum of independent exponentials. In particular, this will show that the first meeting time among $k$ random walkers has mean \approx\mathsf{m}(G)/\bigl({\matrix{k 2}}\bigr).Comment: Published in at http://dx.doi.org/10.1214/12-AOP813 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multi-field Inflation with a Random Potential

Motivated by the possibility of inflation in the cosmic landscape, which may be approximated by a complicated potential, we study the density perturbations in multi-field inflation with a random potential. The random potential causes the inflaton to undergo a Brownian motion with a drift in the D-dimensional field space. To quantify such an effect, we employ a stochastic approach to evaluate the two-point and three-point functions of primordial perturbations. We find that in the weakly random scenario the resulting power spectrum resembles that of the single field slow-roll case, with up to 2% more red tilt. The strongly random scenario, leads to rich phenomenologies, such as primordial fluctuations in the power spectrum on all angular scales. Such features may already be hiding in the error bars of observed CMB TT (as well as TE and EE) power spectrum and can be detected or falsified with more data coming in the future. The tensor power spectrum itself is free of fluctuations and the tensor to scalar ratio is enhanced. In addition a large negative running of the power spectral index is possible. Non-Gaussianity is generically suppressed by the growth of adiabatic perturbations on super-horizon scales, but can possibly be enhanced by resonant effects or arise from the entropic perturbations during the onset of (p)reheating. The formalism developed in this paper can be applied to a wide class of multi-field inflation models including, e.g. the N-flation scenario.Comment: More clarifications and references adde