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We consider one component lattice gases with a local dynamics and a stationary product Bernoulli measure. We give upper and lower bounds on the diffusivity at an equilibrium point depending on the dimension and the local behavior of the macroscopic flux function. We show that if the model is expected to be diffusive, it is indeed diffusive, and, if it is expected to be superdiffusive, it is indeed superdiffusive.
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The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Sepp{a}lainen and Balazs recently proved this conjecture for the nearest neighbor case. We extend their results to general finite range exclusion by proving that the Laplace transform of the diffusivity is of the conjectured order. We also obtain a pointwise upper bound for $D(t)$ the correct order.
In the randomly-oriented Manhattan lattice, every line in $mathbb{Z}^d$ is assigned a uniform random direction. We consider the directed graph whose vertex set is $mathbb{Z}^d$ and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the $d$ legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.
We consider the annealed asymptotics for the survival probability of Brownian motion among randomly distributed traps. The configuration of the traps is given by independent displacements of the lattice points. We determine the long time asymptotics of the logarithm of the survival probability up to a multiplicative constant. As applications, we show the Lifshitz tail effect of the density of states of the associated random Schr{o}dinger operator and derive a quantitative estimate for the strength of intermittency in the Parabolic Anderson problem.
Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.
Kinetically constrained lattice gases (KCLG) are interacting particle systems on the integer lattice $mathbb Z^d$ with hard core exclusion and Kawasaki type dynamics. Their peculiarity is that jumps are allowed only if the configuration satisfies a constraint which asks for enough empty sites in a certain local neighborhood. KCLG have been introduced and extensively studied in physics literature as models of glassy dynamics. We focus on the most studied class of KCLG, the Kob Andersen (KA) models. We analyze the behavior of a tracer (i.e. a tagged particle) at equilibrium. We prove that for all dimensions $dgeq 2$ and for any equilibrium particle density, under diffusive rescaling the motion of the tracer converges to a $d$-dimensional Brownian motion with non-degenerate diffusion matrix. Therefore we disprove the occurrence of a diffusive/non diffusive transition which had been conjectured in physics literature. Our technique is flexible enough and can be extended to analyse the tracer behavior for other choices of constraints.
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