The divergence of fluctuations for shape in first passage percolation

We consider the first passage percolation model on Z ^d for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R ^d and It is well known that if p < p _c , there exists a compact shape B _d ⊂ R ^d such that for all > 0, t B _d (1 − ) ⊂ B(t) ⊂ tB _d (1 + ) and G(t)(1 − ) ⊂ B(t) ⊂ G(t)(1 + ) eventually w.p.1. We denote the fluctuations of B(t) from tB _d and G(t) by In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB _d ) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the paths themselves. This linearity is also independently interesting. Research supported by NSF grant DMS-0405150     

Tree-indexed random walks on groups and first passage percolation

Summary Suppose that i.i.d. random variables are attached to the edges of an infinite tree. When the tree is large enough, the partial sumsS along some of its infinite paths will exhibit behavior atypical for an ordinary random walk. This principle has appeared in works on branching random walks, first-passage percolation, and RWRE on trees. We establish further quantitative versions of this principle, which are applicable in these settings. In particular, different notions of speed for such a tree-indexed walk correspond to different dimension notions for trees. Finally, if the labeling variables take values in a group, then properties of the group (e.g., polynomial growth or a nontrivial Poisson boundary) are reflected in the sample-path behavior of the resulting tree-indexed walk.Partially supported by a grant from the Landau Center for Mathematical AnalysisPartially supported by NSF grant DMS-921 3595     

Biased random walk in a one-dimensional percolation model

We consider random walk with a nonzero bias to the right, on the infinite cluster in the following percolation model: take i.i.d. bond percolation with retention parameter p$p$ on the so-called infinite ladder, and condition on the event of having a bi-infinite path from −∞$-\infty$ to ∞$\infty$. The random walk is shown to be transient, and to have an asymptotic speed to the right which is strictly positive or zero depending on whether the bias is below or above a certain critical value which we compute explicitly.     

On Lundh’s percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable configurations are generated with positive probability, Lundh calls this percolation diffusion. An integral condition for percolation diffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.     

Rates for the probability of large cubes being non-internally spanned in modified bootstrap percolation

Summary We find the exact rate of decay for the probability that a large cube is not internally spanned for the modified bootstrap percolation. It is proven that for cubes of large side the event that the cube is not internally spanned is essentially the same as the event that the cube possesses a completely vacant line.Research partially supported by NSF DMS 9157461 and a grant from the Sloan Foundation     

Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II

This paper deals with the blow-up of the solution to a semilinear second-order parabolic equation with nonlinear boundary conditions. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

Critical phenomenon for a percolation model

We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (1-p), respectively, we obtain a lower bound on the critical intensity of percolation.     

Finite size percolation in regular trees

In the context of percolation in a regular tree, we study the size of the largest cluster and the length of the longest run starting within the first d generations. As d tends to infinity, we prove almost sure and weak convergence results.     

Upper large deviations for the maximal flow in first-passage percolation

We consider the standard first-passage percolation in Z^d${\mathbb{Z}}^d$ for d≥2$d\ge 2$ and we denote by _?nd−1,h(n)${?}_{n^{d-1},h\left(n\right)}$ the maximal flow through the cylinder ]0,n]^d−1×]0,h(n)]${]0,n]}^{d-1}\times ]0,h\left(n\right)]$ from its bottom to its top. Kesten proved a law of large numbers for the maximal flow in dimension 3: under some assumptions, _?nd−1,h(n)/n^d−1${?}_{n^{d-1},h\left(n\right)}/n^{d-1}$ converges towards a constant ν$\nu$. We look now at the probability that _?nd−1,h(n)/n^d−1${?}_{n^{d-1},h\left(n\right)}/n^{d-1}$ is greater than ν+ε$\nu +\epsilon$ for some ε>0$\epsilon >0$, and we show under some assumptions that this probability decays exponentially fast with the volume n^d−1h(n)$n^{d-1}h\left(n\right)$ of the cylinder. Moreover, we prove a large deviation principle for the sequence (_?nd−1,h(n)/n^d−1,n∈N)$({?}_{n^{d-1},h\left(n\right)}/n^{d-1},n\in \mathbb{N})$.     

A semilinear equation with generalized Wentzell boundary condition

In this paper we consider a semilinear equation with a generalized Wentzell boundary condition. We prove the local well-posedness of the problem and derive the conditions of the global existence of the solution and the conditions for finite time blow-up. We also derive an estimate for the blow-up time.     

Mean first passage times of two-dimensional processes with jumps

In this paper, we incorporate a jump component into the model based on a two-dimensional degenerate diffusion process for the remaining lifetime of machines in the recent paper [Lefebvre, M., 2010. Mean first-passage time to zero for wear processes. Stochastic Models 26, 46-53] by the second author. We calculate explicitly the expected value of first passage times associated to the two-dimensional process when the jump component is taken to be a compound Poisson process with exponential jumps and random proportion of jumps.     

A semilinear parabolic system with generalized Wentzell boundary condition

In this paper, we consider a weak coupled semilinear parabolic system with general Wentzell boundary condition. We prove the well-posedness of the problem and derive different conditions in terms of the powers of the nonlinear terms under which the global solution exists and finite time blow-up occurs.     

Exponential convergence for a periodically driven semilinear heat equation

We consider a semilinear heat equation in one space dimension, with a periodic source at the origin. We study the solution, which describes the equilibrium of this system and we prove that, as the space variable tends to infinity, the solution becomes, exponentially fast, asymptotic to a steady state. The key to the proof of this result is a Harnack type inequality, which we obtain using probabilistic ideas.     

Multiplicative asymptotics of solutions of the first boundary value problem on a half-axis for a parabolic equation with a small parameter

In this work we consider the first boundary value problem for a parabolic equation of second order with a small parameter on a half-axis (i.e., we consider the one-dimensional case). We take the zero initial condition. We construct the global (that is, the caustic points are taken into account) asymptotics of a solution for the boundary value problem. The asymptotic solution of this problem has a different structure depending on the sign of the coefficient (the drift coefficient) at the derivative of first order at a boundary point. The constructed asymptotic solutions are justified.     

On the rate of convergence of simple and jump-adapted weak Euler schemes for Lévy driven SDEs

The paper studies the rate of convergence of a weak Euler approximation for solutions to possibly completely degenerate SDEs driven by Lévy processes, with Hölder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes and its robustness to the approximation of the increments of the driving process. A convergence rate is derived for some approximate jump-adapted Euler scheme as well.     

On some growth models with a small parameter

Summary We consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models, when a certain parameter becomes small. The basic example treated is the variant of Richardson's growth model on ^d in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors, at rate 1 if it has exactly 1 occupied neighbor and, of course, at rate 0 if it has no occupied neighbor. Occupied sites remain occupied forever. Starting from a single occupied site, this model has asymptotic speeds of growth in each direction (as time goes to infinity) and these speeds determine an asymptotic shape in the usual sense. It is proven that as tends to 0, the asymptotic speeds scale as ^1/d and the asymptotic shape, when renormalized by dividing it by ^1/d , converges to a cube. Other similar models which are partially oriented are also studied.The work of R.H.S. was supported by the N.S.F. through grant DMS 91-00725. In addition, both authors were supported by the Newton Institute in Cambridge. The authors thank the Newton Institute for its support and hospitality     

Scaling limits for one-dimensional long-range percolation: Using the corrector method

In this paper, by using the corrector method we give another proof of the quenched invariance principle for the random walk on the infinite random graph generated by a one-dimensional long-range percolation under the conditions that the connection probability p(1)=1$p\left(1\right)=1$ and the percolation exponent s>2$s>2$. The key step of the proof is the construction of the corrector. We show that the corrector can be constructed under either s∈(2,3]$s\in (2,3]$ or s>3$s>3$, though the corresponding underlying measures may be different. As an application of the main result we get a new lower bound of the quenched diagonal transition probability for the random walk.     

Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition

Let p>1 and Ω be a smoothly bounded domain in . This paper is concerned with a Cauchy-Neumann problem

     

Sharp estimates of the convergence rate for a semilinear parabolic equation with supercritical nonlinearity

We study the behavior of solutions of the Cauchy problem for a semilinear parabolic equation with supercritical nonlinearity. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its sharp convergence rate for a class of initial data. We also derive a universal lower bound of the convergence rate which implies the optimality of the result. Proofs are given by a comparison method based on matched asymptotics expansion.     

Finite clusters in high-density continuous percolation: Compression and sphericality

Summary A percolation process inR ^d is considered in which the sites are a Poisson process with intensity and the bond between each pair of sites is open if and only if the sites are within a fixed distancer of each other. The distribution of the number of sites in the clusterC of the origin is examined, and related to the geometry ofC. It is shown that when andk are large, there is a characteristic radius such that conditionally on |C|=k, the convex hull ofC closely approximates a ball of radius , with high probability. When the normal volumek/ thatk points would occupy is small, the cluster is compressed, in that the number of points per unit volume in this -ball is much greater than the ambient density . For larger normal volumes there is less compression. This can be compared to Bernoulli bond percolation on the square lattice in two dimensions, where an analog of this compression is known not to occur.Research supported by NSF grant number DMS-9006395