A contribution to shape theory for the ising model at low temperature

Summary We consider low temperature limits of Gibbs states of the ferromagnetic nearest-neighbour Ising Hamiltonian in the positive orthant of the lattice ^d ,d=1, 2,..., under a negative boundary condition and a small positive external fieldh that decreases linearly with the temperatureT. It is shown that positive and negative spins are separated by a staircase-shaped random boundary. Its explicit distribution is computed in the case that the ratio =h/T exceeds some positive ₀. For < ₀, our results do not rule out infinite negative areas.     

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.     

Coupling in the theory of interacting systems

In this paper we consider the construction of couplings for Markovian evolutions on a state space of the formE , with (measurable) and a countable group (^d for example). The evolutions we focus on are mainly systems of linearly interacting diffusions, withE compact. We explain and state properties of such couplings and show how they are used to obtain information on the behaviour of the evolution in finite time and as time tends to infinity. An important property of a coupling is to be a successful coupling. The latter concept is introduced here in the context of interacting systems, which is different from the classical concept for Markov chains or processes with state space ^d. The analysis of the question when a coupling is successful depends heavily on the structure of the interaction term and is investigated in detail. We formulate some open problems and conjectures.The paper puts in perspective the coupling statements appearing in the proofs of various results and is largely based on the works of Cox and Greven, Fleischmann and Greven, Dawson and Greven, Greven, and Cox, Greven and Shiga.     

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     

Properties of the limit shape for some last-passage growth models in random environments

We study directed last-passage percolation on the planar square lattice whose weights have general distributions, or equivalently, queues in series with general service distributions. Each row of the last-passage model has its own randomly chosen weight distribution. We investigate the limiting time constant close to the boundary of the quadrant. Close to the y-axis, where the number of random distributions averaged over stays large, the limiting time constant takes the same universal form as in the homogeneous model. But close to the x-axis we see the effect of the tail of the distribution of the random environment.     

Distributional limits for the symmetric exclusion process

Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and Gaussian distributions for various functionals of the process.     

The Legendre transform of two replicas of the Sherrington-Kirkpatrick spin glass model

Summary We prove a variational inequality linking the values of the free energy per site at different temperatures. This inequality is based on the Legendre transform of the free energy of two replicas of the system. We prove that equality holds when1/ and fails when 1/ <1. We deduce from this that the mean entropy per site of the uniform distribution with respect to the distribution of the coupling _i ¹ _i ² = _i between two replicas is null when 01/ and strictly positive when 1/ <1. We exhibit thus a new secondary critical phenomenon within the high temperature region 01. We given an interpretation of this phenomenon showing that the fluctuations of the law of the coupling with the interactions remains strong in the thermodynamic limit when>1/ . We also use our inequality numerically within the low temperature region to improve (slightly) the best previously known lower bounds for the free energy and the ground state energy per site.     

Reverse Hölder inequalities for singular parabolic equations near the boundary

We show that weak solutions to a singular parabolic partial differential equation globally belong to a higher Sobolev space than assumed a priori. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary. The results extend to singular parabolic systems as well. Motivation for studying reverse Hölder inequalities comes partly from applications to regularity theory.     

Two phase transitions for the contact process on small worlds

In our version of Watts and Strogatz’s small world model, space is a d$d$-dimensional torus in which each individual has in addition exactly one long-range neighbor chosen at random from the grid. This modification is natural if one thinks of a town where an individual’s interactions at school, at work, or in social situations introduce long-range connections. However, this change dramatically alters the behavior of the contact process, producing two phase transitions. We establish this by relating the small world to an infinite “big world” graph where the contact process behavior is similar to the contact process on a tree. We then consider the contact process on a slightly modified small world model in order to show that its behavior is decidedly different from that of the contact process on a tree.     

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.     

The interface between regions where u<0 andu > 0in the porous medium equation

We consider the porous medium equation with sign changes. In particular this equation describes the mixing of fresh and salt groundwater due to mechanical dispersion. The unknown function u, which denotes the velocity of the fluids, may take positive as well as negative values. Our main result is the following : under certain monotonicity hypotheses on the initial function, there exists a time T> 0 after which the regions where u < 0 and u > 0 are separated by an interface x = ζ(t) such that ζ is continuously differentiable on [T,∞]. The method of proof is based on a priori estimates for solutions of regularized problems and for their level lines     

Shock fluctuations in asymmetric simple exclusion

Summary The one dimensional nearest neighbors asymmetric simple exclusion process in used as a microscopic approximation to the Burgers equation. We study the process with rates of jumpsp>q to the right and left, respectively, and with initial product measure with densities < to the left and right of the origin, respectively (with shock initial conditions). We prove that a second class particle added to the system at the origin at time zero identifies microscopically the shock for all later times. If this particle is added at another site, then it describes the behavior of a characteristic of the Burgers equation. For vanishing left density (=0) we prove, in the scale t^1/2, that the position of the shock at timet depends only on the initial configuration in a region depending ont. The proofs are based on laws of large numbers for the second class particle.     

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     

Homogenization and propagation of chaos to a nonlinear diffusion with sticky reflection

Summary We study a process reflecting in a domain. The process follows Wentzell non-sticky boundary conditions while being adsorbed at the boundary at a certain rate with respect to local time and desorbed at a rate with respect to natural time. We show that when the rates go to infinity with a converging ratio, the process converges to a process with sticky reflection having the limit ratio as the sojourn coefficient. We then study a mean-field interacting system of such particles. We show propagation of chaos to a nonlinear diffusion with sticky reflection when we perform this homogenization simultaneously as the number of particles goes to infinity.     

Central limit theorem for a tagged particle in asymmetric simple exclusion

We prove a functional central limit theorem for the position of a tagged particle in the one-dimensional asymmetric simple exclusion process for hyperbolic scaling, starting from a Bernoulli product measure conditioned to have a particle at the origin. We also prove that the position of the tagged particle at time t$t$ depends on the initial configuration, through the number of empty sites in the interval [0,(p−q)αt]$[0,(p-q)\alpha t]$ divided by α$\alpha$, on the hyperbolic time scale and on a longer time scale, namely N^4/3$N^{4/3}$.     

Large deviations for a reaction diffusion model

Summary We obtain large deviation estimates for the empirical measure of a class of interacting particle systems. These consist of a superposition of Glauber and Kawasaki dynamics and are described, in the hydrodynamic limit, by a reaction diffusion equation. We extend results of Kipnis, Olla and Varadhan for the symmetric exclusion process, and provide an approximation scheme for the rate functional. Some physical implications of our results are briefly indicated.     

A system of infinitely many mutually reffecting Brownian balls in ℝ d

Sumamry An infinite system of Skorohod type equations is studied. The unique solution of the system is obtained from a finite case by passing to the limit. It is a diffusion process describing a system of infinitely many Brownian hard balls and has a Gibbs state associated with the hard core pair potential as a reversible measure.On leave of, Department of Mathematics and Informatics, Faculty of Science Chiba University Chiba, 263 JapanSupported by Swiss National Foundation, contract Nr. 20-36305.92