percolation, self-avoiding walk and Ising model
abstract:
We consider a
long-range version of percolation, self-avoiding walk and the Ising
model, which we shall introduce first.
Let D(x,y) denote the transition probabilities for some random walk on the
hypercubic lattice Z^d. We want D to be translation invariant, symmetric,
and invariant under rotations by 90 degrees.
Now consider the related long-range percolation model: For two
lattice points x,y we let a bond {x,y} be occupied with probability z.D(x,y)
(and otherwise vacant), where z is a percolation parameter. A quantity of
interest is the probability G_z(x,y) that the lattice points x and y are
connected with a path of open bonds.
Similarly, we have a self-avoiding walk model by conditioning the ordinary
random walk with step distribution D not to intersect itself. For this model
we let G_z(x,y) be the associated Green's function.
We finally consider a long-range Ising model, where the spin-spin
interaction is related to D.
If D describes nearest-neighbor random walk (i.e., D(x,y)=1/(2d) for |x-y|=1
and 0 otherwise), then we obtain the (classical) nearest-neighbor versions
of SAW, percolation, and the Ising model. These are well-understood in
sufficiently high dimension through the work of Hara & Slade (1990,1992),
and Sakai (2007).
In recent joint work with Remco van der Hofstad and Akira Sakai we consider
the case where D(x,y) is of the order |x-y|^{-d-\alpha}. This is known as
long-range interaction. As a main result we obtain mean-field
behaviour for these models if d > 3.Min{2,\alpha} for percolation, and d
> 2.Min{2,\alpha} for self-avoiding walk and the Ising model. The proof of
this result is based on the lace expansion.
In the talk I will mainly concentrate on the percolation case, though the
method works (to some extent) uniformly for all three models. I shall begin
by discussing long-range percolation in general, and then discuss some
aspects of the proof.
abstract:
We study metastability and nucleation for the two-dimensional lattice gas with Kawasaki dynamics at low temperature and low density. Let $\beta>0$ be the inverse temperature and let $\Lambda_\beta \subset Z^2$ be a large finite box with periodic boundary conditions whose size depends on $\beta$. Particles perform a simple exclusion process on $\Lambda_\beta$, but when they occupy neighboring sites they feel a binding energy $-U<0$ that slows down their dissociation (i.e., the dynamics follows a Metropolis algorithm with an attractive lattice gas Hamiltonian).
The initial configuration is chosen
according to the grand-canonical Gibbs measure on $\Lambda_\beta$ with
activity energy $\Delta\in (U,2U)$ conditioned on all the droplets being
subcritical (i.e., at time zero the particle density equals $\rho =
e^{-\Delta\beta}$ and no stable droplets appear). For large $\beta$, the
system wants to nucleate because of the binding energy. We investigate how
this nucleation takes place under the dynamics. The restriction $\Delta\in
(U,2U)$ corresponds to the situation where the side length of the critical
droplet $l_c = \lceil U/(2U-\Delta)\rceil$ is neither 1 nor $\infty$ (i.e.,
the system is metastable).
Since the density is low, the creation of critical droplets occurs roughly
independently at different sites of $\Lambda_\beta$ and the nucleation is
triggered by one critical droplet that appears after many unsuccessful
attempts (i.e., the system shows homogeneous nucleation).
A key ingredient we use to approach the problem of metastability is the
ideal gas approximation for a two-dimensional rarefied gas under Kawasaki
dynamics. We show that, at fixed temperature and in the limit as the
particle density tends to zero, such a gas evolves in a way that is close to
an ideal gas, where particles have no interaction. In particular, we prove
three theorems showing that particle trajectories are non-superdiffusive and
have a diffusive spread-out property.
Our results are formulated in the more general context of systems of ``quasi
random walks'', of which we show that the lattice gas under Kawasaki
dynamics is an example. We are able to deal with a large class of initial
conditions having no anomalous concentration of particles and with time
horizons that are much larger than the typical particle collision time.