Posta G.
Spectral Gap for an Unrestricted Kawasaki Type Dynamics
(105K, Plain TeX)
ABSTRACT. We give an accurate asymptotic estimate for the gap of the generator of
a particular interacting particle system.
The model we consider may be informally
described as follows. A certain number of charged particles moves on the segment $[1,L]\cap\natural$
according to a Markovian law.
If $\h_k\in\integer$ is the charge at a
site $k\in [1,L]\cap\natural$ one unitary charge, positive or negative, jumps to a neighboring site,
$k\pm1$ at a rate which
depends on the charge at site $k$ and at site $k\pm1$.
The total charge $\sum_{k=1}^L\h_k$ is preserved by the dynamics,
in this sense our dynamics is similar to the Kawasaki dynamics, but in our case there is no
restriction on the maximum charge allowed per site.
The model is
equivalent to an interface dynamics connected with the stochastic Ising model
at very low temperature:
the ``unrestricted solid on solid model''.
Thus the results we obtain may be read as results for this model.
We give necessary and sufficient conditions to ensure that
gap shrinks as $L^{-2}$, independently of
the total charge.
We follow the method outlined in some papers by Yau ([Lu,Ya], [Ya])
where a similar spectral gap is proved for the original Kawasaki dynamics.