5 research outputs found
On the mixing time of the 2D stochastic Ising model with "plus" boundary conditions at low temperature
We consider the Glauber dynamics for the 2D Ising model in a box of side L,
at inverse temperature and random boundary conditions whose
distribution P either stochastically dominates the extremal plus phase (hence
the quotation marks in the title) or is stochastically dominated by the
extremal minus phase. A particular case is when P is concentrated on the
homogeneous configuration identically equal to + (equal to -). For
large enough we show that for any there exists
such that the corresponding mixing time satisfies
. In the non-random case
(or ), this implies that . The same bound holds when the boundary conditions are all
+ on three sides and all - on the remaining one. The result, although still
very far from the expected Lifshitz behaviour , considerably
improves upon the previous known estimates of the form . The techniques are based on induction over length
scales, combined with a judicious use of the so-called "censoring inequality"
of Y. Peres and P. Winkler, which in a sense allows us to guide the dynamics to
its equilibrium measure.Comment: 39 pages, 8 figures; v2: typos corrected, two references added. To
appear on Comm. Math. Phy