Analysis of Critical Phenomena in Random Systems

点击翻译按钮获取中文摘要

We investigated the spectral gap of Glauber dynamics in a finite square in the low temperature case in two dimensions. This is known to have a uniform lower bound independent of the boundary condition when the temperature is higher than the critical point, and goes to zero as the size of the square goes to infinity in the low temperature case.We obtained a sharpe lower bound of the spectral gap for the + boundary condition up to the critical point : gap(L,+)【greater than or equal】exp{-CィイD8LlogLィエD8}, where L is the size of the box. This bound is better than the existing one. It is even better than the estimate which is known only for the low enough temperature.Further, we showed that the uniform spectral gap is equivalent to the strong mixing condition even for unbounded spin systems.