On the equilibrium state of the one-dimensional near-symmetric simple exclusion process

In the simple exclusion process on Z, a particle waits for a unit exponential time, then tries to jump to its right with probability p≥1/2 and left with probability q=1−p. In asymmetric case, the only non-translation invariant extremal stationary measures are πκ together with its shifts, where κ=q/p. In this paper, we investigate the asymptotic behaviour of πκ as κ↑1, and obtain the following results. (a) πκ weakly converges to the Bernoulli measure with density 1/2; (b) Let αk(x) be the occupation probability at site x, then ακ(⋅), after scaling properly, converges to a smooth curve; (c) The position of the leftmost particle, after scaling properly, converges in distribution. Furthermore, we show the explicit expression of the limit distribution.