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

In the simple exclusion process on ZZ, a particle waits for a unit exponential time, then tries to jump to its right with probability p≥1/2p≥1/2 and left with probability q=1−pq=1−p. In asymmetric case, the only non-translation invariant extremal stationary measures are πκπκ together with its shifts, where κ=q/pκ=q/p. In this paper, we investigate the asymptotic behaviour of πκπκ as κ↑1κ↑1, and obtain the following results. (a) πκπκ weakly converges to the Bernoulli measure with density 1/21/2; (b) Let αk(x)αk(x) be the occupation probability at site xx, then ακ(⋅)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.