# Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

### The asymmetric simple exclusion process: Integrable structure and limit theorems

**Mathematical Physics & Probability**

Speaker: | Craig Tracy, UC Davis |

Location: | 2112 MSB |

Start time: | Wed, Dec 3 2008, 4:10PM |

Since its introduction by Frank Spitzer nearly forty years ago, the asymmetric simple exclusion process (ASEP), has become the “default stochastic model for transport phenomena.” Some have called the ASEP the “Ising model for nonequilibrium physics.” In ASEP on the integer lattice Z particles move according to two rules: (1) A particle at x waits an exponential time with parameter one (independently of all the other particles), and then it chooses y with probability p(x, y); (2) If y is vacant at that time it moves to y, while if y is occupied it remains at x and restarts the clock. The adjective “simple” refers to the fact that allowed jumps are one step to the right, p(x, x + 1) = p, or one step to the left, p(x, x − 1) = 1 − p = q. The asymmetric condition means p 6= q so that there is a net drift to either the right or the left. In this lecture we consider ASEP on the integer lattice Z with step initial condition: At time zero the particles are located at Z+ = {1, 2, . . . , } and there is a drift to the left (q > p). If xm(t) denotes the position of the mth particle from the left at time t (so that xm(0) = m), a basic quantity is the distribution function P(xm(t) ≤ x) which describes the “current fluctuations.” Physicists have conjectured that the limiting distribution of xm(t) as m → , t → ∞ with m/t fixed is in the 1+1 KPZ Universality Class. We show that this is indeed the case and describe the limiting distribution function. (This limiting distribution function first appeared in the random matrix theory literature.) This result extends an earlier theorem of Kurt Johansson on the T(totally)ASEP where q = 1 and p = 0. This work is joint work with Harold Widom. The lecture itself is for a general audience.