Integral Formulas for the Asymmetric Simple Exclusion Process a Craig A. Tracy & Harold Widom SPA07 University of Illinois at Urbana-Champaign August 6 10, 2007 a arxiv:0704.2633, supported in part by the National Science Foundation 1
The asymmetric exclusion process (ASEP), introduced in 1970 by Frank Spitzer (1926 1992) in Interaction of Markov Processes, has become the default stochastic model for transport phenomena (H.-T. Yau). Many have called it the Ising model of nonequilibrium statistical physics. 2
Definition of Model The ASEP is a model for interacting particles on a lattice S, say S = Z d. 1. A state η of the system is a map η : S {0, 1} such that 1 if site x S is occupied by a particle, η(x) = 0 if site x S is vacant. States Ω = {0, 1} S. 2. Introduce dynamics: t η t Ω: (a) Particle x S waits an exponential time with parameter 1; (b) at the end of that time, it chooses a y S with probability p(x, y); and (c) if y is vacant, it goes to y, while if y is occupied, it stays at x. 3
Markov Chain Aspects Without the exclusion of part (c), we would have particles moving on S according to independent, continuous time Markov chains on S that have unit exponential holding times. Now take S = Z with p(x, y) = q p if y = x + 1 q if y = x 1 0 otherwise p x 1 x x+1 4
T(totally)ASEP: p = 1. Random matrix connection (Kurt Johansson 2000): Initial configuration Z. Probability that at time t the particle initially at m has moved at least n ( m) times equals C m,n [0,t] m 0 i<j<m(τ i τ j ) 2 i ( t = C m,n det 0 (τ n m i ) τ i+j+n m e τ dτ. e τ i )d m τ = distribution of largest eigenvalue in the unitary Laguerre ensemble. Thus underlying determinantal process Not the case for p < 1. 5
N-particle ASEP: Possible configuration X = {x 1,...,x N }, x 1 < < x N, x i Z. Initial config.: Y = {y 1,...,y N }. We obtain integral formulas for P Y (X; t) = prob. that at time t the system is in configuration X. P(x m (t) = x) = prob. that at time t the mth particle from the left is at x. TASEP: (p = 1) Schütz (1997): P Y (X; t) = N N determinant. Rákos-Schütz (2005): Schütz Johansson (2000). Particles only influenced by those to the right. 6
Differential Equation New notation u(x; t) or u(x) for P Y (X; t). Initial condition: u(x; 0) = δ Y (X). For N = 2, x 1 < x 2, If x 2 x 1 > 1 then d dt u(x 1, x 2 ) = p u(x 1 1, x 2 ) + q u(x 1 + 1, x 2 ) +p u(x 1, x 2 1) + q u(x 1, x 2 + 1) 2 u(x 1, x 2 ). If x 2 x 1 = 1 then d dt u(x 1, x 2 ) = p u(x 1 1, x 2 ) + q u(x 1, x 2 + 1) u(x 1, x 2 ). Formally subtract when x 2 = x 1 + 1: 0 = p u(x 1, x 1 ) + q u(x 1 + 1, x 1 + 1) u(x 1, x 1 + 1). First equation + last condition second equation. 7
General N: X = {x 1,...,x N } Z N. Differential equation (master equation): d dt u(x; t) = i [p u(...,x i 1,...) + q u(...,x i + 1,...) u(x)]. Boundary conditions (i = 1,...,N 1): u(...,x i, x i + 1,...) = p u(..., x i, x i,...) + q u(...,x i + 1, x i + 1,...). Initial condition u(x; 0) = δ Y (X) when x 1 < < x N. Equation + boundary conditions + initial condition u(x; t) = P Y (X; t) when x 1 < < x N. 8
Bethe Ansatz Solution ε(ξ) := p ξ 1 + q ξ 1. For any ξ 1,...,ξ N C\{0} a solution of the DE is ( ) ξ x j j e ε(ξ j) t. For any σ S N, another solution is j j ξ x i σ(j) j e ε(ξ j) t or any linear combination of these, or any integral of a linear combination. Bethe Ansatz: u(x; t) = σ S N F σ (ξ) j ξ x j σ(j) e ε(ξ j) t d N ξ. j Want the boundary conditions to be satisfied. 9
Look for F σ such that the integrand satisfies the BCs pointwise: F σ (p+q ξ σ(i) ξ σ(i+1) ξ σ(i+1) ) (ξ σ(i) ξ σ(i+1) ) xi ξ x j σ(j) = 0. σ S N j i, i+1 Definition: T i σ differs from σ by an intechange of the ith and (i + 1)st entries. If σ = (2 3 1 4) then T 2 σ = (2 1 3 4). Replace σ by T i σ and add. Sufficient conditions F Ti σ F σ = p + qξ σ(i)ξ σ(i+1) ξ σ(i+1) p + qξ σ(i) ξ σ(i+1) ξ σ(i). General solution F σ (ξ) = sgn σ j<k(p + qξ σ(j) ξ σ(k) ξ σ(j) ) ϕ(ξ). Choose ϕ(ξ) so that initial condition is satisfied by σ = id summand. 10
Take F id (ξ) = j ξ y j 1 j C N j since ξ x j y j 1 j d N ξ = δ Y (X). Upshot: If we define A σ = sgnσ i<j (p + qξ σ(i)ξ σ(j) ξ σ(i) ) i<j (p + qξ iξ j ξ i ) then u(x; t) = σ A σ (ξ) i ξ x i σ(i) i ( ξ y i 1 i e ε(ξ i) t ) d N ξ (1) satisfies the master equation + boundary conditions. The σ = id summand satisfies initial condition. 11
TASEP A σ = sgnσ (1 ξ σ(i) ) σ(i) i, ( ) u(x; t) = det (1 ξ) j i ξ x i y j 1 e (ξ 1 1)t dξ. Schütz (1997): If all contours C r with r < 1 the initial conditions are satisfied. Therefore ( ) P Y (X; t) = det (1 ξ) j i ξ x i y j 1 e (ξ 1 1)t dξ. Cr 12
ASEP (TW, 2007) Want contours so that when x 1 < < x N σ id A σ (ξ) i ξ x i σ(i) i ξ y i 1 i d N ξ = 0. N = 2: If both contours are small enough get Schütz s formula. N = 3: Take all contours small. Three integrals are zero, other two are negatives. So for N = 2, 3, P Y (X; t) equals (1) with integrals over C r, r 1. General N: I(σ) = A σ (ξ) i ξ x i σ(i) i ξ y i 1 i d N ξ. Some I(σ) = 0, others come in pairs (σ, σ ) such that I(σ) + I(σ ) = 0. 13
P Y (X; t) := prob. that at time t the system is in configuration X. Theorem (TW): If p 0 and r is small enough then P Y (X; t) = A σ (ξ) ( ξ x i σ(i) ξ y i 1 i e ε(ξ i) t ) d N ξ. Cr N i σ S N i where and satisfies A σ = sgnσ i<j (p + qξ σ(i)ξ σ(j) ξ σ(i) ) i<j (p + qξ iξ j ξ i ) P Y (X; 0) = δ Y (X). 14
Left-Most Particle: Formulas for P(x 1 (t) = x) Since x 1 < < x N set x 1 = x, x 2 = x + z 1,...,x N = x + z 1 + z 2 + z N 1. When r < 1 can sum the formula for P Y (X; t) over z i > 0. Integrand becomes i (ξx y i 1 i e ε(ξi)t ) i<j (p + qξ iξ j ξ i ) sgnσ + qξ σ(i) ξ σ(j) ξ σ(i) ) i<j(p σ ξ σ(2) ξ 2 σ(3) ξn 1 σ(n) (1 ξ σ(2) ξ σ(3) ξ σ(n) )(1 ξ σ(3) ξ σ(n) ) (1 ξ σ(n) ) A miracle! The sum equals p (1 ξ N(N 1)/2 1 ξ N ) i<j (ξ j ξ i ) i (1 ξ i). ). 15
Define I(x, Y, ξ) = i<j ξ j ξ i 1 ξ 1 ξ N p + qξ i ξ j ξ i i (1 ξ i) i (ξ x y i 1 i e ε(ξ i)t ). Theorem (TW): When p 0 and r is small enough P(x 1 (t) = x) = p N(N 1)/2 C N r I(x, Y, ξ) d N ξ. Remark: To show that x= P(x 1(t) = x) = 1, to compute E(x 1 (t)) or to let N, need integrals over C R with R large. Expand contours: Another miracle only poles at ξ i = 1 contribute. 16
For S {1,...,N}, I(x, Y S, ξ) is I(x, Y, ξ) but only indices i S. Theorem (TW): When q 0 and R is large enough P(x 1 (t) = x) = S c S C S R I(x, Y S, ξ) d S ξ, where c S = p σ(s) S q σ(s) S ( S +1)/2, σ(s) = i S i. For initial configuration sum over all finite subsets of Z +. y 1 < y 2 < + 17
Formulas for P(x m (t) = x) x 1 = x v m 1 v 1,, x m 2 = x v 2 v 1, x m 1 = x v 1, x m = x, x m+1 = x + z 1,..., x N = x + z 1 + + z N m. Sum over z i > 0, v i > 0, so take integrals over C r and C R. Take S {1,...,N} with S = m and sum over all σ such that σ({1,...,m}) S, σ({m + 1,...,N}) S c. Use the miraculous formula twice and another one, S =m p + qξ i ξ j ξ i ξ j ξ i i S, j S c (1 j S c ξ j ) = C N,m (1 N j=1 ξ j ). 18
[ ] N m [N] = pn q N, [N]! = [N] [N 1] [1], p q = [N]! [m]! [N m]!, C N,m = q m [ N 1 m (q binomial coefficient), ]. Theorem (TW): When p 0 P(x m (t) = x) = S c <m c N,m,S C S r I(x, Y S, ξ) d S ξ. When q 0 P(x m (t) = x) = S m c m,s C S R I(x, Y S, ξ) d S ξ. Second representation holds for N =. 19
Y = Z + : Define k-dimensional integrand J k (x, ξ) = i j ξ j ξ i p + qξ i ξ j ξ i 1 ξ 1 ξ k (1 ξ i ) (qξ i p) i i ( ξ x 1 i e ε(ξ i)t ). Corollary. When Y = Z + and q 0 P(x m (t) = x) = k m c m,k C k R J k (x, ξ) d k ξ. When p = 0 (left-moving TASEP) only c m,m survives. Get m m Toeplitz determinant ( ) P(x m (t) x) = det ξ i j+x 1 (ξ 1) m e (ξ 1)t dξ. C R This equals the Rákos-Schütz determinant Johansson s result. 20
We acknowledge help from Anne Schilling and Doron Zeilberger. 21
References 1. H.A. Bethe, On the theory of metals, I. Eigenvalues and eigenfunctions of a linear chain of atoms (German), Zeits. Phys. 74, 205 226 (1931). 2. F. Spitzer, Interaction of Markov processes. Adv. Math. 5, 246 290 (1970). 3. G.M. Schütz, Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Physics 88 (1997), 427 445. 4. K. Johansson, Shape fluctuations and random matrices. Comm. Math. Phys. 209, 437 476 (2000). 5. A. Rákos and G.M., Schütz, Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Physics 118, 511 530 (2005). 22