Craig A. Tracy Joint work with Harold Widom

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The asymmetric simple exclusion process (ASEP): Introduced in 1970 by Frank Spitzer in “Interaction of Markov Processes.” The “default stochastic model for transport phenomena”. The ``Ising model of nonequilibrium phenomena’’. ASEP is a model for interacting particles on a lattice. Frank Spitzer Sunday, March 29, 2009 2
ASEP on Integer Lattice q p p≠q Sunday, March 29, 2009 3
Page 1: The Asymmetric Simple Exclusion Pro
Page 5 and 6: ASEP on Integer Lattice q p p≠q
Page 7 and 8: Initial Conditions Sunday, March 29
Page 9 and 10: Initial Conditions Step Initial Con
Page 11 and 12: Growth Processes & ASEP ● ● ●
Page 13 and 14: KPZ Equation & Growth Processes ❘
Page 15 and 16: • Physicists expect KPZ equation
Page 17 and 18: • Physicists expect KPZ equation
Page 19 and 20: Total Current I(x,t) Step Initial C
Page 21 and 22: Current & Position of m th particle
Page 23 and 24: Current & Position of m th particle
Page 25 and 26: N=1 ASEP probability q probability
Page 27 and 28: N=2 ASEP ● ● ● ● ● ●
Page 29 and 30: Bethe Ansatz Solution of Master Equ
Page 31 and 32: Bethe Ansatz Solution of Master Equ
Page 33 and 34: Remarks • Want BC to be satisfied
Page 35 and 36: Remarks • Want BC to be satisfied
Page 37 and 38: Remarks • To recap: We have found
Page 39 and 40: Remarks • To recap: We have found
Page 41 and 42: Theorem (TW): If p ≠ 0 and r is s
Page 43 and 44: Marginal Distributions P(x m (t)≤
Page 45 and 46: First Identity ∑ sgn σ ⎛ ⎝
Page 47 and 48: Prob(x 1 (t)=x) = ∫ p N(N−1)/2
Page 53 and 54:
Remarks Sunday, March 29, 2009 23
Page 55 and 56:
Remarks • To go beyond the left-m
Page 57 and 58:
Second Identity S ranges over subse
Page 59 and 60:
Final series result for case Y = Z
Page 61 and 62:
Final series result for case Y = Z
Page 63 and 64:
Final expression for m th particle
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ξ −→ 1 − τη 1 − η , τ
Page 67 and 68:
Two Preliminary Results Proposition
Page 69 and 70:
Denote by R the resolvent kernel of
Page 71 and 72:
Main Result We set σ = m t ,c 1 =
Page 73 and 74:
Remarks Sunday, March 29, 2009 35
Page 75 and 76:
Remarks • Theorems generalize Kur
Page 77 and 78:
Remarks • Theorems generalize Kur
Page 79 and 80:
● Would like a conceptual underst
Page 81 and 82:
● Would like a conceptual underst
Page 83:
Thanks to Anne Schilling & Doron Ze
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Craig A. Tracy Joint work with Harold Widom

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