Craig A. Tracy Joint work with Harold Widom

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• Physicists expect KPZ equation to describe a large class of stochastically growing interfaces: KPZ Universality Remarks • KPZ eqn mathematically difficult to handle so make discrete approximation in space • ASEP is one discrete version of KPZ; thus expect ``universal behavior’’ in limit theorems • TASEP is ASEP <strong>with</strong> jumps only to left or jumps only to right. TASEP is a simpler model (determinantal process) Sunday, March 29, 2009 7
Total Current I(x,t) Step Initial Condition Take q>p net drift to left I(x,t) = # of particles ≤ x at time t, x ≤ 0 Let η(x,t)=1 if particle at x at time t otherwise 0 so I(x,t)=∑ y≤x η(y,t) Sunday, March 29, 2009 8
Page 1 and 2: The Asymmetric Simple Exclusion Pro
Page 3 and 4: ASEP on Integer Lattice q p p≠q
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: • Physicists expect KPZ equation
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
Page 65 and 66: 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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