Craig A. Tracy Joint work with Harold Widom

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Two Preliminary Results Proposition: Let Γ be any closed curve going around η=1 once counterclockwise <strong>with</strong> η=1/τ on the outside. Then the Fredholm determinant of K(ξ,ξ’) acting on C R has the same Fredholm determinant as K 1 (η,η’)-K 2 (η,η’) acting on Γ. Proposition: Suppose the contour Γ is star-shaped <strong>with</strong> respect to η=0. Then the Fredholm determinant of K 1 acting on Γ is equal to ∞∏ Sunday, March 29, 2009 k=0 (1 − λτ k ) 30
Denote by R the resolvent kernel of K 1 Factor determinant: det(I-λ K)=det(I-λ K 1 ) det(I+K 2 (I+R)) Set λ=τ -m μ so formula for distr. fn becomes ∫ ∞ ∏ k=0 (1 − µτ k ) det ( I + τ −m µK 2 (I + R) ) dµ µ μ runs over a circle of radius > τ Sunday, March 29, 2009 31
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 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
Page 65 and 66: ξ −→ 1 − τη 1 − η , τ
Page 67: Two Preliminary Results Proposition
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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