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In are all distinct. Therefore the RSK correspondence holds as before, except that Q becomes a standard tableau; thus, a word maps to a pair of arbitrary Young tableaux (P, Q), with P semi-standard and Q standard and converse. Also the correspondence with integer matrices is the same, with the extra requirement that the matrix contains 0 and 1’s, with each row containing exactly one 1. Consider now the uniform probability Pn,q on words in Snq ; from the RSK correspondence we have the one-to-one correspondence, given a fixed partition λ, {words in Sqn leading to shape λ} semi-standard tableaux of ⇐⇒ shape λ, filled with integers 1, .

P − q + j)! 1! 2! (p − q − 1)! j=0 This ends the proof of Prop. 5. 2 Percolation Model with Geometrically Distributed Entries Consider the ensemble Mat(p,q) = {p × q matrices M with entries Mij = 0, 1, 2, . . } with independent and geometrically distributed entries, for fixed 0 < ξ < 1, P(Mij = k) = (1 − ξ)ξ k , k = 0, 1, 2, . . 3 (Johansson [57]). Then L(M ) := max all such paths Mij , over right/down paths starting from entry (1, 1) to (p, q) (1, 1) (1, q) (p, 1) (p, q) Fig. 2. 1 Random and Integrable Models in Mathematics and Physics 27 has the following distribution, assuming q ≤ p, (1 − ξ)pq ξ |λ| ˜sλ (1q )˜sλ (1p ) P(L(M ) ≤ l) = λ∈Ymin(q,p) λ1 ≤l q −1 = Zp,q Δq (h1 , .

5. d. random variables ω(x, t), with x ∈ Z, t ∈ Z+ , P(ω(x, t) = k) = (1 − ξ)ξ k , k ∈ Z+ , except ω(x, t) = 0 if t − x is even or |x| > t . Define inductively a growth process, with a height curve h(x, t), with x ∈ Z, t ∈ Z+ , given by h(x, 0) = 0 , h(x, t + 1) = max h(x − 1, t), h(x, t), h(x + 1, t) + ω(x, t + 1) . For this model, see [59, 66, 77]. 6. The height curve at even sites 2x at times 2t − 1 is given by h(2x, 2t − 1) = max V (i, j), all such paths over right/down paths starting from entry (1, 1) to (t+x, t−x) where V (i, j) := ω(i − j, i + j − 1) .

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A Postmodernist Reading of Henrik Ibsen’s Ghosts issue No. 4, 2011 by Noorbakhsh Hooti & Amin Davoodi

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