B. Logan and L. Shepp, A variational problem for random Young tableaux, Adv. in Math., 26, (1977), pp. 206 - 222.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to conjecture that the answer asymptotically approaches c n for some constant c 1. Based on more extensive computations, Baer and Brock ( 1] conjectured that c = 2. Hammersley in [8] proved that =2 c e, and presented heuristic arguments for the value c = 2. Logan and Shepp proved in [9] that c 2, and Pilpel [12] presented an elementary proof of this inequality. Finally, Ver sik and Kerov ( 15] settled the conjecture by proving c 2. The structure of the distribution of L n has been studied in [2] 5] 11] The multidimensional generalization of the monotone subsequence ....

B. Logan and L. Shepp, A variational problem for random Young tableaux, Adv. in Math., 26, (1977), pp. 206 - 222.

....w2Sn is n (w) Elementary arguments show that 1 n E(n) e n; and Hammersley [33, Thm. 4] showed in 1972, using subadditive ergodic theory, that the limit c = lim E(n) exists. Vershik and Kerov [68] with the dicult direction c 2 shown independently by Logan and Shepp [43]) showed in 1977 that c = 2. A partition of the integer n 0, denoted n, is a sequence ( 1 ; 2 ; of nonnegative integers satisfying 1 2 0 and i = n. If k 1 = k 2 = 0, then we also write = 1 ; k ) A standard Young tableau (SYT) of ....

B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206-222.

....= w2Sn is n (w) Elementary arguments show that 1 n E(n) e n; and Hammersley [21, Thm. 4] showed in 1972, using subadditive ergodic theory, that the limit c = lim E(n) exists. Vershik and Kerov [40] with the dicult direction c 2 shown independently by Logan and Shepp [28]) showed in 1977 that c = 2. The proof of Vershik Kerov and Logan Shepp is based on the identity n 1 f 2 ; 5) where = 1 ; 2 ; and f denotes the number of SYT of shape as in Section 3. Equation (5) is due to Craige Schensted [34] and is an immediate consequence ....

B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206-222.

.... holds under the weaker hypothesis that there exists a constant A 0 for which 1 A n and ( A n for all n 1. 11 Given 0, let Par (n) f n : 2 ) n 1 (2 ) and (2 ) n ( 2 ) ng: It is a consequence of the work of Logan and Shepp [5] or Vershik and Kerov [13] see e.g. 1] for much stronger results) that for any 0, 2Par (n) t n ; n 1: Thus not only is the sum N(n; asymptotic to f t n =k as n 1 (as follows from (7) but the terms f contributing to most of the sum are close to f =k . ....

B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206-222.

....In this case, M is called the Plancherel measure which arises in the representation theory of the symmetric group S. Denote by L ) the random variable under the Plancherel measure M, and set L = L . In 1977, the limiting expected shape of under M is obtained in [48] also independently in [35] for the so called Poissonized Plancherel measure. In particular, it is shown that lim E(Ln) 2. 1.3) A central limit theorem for L is then obtained by [3] where F2 is he so called Tracy widom distribution function, which is expressed in erms of a solution o he Painlev II equation (see ....

B. Logan and L. Shepp. A variational problem for random Young tableaux. Adv. in Math., 26:206-222, 1977.

....give the full conjecture. 2. Average behavior Ulam [15] was apparently the first one to ask about the distribution of L n ,the length of the longest increasing subsequence in a permutation of n distinct real numbers. After initial work of Baer and Brock [1] and Hammersley [6] Logan and Shepp [9] and Vershik and Kerov [16] proved the conjecture that L n tends to 2n in probability ##. Later it was shown by Frieze [5] that the distribution of L n is concentrated near its mean. Frieze s result was subsequently sharpened by Bollob as and Brightwell [2] and Talagrand [14] Some of the fine ....

B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances Math. 26 (1977), 206--222.

....we use the same notation . In this identi cation, l n ( is the length of the longest increasing subsequence of a random permutation. The longest increasing subsequence has been of great interest for a long time (see e.g. 2] 32] 4] Especially as n 1, it is known that E(l n ) 2 p n [30], 45, 46] also [1, 37, 26] and Var(l n ) c 0 n 1=3 [4] with some numerical constant c 0 = 0:8132 . Moreover, the limiting distribution of l n after proper scaling is obtained in [4] in terms of the solution to the Painlev e II equation (see Section 3 for precise statement) An ....

....uctuation in the limit by Theorem 4.1. In fact, it is shown in [33, 10, 27] that, for the case of = 2, in the large n limit, the number of boxes in the k th of a random Young diagram corresponds to the k th largest eigenvalue of a random GUE matrix. Also the typical shape of is obtained in [30] and [45] which is related to Wigner s semicircle law [28] On the other hand, the second row of a random Young diagram for = 1 is discussed in [8] implying that it corresponds to the second eigenvalue of a random GOE matrix. It would be interesting to obtain similar results for general row ....

B. Logan and L. Shepp. A variational problem for random Young tableaux. Adv. in Math., 26:206-222, 1977.

.... matrices in terms of the 2 D Coulomb gas [15] Wigner s construction [41] is at the center of a recent surge of research on random matrix theory, relating several diverse areas such as zeroes of Riemann zeta functions [35] 18] Riemann Hilbert problems [7] representations of the symmetric group [36], 32] non commutative probability theory [14] 40] mathematical physics [8] 37] 10] analysis of noisy data [38] and inverse scattering theory [39] After some simple manipulations, we obtain ZN = Z R 2 dz 1 : Z R 2 dz N N Y j=1 exp( r 2 j ) 0 Y j k l jk 1 A 2 =2 ....

B. Logan and L. A. Shepp, A variational problem for random Young tableaux, Adv. Math., 26, 206-222, 1977.

....was supported in part by the Swedish Natural Research Council (NFR) 1 2 JINHO BAIK, PERCY DEIFT, AND KURT JOHANSSON Subsequent numerical work by Baer and Brock [BB] in late 60 s suggested that value of c is 2. The existence of the limit was rigorously established by Hammersley [Ha] in 1972. In [LS], Logan and Shepp proved that c 2 and simultaneously Vershik and Kerov [VK1] see also [VK2] showed that c = 2, thus settling Ulam s problem. Alternative proofs of Ulam s problem are due to Aldous and Diaconis [AD] Sepp al ainen [Se1] and Johansson [Jo1] Over the years, various conjectures ....

....p N log P rob l N x p N = I(x) 1.20) 6 JINHO BAIK, PERCY DEIFT, AND KURT JOHANSSON For the lower tail the large deviation formula for l N is not the same as for L( the Poissonized case. Deuschel and Zeitouni in [DeZe2] use combinatorial and variational ideas from Logan and Shepp [LS] to prove that lim N 1 1 N log P rob l N x p N = H(x) 1.21) if 0 x 2, where H(x) 1 2 x 2 8 log x 2 1 x 2 4 log 2x 2 4 x 2 : 1.22) For the lower tail we have no analogue of (1.19) The rate functions U and H are related via a Legendre transform, see ....

B.F.Logan and L.A.Shepp, A variational problem for random Young tableaux, Advances in Math., 26, 206-222, (1977).

....that there exists a constant A 0 for which n 1 A p n and ( n ) A p n for all n 1. 11 Given ffl 0, let Par ffl (n) f n : 2 Gamma ffl) p n 1 (2 ffl) p n and (2 Gamma ffl) p n ( 2 ffl) p ng: It is a consequence of the work of Logan and Shepp [5] or Vershik and Kerov [13] see e.g. 1] for much stronger results) that for any ffl 0, X 2Par ffl (n) f t n ; n 1: Thus not only is the sum N(n; ff) P n f =ff asymptotic to f ff t n =k as n 1 (as follows from (7) but the terms f =ff contributing to most of the sum ....

B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206--222.

....the singularity is large. The generating function is entire, and so the singularity is at infinity. Example 10.5. Permutations without long increasing subsequences. Let u k (n) be the number of permutations of 1, 2, n that have no increasing subsequence of length k. Logan and Shepp [257] and Vershik and Kerov [370] established by calculus of variations and combinatorics that the average value of the longest increasing subsequence in a random permutation is asymptotic to 2n 1 2 . Frieze [149] has proved recently, using probabilistic methods, a stronger result, namely that almost ....

B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances Math., 26 (1977), 206--222. 178

....to conjecture that the answer asymptotically approaches c p n for some constant c 1. Based on more extensive computations, Baer and Brock ( 1] conjectured that c = 2. Hammersley in [8] proved that =2 c e, and presented heuristic arguments for the value c = 2. Logan and Shepp proved in [9] that c 2, and Pilpel [12] presented an elementary proof of this inequality. Finally, Ver sik and Kerov ( 15] settled the conjecture by proving c 2. The structure of the distribution of L n has been studied in [2] 5] 11] The multidimensional generalization of the monotone subsequence ....

B. Logan and L. Shepp, A variational problem for random Young tableaux, Adv. in Math., 26, (1977), pp. 206 - 222.

....w2Sn is n (w) Elementary arguments show that 1 2 p n E(n) e p n; and Hammersley [21, Thm. 4] showed in 1972, using subadditive ergodic theory, that the limit c = lim n 1 E(n) p n exists. Vershik and Kerov [37] with the dicult direction c 2 shown independently by Logan and Shepp [26]) showed in 1977 that c = 2. The proof of Vershik Kerov and Logan Shepp is based on the identity E(n) 1 n X n 1 f 2 ; 5) where = 1 ; 2 ; and f denotes the number of SYT of shape as in Section 3. Equation (5) is due to Craige Schensted [31] and is an immediate ....

B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206-222.

....DIFFERENTIAL MODEL OF GROWTH OF YOUNG DIAGRAMS S. Kerov 1. Introduction The starting point of the present paper is the result on the asymptotics of random growth of Young diagrams obtained in [1] [9]. The subject of these articles was a discrete time Markov process with the set of Young diagrams as a state space. At each stage of the process a new random box is attached to the current diagram. The definition of transition probabilities was motivated by asymptotic problems of representation ....

....Growth Process. In the process of growth, the area of a diagram increases to infinity. If a diagram is contracted uniformly, so that its area is normalized to unity, the edge of the diagram looks more and more like a continuous curve. The essential part of the result discovered and proved in [1] [9] is as follows: in the course of Plancherel Growth Process almost all of Young diagrams become uniformly close (assuming the area is normalized) to a common universal curve. In a natural coordinate system, the equation of the curve is (1) Omega Gamma u) ae 2 (u arcsin u 2 p 4 ....

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B.F.Logan, L.A.Shepp, A variational problem for random Young tableaux, Adv. Math. 26 (1977), 206 - 222.

....groups. There are two main results. It is shown in Section 3.3 that the transition probabilities for the Plancherel measure of the group S1 approach the distribution with semi circle density in the limit of infinitely growing Young diagrams. To this end we make essential use of the theorem of [3,11] on the limiting shape of large random diagrams. Section 4 contains an algorithm for the random division of an interval, controlled by the diagram (or, equivalently, by the function r) The algorithm produces a random variable with the distribution related to the function r by the equation ....

....x k Gamma xm ; it follows that c Gamma1 Y i=1 h ri h ri 1 = k Gamma1 Y m=1 x k Gamma ym x k Gamma xm : In the same way, r Gamma1 Y j=1 h jc h jc 1 = n Y m=k 1 x k Gamma ym Gamma1 x k Gamma xm : 11 3.3 The asymptotics for large Young diagrams. It was proved in [3,11,5] that in the course of random growth almost all Young diagrams become close (after suitable renormalization of the area) to the universal shape; actually to that of the diagram Omega of Example 1, formula (2.2.2) More precisely, the following result holds: Theorem [3,11] For almost all Young ....

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B.F.Logan, L.A.Shepp, A variational problem for random Young tableaux, Adv. Math. 26 (1977), 206 - 222.

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B. Logan and L. Shepp, A variational problem for random Young tableaux, Adv. in Math., 26, (1977), pp. 206 - 222.

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B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206--222.

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B.F.Logan and L.A.Shepp, A variational problem for random Young tableaux, Advances in Math., 26, 206-222, (1977).

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B.F.Logan and L.A.Shepp, A variational problem for random Young tableaux, Advances in Math., 26, 206-222, (1977).

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B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances Math. 26 (1977), 206--222.

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B.F. Logan and L.A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206-222.

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B.F. Logan and L.A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206-222.

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B.F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math., 26 (1977), pp. 206-222.

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B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), 206--222.

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B.F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math., 26 (1977), pp. 206-222.

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