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Random planar lattices and integrated superBrownian excursion
 Probab. Th. Rel. Fields
"... Abstract. In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scalin ..."
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Cited by 63 (3 self)
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Abstract. In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r = R−L of the support of the onedimensional ISE, or precisely: n −1/4 rn law − → (8/9) 1/4 r. More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive halfline, reminiscent of Cori and Vauquelin’s well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat’s construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks (e (n) , ˆ W (n) ) to the Brownian snake description (e, ˆ W) of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in twodimensional pure quantum gravity. 1.
An approximation algorithm for counting contingency tables
, 2008
"... We present a randomized approximation algorithm for counting contingency tables, m × n nonnegative integer matrices with given row sums R = (r1,..., rm) and column sums C = (c1,..., cn). We define smooth margins (R, C) in terms of the typical table and prove that for such margins the algorithm has ..."
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Cited by 2 (1 self)
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We present a randomized approximation algorithm for counting contingency tables, m × n nonnegative integer matrices with given row sums R = (r1,..., rm) and column sums C = (c1,..., cn). We define smooth margins (R, C) in terms of the typical table and prove that for such margins the algorithm has quasipolynomial N O(ln N) complexity, where N = r1 + · · · + rm = c1 + · · · + cn. Various classes of margins are smooth, e.g., when m = O(n), n = O(m) and the ratios between the largest and the smallest row sums as well as between the largest and the smallest column sums are strictly smaller than the golden ratio (1 + √ 5)/2 ≈ 1.618. The algorithm builds on Monte Carlo integration and sampling algorithms for logconcave densities, the matrix scaling algorithm, the permanent approximation algorithm, and an integral representation for the number of contingency tables.
Descent Functions and Random Young Tableaux
 Combin., Prob., and Computing
, 2000
"... The expectation of the descent number of a random Young tableau of a fixed shape is given, and concentration around the mean is shown. This result is generalized to the major index and to other descent functions. The proof combines probabilistic arguments together with combinatorial character theory ..."
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Cited by 1 (0 self)
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The expectation of the descent number of a random Young tableau of a fixed shape is given, and concentration around the mean is shown. This result is generalized to the major index and to other descent functions. The proof combines probabilistic arguments together with combinatorial character theory. Connections with Hecke algebras are mentioned. 1
Low rank approximations of symmetric polynomials and asymptotic counting of contingency tables, preprint arXiv math.CO/0503170
, 2005
"... We represent the number of m × n nonnegative integer matrices (contingency tables) with prescribed row sums and column sums as the expected value of the permanent of a nonnegative random matrix with exponentially distributed entries. We bound the variance of the obtained estimator, from which it f ..."
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Cited by 1 (1 self)
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We represent the number of m × n nonnegative integer matrices (contingency tables) with prescribed row sums and column sums as the expected value of the permanent of a nonnegative random matrix with exponentially distributed entries. We bound the variance of the obtained estimator, from which it follows that if the row and column sums are bounded by a constant fixed in advance, we get a polynomial time approximation scheme for counting contingency tables. We show that the complete symmetric polynomial of a fixed degree in n variables can be ǫapproximated coefficientwise by a sum of powers of O(log n) linear forms, from which it follows that if the row sums (but not necessarily column sums) are bounded by a constant, there is a deterministic approximation algorithm of m O(log n) complexity to compute the logarithmic asymptotic of the number of tables. 1. Introduction and
Discrete and Continuous: Two sides of the same?
"... How deep is the dividing line between discrete and continuous mathematics? Basic structures and methods of both sides of our science are quite different. But on a deeper level, there is a more solid connection than meets the eye. ..."
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Cited by 1 (0 self)
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How deep is the dividing line between discrete and continuous mathematics? Basic structures and methods of both sides of our science are quite different. But on a deeper level, there is a more solid connection than meets the eye.
Saturating Constructions for Normed Spaces
, 2004
"... We prove several results of the following type: given finite dimensional normed space V there exists another space X with log dimX = O(log dim V) and such that every subspace (or quotient) of X, whose dimension is not “too small, ” contains a further subspace isometric to V. This sheds new light on ..."
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We prove several results of the following type: given finite dimensional normed space V there exists another space X with log dimX = O(log dim V) and such that every subspace (or quotient) of X, whose dimension is not “too small, ” contains a further subspace isometric to V. This sheds new light on the structure of such large subspaces or quotients (resp., large sections or projections of convex bodies) and allows to solve several problems stated in the 1980s by V. Milman. 1