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Boltzmann Samplers For The Random Generation Of Combinatorial Structures
 Combinatorics, Probability and Computing
, 2004
"... This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combina ..."
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Cited by 107 (3 self)
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This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class  an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on realarithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.
Bins And Balls: Large Deviations Of The Empirical Occupancy Process
 KWABENA DOKUAMPONSAH AND PETER M ÖRTERS
, 2000
"... . In the random allocation model, balls are sequentially inserted at random into n exchangeable bins. The occupancy score of a bin denotes the number of balls inserted in this bin. The (random) distribution of occupancy scores denes the object of this paper: the empirical occupancy measure which is ..."
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Cited by 14 (0 self)
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. In the random allocation model, balls are sequentially inserted at random into n exchangeable bins. The occupancy score of a bin denotes the number of balls inserted in this bin. The (random) distribution of occupancy scores denes the object of this paper: the empirical occupancy measure which is a probability measure over the integers. This measurevalued random variable packages many useful statistics. This paper characterizes the Large Deviations of the ow of empirical occupancy measures when n goes to innity while the number of inserted balls remains proportional to n. The main result is a Sanovlike theorem for the empirical occupancy measure when the set of probability measures over the integers is endowed with metrics that are slightly stronger than the total variation distance. Thanks to a coupling argument, this result applies to the degree distribution of sparse random graphs. 1. Introduction Consider the following classical model in random combinatorics. At each time ...
Random Sampling from Boltzmann Principles
, 2002
"... This extended abstract proposes a surprisingly simple framework for the random generation of combinatorial configurations based on Boltzmann models. Random generation of possibly complex structured objects is performed by placing an appropriate measure spread over the whole of a combinatorial class. ..."
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Cited by 13 (2 self)
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This extended abstract proposes a surprisingly simple framework for the random generation of combinatorial configurations based on Boltzmann models. Random generation of possibly complex structured objects is performed by placing an appropriate measure spread over the whole of a combinatorial class. The resulting algorithms can be implemented easily within a computer algebra system, be analysed mathematically with great precision, and, when suitably tuned, tend to be efficient in practice, as they often operate in linear time.
The limit shape and fluctuations of random partitions of naturals with fixed number of summands
 Moscow Math. J
, 2001
"... Dedicated to Robert Minlos on the occasion of his 70 th birthday Abstract. We consider the uniform distribution on the set of partitions of integer n with c √ n numbers of summands, c> 0 is a positive constant. We calculate the limit shape of such partitions, assuming c is constant and n tends to ..."
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Dedicated to Robert Minlos on the occasion of his 70 th birthday Abstract. We consider the uniform distribution on the set of partitions of integer n with c √ n numbers of summands, c> 0 is a positive constant. We calculate the limit shape of such partitions, assuming c is constant and n tends to infinity. If c → ∞ then the limit shape tends to known limit shape for unrestricted number of summands (see reference [11]). If the growth is slower than √ n then the limit shape is universal (e −t). We prove the invariance principle (central limit theorem for fluctuations around the limit shape) and find precise expression for correlation functions. These results can be interpreted in terms of statistical physics of ideal gas, from this point of view the limit shape is a limit distribution of the energy of two dimensional ideal gas with respect to the energy of particles. The proof of the limit theorem uses partially inversed Fourier transformation of the characteristic function and refines the methods of the previous papers of authors (see references).
Kronecker products, characters, partitions, and the tensor square conjectures
"... Abstract. We study the remarkable Saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups Sn contain all irreducibles as their constituents. Our main result is that they contain representations corresponding to hooks and two row Young diagrams. ..."
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Cited by 7 (7 self)
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Abstract. We study the remarkable Saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups Sn contain all irreducibles as their constituents. Our main result is that they contain representations corresponding to hooks and two row Young diagrams. For that, we develop a new sufficient condition for the positivity of Kronecker coefficients in terms of characters, and use combinatorics of rim hook tableaux combined with known results on unimodality of certain partition functions. We also present connections and speculations on random characters of Sn. 1. Introduction and
The nature of partition bijections II. Asymptotic stability
"... We introduce a notion of asymptotic stability for bijections between sets of partitions and a class of geometric bijections. We then show that a number of classical partition bijections are geometric and that geometric bijections under certain conditions are asymptotically stable. ..."
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Cited by 5 (3 self)
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We introduce a notion of asymptotic stability for bijections between sets of partitions and a class of geometric bijections. We then show that a number of classical partition bijections are geometric and that geometric bijections under certain conditions are asymptotically stable.
Hydrodynamic limit for an evolutional model of twodimensional Young diagrams
, 2009
"... We construct dynamics of twodimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik [17], are uni ..."
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Cited by 4 (2 self)
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We construct dynamics of twodimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik [17], are uniform measures under conditioning on their size (or equivalently, area). We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain nonlinear partial differential equation under a proper hydrodynamic scaling. Furthermore, the stationary solution of the limit equation is identified with the socalled Vershik curve. We discuss both uniform and restricted uniform statistics for the Young diagrams. 1
The Wulff construction in statistical mechanics and in combinatorics. Russ
 Math. Surv
, 2001
"... We present the geometric solutions to some variational problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, we discuss the construction which exhibit the shape of a typical Young diagram and of a typical skyscraper. ..."
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Cited by 3 (1 self)
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We present the geometric solutions to some variational problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, we discuss the construction which exhibit the shape of a typical Young diagram and of a typical skyscraper.
Fluctuations of the BoseEinstein condensate
 J. Phys. A
, 2014
"... Abstract. This article gives a rigorous analysis of the fluctuations of the BoseEinstein condensate for a system of noninteracting bosons in an arbitrary potential, assuming that the system is governed by the canonical ensemble. As a result of the analysis, we are able to tell the order of fluctua ..."
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Abstract. This article gives a rigorous analysis of the fluctuations of the BoseEinstein condensate for a system of noninteracting bosons in an arbitrary potential, assuming that the system is governed by the canonical ensemble. As a result of the analysis, we are able to tell the order of fluctuations of the condensate fraction as well as its limiting distribution upon proper centering and scaling. This yields interesting results. For example, for a system of n bosons in a 3D harmonic trap near the transition temperature, the order of fluctuations of the condensate fraction is n−1/2 and the limiting distribution is normal, whereas for the 3D uniform Bose gas, the order of fluctuations is n−1/3 and the limiting distribution is an explicit nonnormal distribution. For a 2D harmonic trap, the order of fluctuations is n−1/2(logn)1/2, which is larger than n−1/2 but the limiting distribution is still normal. All of these results come as easy consequences of a general theorem. 1.
unknown title
, 1999
"... Geometric variational problems of statistical mechanics and of combinatorics ..."
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Geometric variational problems of statistical mechanics and of combinatorics