scholarly journals Limit shapes of Gibbs distributions on the set of integer partitions: The expansive case

2008 ◽  
Vol 44 (5) ◽  
pp. 915-945 ◽  
Author(s):  
Michael M. Erlihson ◽  
Boris L. Granovsky
Keyword(s):  
Integer Partitions ◽  
Gibbs Distributions ◽  
Limit Shapes

scholarly journals Integer partitions and exclusion statistics: limit shapes and the largest parts of Young diagrams

2007 ◽  
Vol 2007 (10) ◽  
pp. P10001-P10001 ◽  
Author(s):  
Alain Comtet ◽  
Satya N Majumdar ◽  
Stéphane Ouvry ◽  
Sanjib Sabhapandit
Keyword(s):  
Integer Partitions ◽  
Young Diagrams ◽  
Limit Shapes

scholarly journals Limit Shapes via Bijections

2018 ◽  
Vol 28 (2) ◽  
pp. 187-240 ◽  
Author(s):  
STEPHEN DeSALVO ◽  
IGOR PAK
Keyword(s):  
Scaling Functions ◽  
Integer Partitions ◽  
Limit Shape ◽  
Limit Shapes

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined in [43], and extend them to include limit shapes with different scaling functions.

scholarly journals Counting monster potentials

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Riccardo Conti ◽  
Davide Masoero
Keyword(s):  
Excited States ◽  
Ground State ◽  
Hermite Polynomials ◽  
Large Momentum ◽  
Integer Partitions ◽  
Complex Equilibria ◽  
State Potential

Abstract We study the large momentum limit of the monster potentials of Bazhanov-Lukyanov-Zamolodchikov, which — according to the ODE/IM correspondence — should correspond to excited states of the Quantum KdV model.We prove that the poles of these potentials asymptotically condensate about the complex equilibria of the ground state potential, and we express the leading correction to such asymptotics in terms of the roots of Wronskians of Hermite polynomials.This allows us to associate to each partition of N a unique monster potential with N roots, of which we compute the spectrum. As a consequence, we prove — up to a few mathematical technicalities — that, fixed an integer N , the number of monster potentials with N roots coincides with the number of integer partitions of N , which is the dimension of the level N subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

scholarly journals Coagulation Processes with Gibbsian Time Evolution

2012 ◽  
Vol 49 (03) ◽  
pp. 612-626
Author(s):  
Boris L. Granovsky ◽  
Alexander V. Kryvoshaev
Keyword(s):  
Stochastic Process ◽  
Recurrence Relation ◽  
Time Evolution ◽  
Nonnegative Function ◽  
Gibbs Distribution ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Gibbs Distributions ◽  
Giant Cluster ◽  
Positive Integers

We prove that a stochastic process of pure coagulation has at any timet≥ 0 a time-dependent Gibbs distribution if and only if the rates ψ(i,j) of single coagulations are of the form ψ(i;j) =if(j) +jf(i), wherefis an arbitrary nonnegative function on the set of positive integers. We also obtain a recurrence relation for weights of these Gibbs distributions that allow us to derive the general form of the solution and the explicit solutions in three particular cases of the functionf. For the three corresponding models, we study the probability of coagulation into one giant cluster by timet> 0.

scholarly journals Probabilistic Divide-and-Conquer: A New Exact Simulation Method, With Integer Partitions as an Example

2016 ◽  
Vol 25 (3) ◽  
pp. 324-351 ◽  
Author(s):  
RICHARD ARRATIA ◽  
STEPHEN DeSALVO
Keyword(s):  
Success Probability ◽  
Simulation Method ◽  
New Method ◽  
Divide And Conquer ◽  
Integer Partitions ◽  
Rejection Sampling ◽  
Substantial Fraction ◽  
Recursive Scheme ◽  
Rejection Cost ◽  
Sampling Cost

We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from asymptotically order n3/4 to a constant. We show other examples for which a non-recursive, one-time application of probabilistic divide-and-conquer removes a substantial fraction of the rejection sampling cost.We also present a variation of probabilistic divide-and-conquer for generating i.i.d. samples that exploits features of the coupon collector's problem, in order to obtain a cost that is sublinear in the number of samples.

scholarly journals How Gibbs distributions may naturally arise from synaptic adaptation mechanisms

2009 ◽  
Vol 10 (S1) ◽  
Author(s):  
Juan C Vasquez ◽  
Bruno Cessac ◽  
Horacio Rostro-Gonzalez ◽  
Thierry Vieville
Keyword(s):  
Adaptation Mechanisms ◽  
Gibbs Distributions

scholarly journals Limit Shapes for Gibbs Ensembles of Partitions

2018 ◽  
Vol 172 (6) ◽  
pp. 1545-1563
Author(s):  
Ibrahim Fatkullin ◽  
Valeriy Slastikov
Keyword(s):  
Limit Shapes
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