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Introduction Framework and basic properties

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38 pages
Introduction Framework and basic properties Case of general densities More precise results for smooth densities Application to Random Placement of Non-Overlapping Balls The problem of hard spheres Abstract probabilistic setting Geometric Analysis of Metropolis Algorithm on Bounded Domain L. Michel (joint work with P. Diaconis and G. Lebeau ) Laboratoire J.-A. Dieudonne Universite de Nice L. Michel (joint work with P. Diaconis and G. Lebeau ) Geometric Analysis of Metropolis Algorithm on Bounded Domain

  • metropolis algorithm

  • spheres abstract

  • probabilistic setting

  • markov kernel

  • phase transition studies

  • borel measure


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Introduction Framework and basic properties Case of general densities More precise results for smooth densities Application to Random Placement of Non-Overlapping Balls
Geometric
(joint
The problem of hard spheres Abstract probabilistic setting
Analysis of Metropolis Bounded Domain
work
with
L. Michel P. Diaconis and
G.
Laboratoire J.-A. Dieudonne ´ Universite´deNice
L. Michel (joint work with P. Diaconis and G. Lebeau )
Algorithm
Lebeau
)
on
Geometric Analysis of Metropolis Algorithm on Bounded Domain
Introduction Framework and basic properties Case of general densities More precise results for smooth densities Application to Random Placement of Non-Overlapping Balls
The problem of hard spheres Abstract probabilistic setting
Letµ=ρ(x)dxbe a probability measure on [a,b] and letfbe a regular function on [a,b]. We want to compute numerically the quantityI=b1aRbaf(x)dµ(x). Standard ”deterministic” method consist to divide [a,b] into Ninterval and to approximateIbyPkN=1AkwhereAkis the area corresponding to thekth interval. Probabilist approach: let (xn)nNbe a sequence of numbers in [a,b] such thatxnis choosen at random with respect toµ. Then, the quantityN1PNn=1f(xn) provides a good approximation ofI. A priori, ”choose a point at random with respect toµ” is not a simpler problem than ”computeI”. The Metropolis Algorithm provides an efficient procedure to sample fromµ.
L. Michel (joint work with P. Diaconis and G. Lebeau )
Geometric Analysis of Metropolis Algorithm on Bounded Domain
Introduction Framework and basic properties Case of general densities More precise results for smooth densities Application to Random Placement of Non-Overlapping Balls
The problem of hard spheres
The problem of hard spheres Abstract probabilistic setting
Consider a fixed box inRd,B=]1,1[d consider the. We problem of placement ofNballs of radius >0 with centers inB under the condition thatthe balls do not overlap. We denote ON,BN Wethe set of all possible configurations. endoweON, with the Lebesgue measuredL.
Problem: Build a sample of pointsX1, . . . ,Xr∈ ON, with respect todL.
distributed uniformly
This problem occurs in statistical physics in phase transition studies. It can be formulated in a more abstract setting.
L. Michel (joint work with P. Diaconis and G. Lebeau )
Geometric Analysis of Metropolis Algorithm on Bounded Domain