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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
The AizenmanSimsStarr and Guerra’s schemes for the SK model with multidimensional spins
, 2008
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Global divergence of spatial coalescents
 In preparation
, 2008
"... A class of processes called spatial Λcoalescents was recently introduced by Limic and Sturm (2006). In these models particles perform independent random walks on some underlying graph G. In addition, particles on the same site merge randomly according to some given coalescing mechanism. The goal of ..."
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A class of processes called spatial Λcoalescents was recently introduced by Limic and Sturm (2006). In these models particles perform independent random walks on some underlying graph G. In addition, particles on the same site merge randomly according to some given coalescing mechanism. The goal of the current work is to obtain several asymptotic results for these processes. If G = Z d, and the coalescing mechanism is Kingman’s coalescent, then starting with N particles at the origin, the number of particles is of order (log ∗ N) d at any fixed time (where log ∗ is the inverse tower function). At sufficiently large times this number is of order (log ∗ N) d−2. Betacoalescents behave similarly, with log log N in place of log ∗ N. Moreover, it is shown that on any graph and for general Λcoalescent, starting with infinitely many particles at a single site, the total number of particles will remain infinite at all times, almost surely.
ASYMPOTIC BEHAVIOR OF THE TOTAL LENGTH OF EXTERNAL BRANCHES FOR BETACOALESCENTS
"... Abstract. In this paper, we consider the Beta(2 − α,α)coalescents with 1 < α < 2 and study the moments of external branches, in particular the total external branch length L (n) ext of an initial sample of n individuals. For this class of coalescents, it has been proved that n α−1 (n) (d) T → ..."
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Abstract. In this paper, we consider the Beta(2 − α,α)coalescents with 1 < α < 2 and study the moments of external branches, in particular the total external branch length L (n) ext of an initial sample of n individuals. For this class of coalescents, it has been proved that n α−1 (n) (d) T → T, where T (n) is the length of an external branch chosen at random, and T is a known non negative random variable. We get the asymptotic behaviour of several moments of L (n) ext. As a consequence, we obtain that for Beta(2−α,α)coalescents with 1 < α < 2, lim n→+ ∞ n3α−5 E[(L (n) ext −n2−α E[T]) 2] =
Generalized Random Energy Model at complex temperatures
, 2014
"... Abstract. Motivated by the Lee–Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature β. We compute the limiting logpartition function and describe the fluctuations of the partition function. For the GREM with ..."
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Abstract. Motivated by the Lee–Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature β. We compute the limiting logpartition function and describe the fluctuations of the partition function. For the GREM with d levels, in total, there are 1 2 (d+ 1)(d+ 2) phases, each of which can symbolically be encoded as Gd1F d2Ed3 with d1, d2, d3 ∈ N0 such that d1 + d2 + d3 = d. In phase Gd1F d2Ed3, the first d1 levels (counting from the root of the GREM tree) are in the glassy phase (G), the next d2 levels are dominated by fluctuations (F), and the last d3 levels are dominated by the expectation (E). Only the phases of the form Gd1Ed3 intersect the real β axis. We describe the limiting distribution of the zeros of the partition function in the complex β plane ( = Fisher zeros). It turns out that the complex zeros densely touch the positive real axis at d points at which the GREM is known to undergo phase transitions. Our results confirm rigorously and considerably extend the replicamethod predictions from the physics literature. Figure 1. Phase diagram of the GREM in the complex β plane together with the level lines of the limiting logpartition function. See Figure 4 for details.
Fluctuations of the partition function in the GREM with external field
, 2008
"... We study Derrida’s generalized random energy model in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchica ..."
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We study Derrida’s generalized random energy model in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarsegraining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarsegrained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetization, as prescribed by strength of external field and by parameters of the GREM.
2 ASYMPOTIC BEHAVIOR OF THE TOTAL LENGTH OF EXTERNAL BRANCHES FOR BETA COALESCENTS
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