The multiplicative coalescent is a Markov process taking values in ordered @math . It is a mean-field process in which any pair of blocks coalesces at rate proportional to the product of their masses. In Aldous and Limic (1998) each extreme eternal version @math of the multiplicative coalescent was described in three different ways. One of these specifications matches the (marginal) law of @math to that of the ordered excursion lengths above past minima of @math , where @math is a certain Levy-type process which (modulo shift and scaling) has infinitesimal drift @math at time @math . Using a modification of the breadth-first-walk construction from Aldous (1997) and Aldous and Limic (1998), and some new insight from the thesis by Uribe (2007), this work settles an open problem (3) from Aldous (1997), in the more general context of Aldous and Limic (1998). Informally speaking, @math is entirely encoded by @math , and contrary to Aldous' original intuition, the evolution of time for @math does correspond to the linear increase in the constant part of the drift of @math . In the "standard multiplicative coalescent" context of Aldous (1997), this result was first announced by Armendariz in 2001, and obtained in a recent preprint by Broutin and Marckert, who simultaneously account for the process of excess edge counts (or marks). The novel argument presented here is based on a sequence of relatively elementary observations. Some of its components (for example, the new dynamic random graph construction via "simultaneous" breadth-first walks) are of independent interest, and may be useful for obtaining more sophisticated asymptotic results on near critical random graphs and related processes.
The present approach to Theorem is of independent interest even in the standard setting (where Section would simplify further, since @math , and already Lemma 8 from @cite_14 would be sufficient for making conclusions in Section ). In addition, it may prove useful for continued analysis of the s, as well as various other processes in the domain of attraction''.
@cite_14
2011017194 2963702702 1971206237 2053661713
As Bleher (J. Stat. Phys. 66(1):315–373, 1992) observed the free flight vector of the planar, infinite horizon, periodic Lorentz process S n ∣n=0,1,2,… belongs to the non-standard domain of attraction of the Gaussian law—actually with the ( n n ) scaling. Our first aim is to establish his conjecture that, indeed, ( S_ n n n ) converges in distribution to the Gaussian law (a Global Limit Theorem). Here the recent method of Balint and Gouezel (Commun. Math. Phys. 263:461–512, 2006), helped us to essentially simplify the ideas of our earlier sketchy proof (Szasz, D., Varju, T. in Modern dynamical systems and applications, pp. 433–445, 2004). Moreover, we can also derive (a) the local version of the Global Limit Theorem, (b) the recurrence of the planar, infinite horizon, periodic Lorentz process, and finally (c) the ergodicity of its infinite invariant measure. We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. (Random Str Algorithm 37(1):85–99, 2010) asked whether there exists an increasing (M )-Lipschitz embedding from one i.i.d. Bernoulli sequence into an independent copy with positive probability. We give a positive answer for large enough (M ). A closely related problem is to show that two independent Poisson processes on ( R ) are roughly isometric (or quasi-isometric). Our approach also applies in this case answering a conjecture of Szegedy and of Peled (Ann Appl Probab 20:462–494, 2010). Our theorem also gives a new proof to Winkler’s compatible sequences problem. Our approach does not explicitly depend on the particular geometry of the problems and we believe it will be applicable to a range of multi-scale and random embedding problems. The main result of the paper consists of the theorem that under certain, natural assumptions the local conditional maximum @math of the function f on the set [ A = x R^n | _i (x) 0, _j (x) = 0,i = 1, ,k,j = 1, ,l ] is identical with the unconditional maximum of the potential function [ p(x, ) = f(x) + _ i = 1 ^k neg ( _i (x)) - _ j = 1 ^l | (x) | , x R^n , 0, ] for @math sufficiently small. There is also provided a draft of a modified gradient procedure for maximizing the potential @math since it is generally nonsmooth even for differentiable f, @math and @math .