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.
As already mentioned, Broutin and Marckert @cite_50 obtain Theorem in the standard case, via Prim's algorithm construction invented for the purpose of their study, and notably different from the approach presented here. Before them @cite_9 @cite_60 proved f.d.d. convergence for models similar to Erd "os-R 'enyi random graph. For the standard additive coalescent, analogous results were obtained rather early by Bertoin @cite_31 @cite_22 and Chassaing and Louchard @cite_49 , and are rederived in @cite_50 , again via an appropriate Prim's algorithm representation.
@cite_22 @cite_60 @cite_9 @cite_50 @cite_49 @cite_31
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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 revisit the discrete additive and multiplicative coalescents, starting with n particles with unit mass. These cases are known to be related to some “combinatorial coalescent processes”: a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process ((G(n,p))_p ) in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as (n + ). We propose to use the Prim order on the vertices instead of the classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows one to answer to a stronger version of an open question of Aldous (Ann Probab 25:812–854, 1997): we prove that not only the sequence of (rescaled) masses, seen as a process indexed by the time ( ), converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift ((B_t+ t - t^2 2, t 0) ), but we also construct a version of the standard augmented multiplicative coalescent of (Probab Theory Relat, 2013) using an additional Poisson point process. In this article, we show several results obtained by combining the use of stable distributions with pseudorandom generators for bounded space. In particular:---We show that, for any p ∈ (0, 2], one can maintain (using only O(log n e2) words of storage) a sketch C(q) of a point q ∈ lnp under dynamic updates of its coordinates. The sketch has the property that, given C(q) and C(s), one can estimate Vq − sVp up to a factor of (1 p e) with large probability. This solves the main open problem of [1999].---We show that the aforementioned sketching approach directly translates into an approximate algorithm that, for a fixed linear mapping A, and given x ∈ ℜn and y ∈ ℜm, estimates VAx − yVp in O(n p m) time, for any p ∈ (0, 2]. This generalizes an earlier algorithm of Wasserman and Blum [1997] which worked for the case p e 2.---We obtain another sketch function C′ which probabilistically embeds ln1 into a normed space lm1. The embedding guarantees that, if we set m e log(1 Δ)O(1 e), then for any pair of points q, s ∈ ln1, the distance between q and s does not increase by more than (1 p e) with constant probability, and it does not decrease by more than (1 − e) with probability 1 − Δ. This is the only known dimensionality reduction theorem for the l1 norm. In fact, stronger theorems of this type (i.e., that guarantee very low probability of expansion as well as of contraction) cannot exist [Brinkman and Charikar 2003].---We give an explicit embedding of ln2 into lnO(log n)1 with distortion (1 p 1 nΘ(1)). Recently, in [Random Struct Algorithm 41 2012, 441-450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 2007, 133-142], in turn based on ideas of Martin-Lof [J Appl Probab 23 1986, 265-282], Karp [Random Struct Alg 1 1990, 73-93] and Aldous [Ann Probab 25 1997, 812-854], to prove asymptotic normality of the number L1 of vertices in the largest component L1 of the random r-uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L1, and joint asymptotic normality of L1 and the number M1 of edges of L1 in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 2016, 21-75], where we enumerate sparsely connected hypergraphs asymptotically. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325-352, 2017