2016/01/06 by Vlada Limic, Limic, Vlada
Computer Science · Mathematics · Physics and Astronomy · #60 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1601.01325
openalex publication_date 2016/01/06 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
The multiplicative coalescent is a Markov process taking values in ordered\nl2. It is a mean-field process in which any pair of blocks coalesces at rate\nproportional to the product of their masses. In Aldous and Limic (1998) each\nextreme eternal version (\X(t),- \∞ < t < \∞) of the\nmultiplicative coalescent was described in three different ways. One of these\nspecifications matches the (marginal) law of \X(t) to that of the\nordered excursion lengths above past minima of L\X(s) +ts, ,s\n\≥ 0 , where L\X is a certain L 'evy-type process which\n(modulo shift and scaling) has infinitesimal drift -s at time s.\n Using a modification of the breadth-first-walk construction from Aldous\n(1997) and Aldous and Limic (1998), and some new insight from the thesis by\nUribe (2007), this work settles an open problem (3) from Aldous (1997), in the\nmore general context of Aldous and Limic (1998). Informally speaking,\n\X is entirely encoded by L\X, and contrary to Aldous'\noriginal intuition, the evolution of time for \X does correspond to\nthe linear increase in the constant part of the drift of L\X. In\nthe "standard multiplicative coalescent" context of Aldous (1997), this result\nwas first announced by Armend 'ariz in 2001, and obtained in a recent preprint\nby Broutin and Marckert, who simultaneously account for the process of excess\nedge counts (or marks).\n The novel argument presented here is based on a sequence of relatively\nelementary observations. Some of its components (for example, the new dynamic\nrandom graph construction via "simultaneous" breadth-first walks) are of\nindependent interest, and may be useful for obtaining more sophisticated\nasymptotic results on near critical random graphs and related processes.\n