A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase (γ < √ 2d) and that the limit is universal (i.e., the limiting measure is independent of the regularisation of the underlying field).
We consider a system of particles which perform branching Brownian motion
with negative drift and are killed upon reaching zero, in the near-critical
regime where the total population stays roughly constant with approximately N
particles. We show that the characteristic time scale for the evolution of this
population is of order $(\log N)^3$, in the sense that when time is measured in
these units, the scaled number of particles converges to a variant of Neveu's
continuous-state branching process. Furthermore, the genealogy of the particles
is then governed by a coalescent process known as the Bolthausen-Sznitman
coalescent. This validates the nonrigorous predictions by Brunet, Derrida,
Muller and Munier for a closely related model.Comment: Published in at http://dx.doi.org/10.1214/11-AOP728 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Coalescents with multiple collisions, also known as $\Lambda$-coalescents,
were introduced by Pitman and Sagitov in 1999. These processes describe the
evolution of particles that undergo stochastic coagulation in such a way that
several blocks can merge at the same time to form a single block. In the case
that the measure $\Lambda$ is the $\operatorname {Beta}(2-\alpha,\alpha)$
distribution, they are also known to describe the genealogies of large
populations where a single individual can produce a large number of offspring.
Here, we use a recent result of Birkner et al. to prove that Beta-coalescents
can be embedded in continuous stable random trees, about which much is known
due to the recent progress of Duquesne and Le Gall. Our proof is based on a
construction of the Donnelly--Kurtz lookdown process using continuous random
trees, which is of independent interest. This produces a number of results
concerning the small-time behavior of Beta-coalescents. Most notably, we
recover an almost sure limit theorem of the present authors for the number of
blocks at small times and give the multifractal spectrum corresponding to the
emergence of blocks with atypical size. Also, we are able to find exact
asymptotics for sampling formulae corresponding to the site frequency spectrum
and the allele frequency spectrum associated with mutations in the context of
population genetics.Comment: Published in at http://dx.doi.org/10.1214/009117906000001114 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Consider a Λ-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number Nt of blocks at any positive time t > 0). We exhibit a deterministic function v : (0, ∞) → (0, ∞) such that Nt/v(t) → 1, almost surely, and in L p for any p ≥ 1, as t → 0. Our approach relies on a novel martingale technique.
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