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Register a new userA phase transition in excursions from infinity of the fast fragmentation-coalescence process
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An important property of Kingmans coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as `coming down from infinity. Moreover, of the many different (exchangeable) stochastic coalescent models, Kingmans coalescent is the `fastest to come down from infinity. In this article we study what happens when we counteract this `fastest coalescent with the action of an extreme form of fragmentation. We augment Kingmans coalescent, where any two blocks merge at rate $c>0$, with a fragmentation mechanism where each block fragments at constant rate, $lambda>0$, into its constituent elements. We prove that there exists a phase transition at $lambda=c/2$, between regimes where the resulting `fast fragmentation-coalescence process is able to come down from infinity or not. In the case that $lambda<c/2$ we develop an excursion theory for the fast fragmentation-coalescence process out of which a number of interesting quantities can be computed explicitly.
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For $alpha >0$, the $alpha$-Lipschitz minorant of a function $f : mathbb{R} rightarrow mathbb{R}$ is the greatest function $m : mathbb{R} rightarrow mathbb{R}$ such that $m leq f$ and $vert m(s) - m(t) vert leq alpha vert s-t vert$ for all $s,t in mathbb{R}$, should such a function exist. If $X=(X_t)_{t in mathbb{R}}$ is a real-valued Levy process that is not a pure linear drift with slope $pm alpha$, then the sample paths of $X$ have an $alpha$-Lipschitz minorant almost surely if and only if $mathbb{E}[vert X_1 vert]< infty$ and $vert mathbb{E}[X_1]vert < alpha$. Denoting the minorant by $M$, we consider the contact set $mathcal{Z}:={ t in mathbb{R} : M_t = X_t wedge X_{t-}}$, which, since it is regenerative and stationary, has the distribution of the closed range of some subordinator made stationary in a suitable sense. We provide a description of the excursions of the Levy process away from its contact set similar to the one presented in It^o excursion theory. We study the distribution of the excursion on the special interval straddling zero. We also give an explicit path decomposition of the other generic excursions in the case of Brownian motion with drift $beta$ with $vert beta vert < alpha$. Finally, we investigate the progressive enlargement of the Brownian filtration by the random time that is the first point of the contact set after zero.
We show that in homogeneous fragmentation processes the largest fragment at time $t$ has size $e^{-t Phi(bar{p})}t^{-frac32 (log Phi)(bar{p})+o(1)},$ where $Phi$ is the Levy exponent of the fragmentation process, and $bar{p}$ is the unique solution of the equation $(log Phi)(bar{p})=frac1{1+bar{p}}$. We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.
We show the result that is stated in the title of the paper, which has consequences about decomposition of Brownian loop-soup clusters in two dimensions.
Growth-fragmentation processes describe the evolution of systems in which cells grow slowly and fragment suddenly. Despite originating as a way to describe biological phenomena, they have recently been found to describe the lengths of certain curves in statistical physics models. In this note, we describe a new growth-fragmentation connected to random planar maps with faces of large degree, having as a key ingredient the ricocheted stable process recently discovered by Budd. The process has applications to the excursions of planar Brownian motion and Liouville quantum gravity.
We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension $dgeq 2,$ and a connectivity phase transition whenever $dgeq 4.$ We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point. A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this restriction results in a fractal ellipsoid model which we describe in detail, as it is key to obtaining our main results. In addition we also determine the almost sure Hausdorff dimension of the fractal set.
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