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Register a new userA growth-fragmentation connected to the ricocheted stable process
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Alexander R. Watson
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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.
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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 study the stochastic growth process in discrete time $x_{i+1} = (1 + mu_i) x_i$ with growth rate $mu_i = rho e^{Z_i - frac12 var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $dZ_t = - gamma Z_t dt + sigma dW_t$ sampled on a grid of uniformly spaced times ${t_i}_{i=0}^n$ with time step $tau$. Using large deviation theory methods we compute the asymptotic growth rate (Lyapunov exponent) $lambda = lim_{nto infty} frac{1}{n} log mathbb{E}[x_n]$. We show that this limit exists, under appropriate scaling of the O-U parameters, and can be expressed as the solution of a variational problem. The asymptotic growth rate is related to the thermodynamical pressure of a one-dimensional lattice gas with attractive exponential potentials. For $Z_t$ a stationary O-U process the lattice gas coincides with a system considered previously by Kac and Helfand. We derive upper and lower bounds on $lambda$. In the large mean-reversion limit $gamma n tau gg 1$ the two bounds converge and the growth rate is given by a lattice version of the van der Waals equation of state. The predictions are tested against numerical simulations of the stochastic growth model.
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.
We consider a discrete-time Markov chain, called fragmentation process, that describes a specific way of successively removing objects from a linear arrangement. The process arises in population genetics and describes the ancestry of the genetic material of individuals in a population experiencing recombination. We aim at the law of the process over time. To this end, we investigate sets of realisations of this process that agree with respect to a specific order of events and represent each such set by a rooted (binary) tree. The probability of each tree is, in turn, obtained by Mobius inversion on a suitable poset of all rooted forests that can be obtained from the tree by edge deletion; we call this poset the textit{pruning poset}. Dependencies within the fragments make it difficult to obtain explicit expressions for the probabilities of the trees. We therefore construct an auxiliary process for every given tree, which is i.i.d. over time, and which allows to give a pathwise construction of realisations that match the tree.
The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.
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