I am a probabilist interested mainly but not only in random trees, discrete or continuous, the stochastic processes revolving around them, and their applications to evolutionary biology:
I carried out a doctoral thesis under the supervision of Amaury Lambert in LPSM (Sorbonne Université) and in the SMILE group hosted at Collège de France. The defence took place on December 2, 2019, and the manuscript can be found here.
SMILE, CIRB, Collège de France
11 place Marcelin Berthelot
75231 Paris
Similarly as in (Electron. J. Probab. 23 (2018)) where nested coalescent processes are studied, we generalize the definition of partition-valued homogeneous Markov fragmentation processes to the setting of nested partitions, i.e. pairs of partitions $\left(\zeta ,\xi \right)$ where $\zeta $ is finer than $\xi $. As in the classical univariate setting, under exchangeability and branching assumptions, we characterize the jump measure of nested fragmentation processes, in terms of erosion coefficients and dislocation measures. Among the possible jumps of a nested fragmentation, three forms of erosion and two forms of dislocation are identified – one being specific to the nested setting and relating to a bivariate paintbox process.
Poursuivant l’idée de (Electron. J. Probab. 23 (2018)) où les processus de coalescence emboîtés sont étudiés, nous étendons ici la définition des processus de fragmentation markoviens homogènes aux processus de fragmentation à valeurs dans les partitions emboîtées, c’est-à-dire les paires de partitions $\left(\zeta ,\xi \right)$ telles que $\zeta $ soit plus fine que $\xi $. Comme dans le contexte classique (dit univarié), sous des hypothèses d’échangeabilité et de branchement, nous caractérisons la mesure de saut des processus de fragmentation emboîtés en termes de coefficients d’érosion et de mesures de dislocation. Les sauts d’une fragmentation emboîtée peuvent être de plusieurs natures différentes : nous distinguons trois formes d’érosions et deux formes de dislocations, l’une d’elles étant spécifique au contexte des partitions emboîtées et étant générée par un processus de pots de peinture bivarié.
For the random interval partition of $[0,1]$ generated by the uniform stick-breaking scheme known as GEM$\left(1\right)$, let ${u}_{k}$ be the probability that the first $k$ intervals created by the stick-breaking scheme are also the first $k$ intervals to be discovered in a process of uniform random sampling of points from $[0,1]$. Then ${u}_{k}$ is a renewal sequence. We prove that ${u}_{k}$ is a rational linear combination of the real numbers $1,\zeta \left(2\right),\dots ,\zeta \left(k\right)$ where $\zeta $ is the Riemann zeta function, and show that ${u}_{k}$ has the limit $1/3$ as $k\to \u202f\infty $. Related results provide probabilistic interpretations of some multiple zeta values in terms of a Markov chain derived from the interval partition. This Markov chain has the structure of a weak record chain. Similar results are given for the GEM$\left(\theta \right)$ model, with beta$\left(1,\theta \right)$ instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function.
We consider the compact space of pairs of nested partitions of $\mathbb{N}$, where by analogy with models used in molecular evolution, we call “gene partition” the finer partition and “species partition” the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of $\Lambda $-coalescents to nested partitions. We characterize the law of SNEC processes as follows. In the absence of gene coalescences, species blocks undergo $\Lambda $-coalescent type events and in the absence of species coalescences, gene blocks lying in the same species block undergo i.i.d. $\Lambda $-coalescents. Simultaneous coalescence of the gene and species partitions are governed by an intensity measure ${\nu}_{s}$ on $\left(0,1]\times {\mathcal{M}}_{1}\right([0,1])$ providing the frequency of species merging and the law in which are drawn (independently) the frequencies of genes merging in each coalescing species block. As an application, we also study the conditions under which a SNEC process comes down from infinity.
Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure (infinitely-many-allele model) on the skeleton of the tree. The partition of the boundary into distinct alleles is the so-called allelic partition. In this paper, we are interested in the infinite trees generated by supercritical, possibly time-inhomogeneous, binary branching processes, and in their boundary, which is the set of particles “coexisting at infinity”. We prove that any such tree can be mapped to a random, compact ultrametric tree called the coalescent point process, endowed with a “uniform” measure on its boundary which is the limit as $t\to \u202f\infty $ of the properly rescaled counting measure of the population at time $t$. We prove that the clonal (i.e., carrying the same allele as the root) part of the boundary is a regenerative set that we characterize. We then study the allelic partition of the boundary through the measures of its blocks. We also study the dynamics of the clonal subtree, which is a Markovian increasing tree process as mutations are removed.
We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy-Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.
Starting from any graph on $\{1,\dots ,n\}$, consider the Markov chain where at each time-step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a stationary distribution whose support is the set of non-empty forests on $\{1,\dots ,n\}$. The random forest corresponding to this stationary distribution has interesting connections with the uniform rooted labeled tree and the uniform attachment tree. We fully characterize its degree distribution, the distribution of its number of trees, and the limit distribution of the size of a tree sampled uniformly. We also show that the size of the largest tree is asymptotically $\alpha logn$, where $\alpha ={(1-\u202flog\u202f(e-1)\u202f)}^{-1}\approx 2.18$, and that the degree of the most connected vertex is asymptotically $logn/\u202flog\u202flogn$.