Random real trees

We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in particular of the relations between discrete Galton-Watson trees and continuous random trees. We then discuss the particular class of self-similar random real trees called stable trees, which generalize the CRT. We review several important results concerning stable trees, including their branching property, which is analogous to the well-known property of Galton-Watson trees, and the calculation of their fractal dimension. We then consider spatial trees, which combine the genealogical structure of a real tree with spatial displacements, and we explain their connections with superprocesses. In the last section, we deal with a particular conditioning problem for spatial trees, which is closely related to asymptotics for random planar quadrangulations.Comment: 25 page

Bessel processes, the Brownian snake and super-Brownian motion

We prove that, both for the Brownian snake and for super-Brownian motion in dimension one, the historical path corresponding to the minimal spatial position is a Bessel process of dimension -5. We also discuss a spine decomposition for the Brownian snake conditioned on the minimizing path.Comment: Submitted to the special volume of S\'eminaire de Probabilit\'es in memory of Marc Yo

Tandemly repeated C-C-C-C-A-A hexanucleotide of Tetrahymena rDNA is present elsewhere in the genome and may be related to the alteration of the somatic genome.

The ribosomal RNA genes of the Tetrahymena macronucleus exist as extrachromosomal, linear molecules. The termini of these molecules have been shown to contain the tandemly repeated hexanucleotide (C-C-C-C-A-A)n. In this study the same or related sequences were found in other locations of the genome. Using the depurination method, we showed that macronuclear DNA contained this sequence even after rDNA had been removed. The sequence was found mainly in the repetitive fraction of the DNA. The presence of this sequence in both the macronucleus and the micronucleus was also shown by Southern hybridization using C-C-C-C-A-A repeat as a probe. Comparison between the hybridization patterns of macronuclei and micronuclei reveals interesting differences. Whereas the two nuclei share the same genetic origin, the majority of the restriction enzyme digestion sites flanking the C-C-C-C-A-A repeat appear to be different. Such a difference was found to be specific for this sequence, because it was not detected when other sequences were used for hybridization. These results suggest that some kind of alteration has occurred in the genome during the formation of the macronucleus, and that the C-C-C-C-A-A repeat may be related to this process

Feller property and infinitesimal generator of the exploration process

We consider the exploration process associated to the continuous random tree (CRT) built using a Levy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale

Shaping the dust mass - star-formation rate relation

There is a remarkably tight relation between the observationally inferred dust masses and star-formation rates (SFRs) of SDSS galaxies, Mdust $\propto$ SFR$^{1.11}$ (Da Cunha et al. 2010). Here we extend the Mdust-SFR relation to the high end and show that it bends over at very large SFRs (i.e., dust masses are lower than predicted for a given SFR). We identify several distinct evolutionary processes in the diagram: (1) A star-bursting phase in which dust builds up rapidly at early times. The maximum attainable dust mass in this process is the cause of the bend-over of the relation. A high dust-formation efficiency, a bottom-light initial mass function, and negligible supernova shock dust destruction are required to produce sufficiently high dust masses. (2) A quiescent star-forming phase in which the subsequent parallel decline in dust mass and SFR gives rise to the Mdust-SFR relation, through astration and dust destruction. The dust-to-gas ratio is approximately constant along the relation. We show that the power-law slope of the Mdust-SFR relation is inversely proportional to the global Schmidt-Kennicutt law exponent (i.e., $\sim 0.9$) in simple chemical evolution models. (3) A quenching phase which causes star formation to drop while the dust mass stays roughly constant or drops proportionally. Combined with merging, these processes, as well as the range in total baryonic mass, give rise to a complex population of the diagram which adds significant scatter to the original Mdust-SFR relation. (4) At very high redshifts, a population of galaxies located significantly below the local relation is predicted.Comment: 5 pages, 1 figure, ApJL, in pres

The topological structure of scaling limits of large planar maps

We discuss scaling limits of large bipartite planar maps. If p is a fixed integer strictly greater than 1, we consider a random planar map M(n) which is uniformly distributed over the set of all 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space M(n) equipped with the graph distance rescaled by the factor n to the power -1/4 converges in distribution as n tends to infinity towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.Comment: 45 pages Second version with minor modification

Work-rate of substitutes in elite soccer: A preliminary study

The aim of this study was to investigate the work-rate of substitutes in professional soccer. A computerised player tracking system was used to assess the work-rates of second-half substitutes (11 midfielders and 14 forwards) in a French Ligue 1 club. Total distance, distance covered in five categories of movement intensity and recovery time between high-intensity efforts were evaluated. First- and second-half work-rates of the replaced players were compared. The performance of substitutes was compared to that of the players they replaced, to team-mates in the same position who remained on the pitch after the substitution and in relation to their habitual performances when starting games. No differences in work-rate between first- and second-halves were observed in all players who were substituted. In the second-half, a non-significant trend was observed in midfield substitutes who covered greater distances than the player they replaced whereas no differences were observed in forwards. Midfield substitutes covered a greater overall distance and distance at high-intensities (p<0.01) and had a lower recovery time between high-intensity efforts (p<0.01) compared to other midfield team-mates who remained on the pitch. Forwards covered less distance (p<0.01) in their first 10-minutes as a substitute compared to their habitual work-rate profile in the opening 10-minutes when starting matches while this finding was not observed in midfielders. These findings suggest that compared to midfield substitutes, forward substitutes did not utilise their full physical potential. Further investigation is warranted into the reasons behind this finding in order to optimise the work-rate contributions of forward substitutes