the bolthausen sznitman coalescent and the genealogy of continuous state branching processes we use bochner s subordination to give a representation of the genealogical structure associated with general continuous state branching processes we then apply this representation to connections between a branching process introduced by neveu and the coalescent process recently investigated by bolthausen sznitman and others

thermodynamical limit for correlated gaussian random energy models let e n n be a family of n 2n centered unit gaussian random variables defined by the covariance matrix cn of elements cn av e n e n and the corresponding random hamiltonian then the quenched thermodynamical limit exists if for every decomposition n n1 n2 and all pairs n n where k k 1 2 are the projections of n into nk the condition is explicitly verified for the sherrington kirkpatrick the even p spin the derrida rem and the derrida gardner grem models

the effects of model selection on confidence intervals for the size of a closed population one encounters in the literature estimates of some rates of genetic and congenital disorders based on log linear methods to model possible interactions among sources often the analyst chooses the simplest model consistent with the data for estimation of the size of a closed population and calculates confidence intervals on the assumption that this simple model is correct however despite an apparent excellent fit of the data to such a model we note here that the resulting confidence intervals may well be misleading in that they can fail to provide an adequate coverage probability we illustrate this with a simulation for a hypothetical population based on data reported in the literature from three sources the simulated nominal 95 per cent confidence intervals contained the modelled population size only 30 per cent of the time only if external considerations justify the assumption of plausible interactions of sources would use of the simpler model s interval be justified

the λ coalescent speed of coming down from infinity consider a coalescent that comes down from infinity meani ng 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

speed and fluctuations of n particle branching brownian motion with spatial selection we consider branching brownian motion on the real line with the following selection mechanism every time the number of particles exceeds a large given number n only the n right most particles are kept and the others killed after rescaling time by log 3n we show that the properly recentred position of the lceil alpha n rceil th particle from the right alpha in 0 1 converges in law to an explicitly given spectrally positive levy process this behaviour has been predicted to hold for a large class of models falling into the universality class of the fkpp equation with weak multiplicative noise brunet et al in phys rev e 73 5 056126 2006 and is proven here for the first time for such a model

fragmentation associated with levy processes using snake we consider the height process of a levy process with no negative jumps and its associated continuous tree representation using levy snake tools developed by le gall le jan and duquesne le gall with an underlying poisson process we construct a fragmentation process which in the stable case corresponds to the self similar fragmentation described by miermont for the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment we also give a special markov property for the snake which is of its own interest

stochastic flows associated to coalescent processes we study a class of stochastic flows connected to the coalescent processes that have been studied recently by mohle pitman sagitov and schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size we define a bridge to be a right continuous process b r r 0 1 with nondecreasing paths and exchangeable increments such that b 0 0 and b 1 1 we show that flows of bridges are in one to one correspondence with the so called exchangeable coalescents this yields an infinite dimensional version of the classical kingman representation for exchangeable partitions of we then propose a poissonian construction of a general class of flows of bridges and identify the associated coalescents we also discuss an important auxiliary measure valued process which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized fleming viot process

a class of path valued markov processes and its applications to superprocesses let s be a continuous markov process satisfying certain regularity assumptions we introduce a path valued strong markov process associated with s which is closely related to the so called superprocess with spatial motion s in particular a subseth of the state space of s intersects the range of the superprocess if and only if the set of paths that hith is not polar for the path valued process the latter property can be investigated using the tools of the potential theory of symmetric markov processes a set is not polar if and only if it supports a measure of finite energy the same approach can be applied to study sets that are polar for the graph of the superprocess in the special case when s is a diffusion process we recover certain results recently obtained by dynkin

universal tree structures in directed polymers and models of evolving populations by measuring or calculating coalescence times for several models of coalescence or evolution with and without selection we show that the ratios of these coalescence times become universal in the large size limit and we identify a few universality classes

the brownian snake and solutions of δu u2 in a domain we investigate the connections between the path valued process called the brownian snake and nonnegative solutions of the partial differential equation u u2 in a domain of d in particular we prove two conjectures recently formulated by dynkin the first one gives a complete characterization of the boundary polar sets which correspond to boundary removable singularities for the equation u u2 the second one establishes a one to one correspondence between nonnegative solutions that are bounded above by a harmonic function and finite measures on the boundary that do not charge polar sets this correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the brownian snake our proofs combine probabilistic and analytic arguments an important role is played by a new version of the special markov property which is of independent interest

exact packing measure of the range of psi super brownian motions we consider super processes whose spatial motion is the d dimensional brownian motion and whose branching mechanism psi is critical or subcritical such processes are called psi super brownian motions if d 2 varvec gamma varvec gamma 1 where varvec gamma in 1 2 is the lower index of psi at infty then the total range of the psi super brownian motion has an exact packing measure whose gauge function is g r log log 1 r varphi 1 1 r log log 1 r 2 where varphi psi prime circ psi 1 more precisely we show that the occupation measure of the psi super brownian motion is the g packing measure restricted to its total range up to a deterministic multiplicative constant only depending on d and psi this generalizes the main result of duquesne ann probab 37 6 2431 2458 2009 that treats the quadratic branching case for a wide class of psi the constant 2 varvec gamma varvec gamma 1 is shown to be equal to the packing dimension of the total range

beta coalescents and continuous stable random trees coalescents with multiple collisions also known as a 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 a is the beta 2 a a 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

characterization of invariant measures at the leading edge for competing particle systems we study systems of particles on a line which have a maximum are locally finite and evolve with independent increments quasi stationary states are defined as probability measures on the algebra generated by the gap variables for which joint distribution of gaps between particles is invariant under the time evolution examples are provided by poisson processes with densities of the form p d x e sx s dx with s 0 and linear superpositions of such measures we show that conversely any quasi stationary state for the independent dynamics with an exponentially bounded integrated density of particles corresponds to a superposition of poisson processes with densities p dx e sx s dx with s 0 restricted to the relevant algebra among the systems for which this question is of some relevance are spin glass models of statistical mechanics where the point process represents the collection of the free energies of distinct pure states the time evolution corresponds to the addition of a spin variable and the poisson measures described above correspond to the so called rem states

effect of selection on ancestry an exactly soluble case and its phenomenological generalization we consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves for one particular model that we shall call the exponential model the properties of the traveling wave front can be calculated exactly as well as the statistics of the genealogy of the population one striking result is that for this particular model the genealogical trees have the same statistics as the trees of replicas in the parisi mean field theory of spin glasses we also find that in the exponential model the coalescence times along these trees grow like the logarithm of the population size a phenomenological picture of the propagation of wave fronts that we introduced in a previous work as well as our numerical data suggest that these statistics remain valid for a larger class of models while the coalescence times grow like the cube of the logarithm of the population size

a classification of coalescent processes for haploid exchangeable population models we consider a class of haploid population models with nonoverlapping generations and fixed population size n assuming that the family sizes within a generation are exchangeable random variables a weak convergence criterion is established for a properly scaled ancestral process as n oo it results in a full classification of the coalescent generators in the case of exchangeable reproduction in general the coalescent process allows for simultaneous multiple mergers of ancestral lines

exit times for an increasing levy tree valued process we give an explicit construction of the increasing tree valued process introduced by abraham and delmas using a random point process of trees and a grafting procedure this random point process will be used in companion papers to study record processes on levy trees we use the poissonian structure of the jumps of the increasing tree valued process to describe its behavior at the first time the tree grows higher than a given height using a spinal decomposition of the tree similar to the classical bismut and williams decompositions we also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree valued process

random overlap structures properties and applications to spin glasses random overlap structures rost s are random elements on the space of probability measures on the unit ball of a hilbert space where two measures are identified if they differ by an isometry in spin glasses they arise as natural limits of gibbs measures under the appropriate algebra of functions we prove that the so called cavity mapping on the space of rost s is continuous leading to a proof of the stochastic stability conjecture for the limiting gibbs measures of a large class of spin glass models similar arguments yield the proofs of a number of other properties of rost s that may be useful in future attempts at proving the ultrametricity conjecture lastly assuming that the ultrametricity conjecture holds the setup yields a constructive proof of the parisi formula for the free energy of the sherrington kirkpatrick model by making rigorous a heuristic of aizenman sims and starr

branching processes in levy processes laplace functionals of snakes and superprocesses we use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism this construction depends on a path valued process called the levy snake which is of independent interest our method of proof involves a calculation of the laplace functional of the occupation field of the levy snake this calculation relies on an evaluation of the corresponding moment functionals which requires precise information about the underlying genealogical structure

on ruelle s probability cascades and an abstract cavity method we construct in this work a markov process which describes a clustering mechanism through which equivalence classes on are progressively lumped together this clustering process gives a new description of ruelle s continuous probability cascades it also enables to introduce an abstract cavity method which mimicks certain features of the cavity method developed by physicists in the context of the sherrington kirkpatrick model

a fragmentation process connected to brownian motion let b s s 0 be a standard brownian motion and t 1 its first passage time at level 1 for every t 0 we consider ladder time set t of the brownian motion with drift t b t s b s ts and the decreasing sequence f t f 1 t f 2 t of lengths of the intervals of the random partition of 0 t 1 induced by t the main result of this work is that f t t 0 is a fragmentation process in the sense that for 0 t t f t is obtained from f t by breaking randomly into pieces each component of f t according to a law that only depends on the length of this component and independently of the others we identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by aldous and pitman 3

on mittag leffler distributions and related stochastic processes random variables with mittag leffler distribution can take values either in the set of non negative integers or in the positive real line they can be of two different types one type 1 heavy tailed with index 0 1 the other type 2 possessing all its moments we investigate various stochastic processes where they play a key role among which the discrete space time neveu branching process the discrete space continuous time neveu branching process the continuous space time neveu branching process csbp and renewal processes with rare events its relation to discrete or continuous self decomposability and branching processes with immigration is emphasized special attention will be paid to the neveu csbp for its connection with the bolthausen sznitman coalescent in this context and following a recent work of mohle 2015 a type 2 mittag leffler process turns out to be the siegmund dual to neveu s csbp block counting process arising in sampling from p d e t 0 further combinatorial developments of this model are investigated

stochastic equations flows and measure valued processes we first prove some general results on pathwise uniqueness comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and poisson random measures the results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions that is generalized fleming viot flows and flows of continuous state branching processes with immigration one of them unifies the different treatments of three kinds of flows in bertoin and le gall ann inst h poincare probab statist 41 2005 307 333 two scaling limit theorems for the generalized fleming viot flows are proved which lead to sub critical branching immigration superprocesses from those theorems we derive easily a generalization of the limit theorem for finite point motions of the flows in bertoin and le gall illinois j math 50 2006 147 181

spin glass computations and ruelle s probability cascades we study the parisi functional appearing in the parisi formula for the pressure of the sk model as a functional on ruelle s probability cascades rpc computation techniques for the rpc formulation of the functional are developed they are used to derive continuity and monotonicity properties of the functional retrieving a theorem of guerra we also detail the connection between the aizenman sims starr variational principle and the parisi formula as a final application of the techniques we rederive the almeida thouless line in the spirit of toninelli but relying on the rpc structure

a remark on the infinite volume gibbs measures of spin glasses in this note we point out that infinite volume gibbs measures of spin glass models on the hypercube can be identified as random probability measures on the unit ball of a hilbert space this simple observation follows from a result of dovbysh and sudakov on weakly exchangeable random matrices limiting gibbs measures can then be studied as single well defined objects this approach naturally extends the space of random overlap structures as defined by aizenman et al we discuss the ruelle probability cascades and the stochastic stability within this framework as an application we use an idea of parisi and talagrand to prove that if a sequence of finite volume gibbs measures satisfies the ghirlanda guerra identities then the infinite volume measure must be singular as a measure on a hilbert space

smaller population size at the mrca time for stationary branching processes we consider an elementary model of random size varying population governed by a stationary continuous state branching process we compute the distributions of various variables related to the most recent common ancestor mrca the time to the mrca the size of the current population and the size of the population just before the mrca in particular we observe a natural mild bottleneck effect as the size of the population just before the mrca is stochastically smaller than the size of the current population we also compute the number of individuals involved in the last coalescent event of the genealogical tree that is the number of individuals at the time of the mrca who have descendants in the current population by studying more precisely the genealogical structure of the population we get asymptotics for the number of ancestors just before the current time we give explicit computations in the case of the quadratic branching mechanism in this case the size of the population at the mrca is in mean 2 3 of the size of the current population we also provide in this case the fluctuations for the renormalized number of ancestors

branching processes in levy processes the exploration process the main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure the exploration process appears as a simple local time functional of a levy process with no negative jumps whose laplace exponent coincides with the branching mechanism function this new relation between spectrally positive levy processes and continuous branching processes provides a unified perspective on both theories in particular we derive the adequate formulation of the classical ray knight theorem for such levy processes as a consequence of this theorem we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process

the uniform random tree in a brownian excursion to any brownian excursione with duration e and anyt 1 t p 0 e we associate a branching tree withp branches denoted byt p e t 1 t p which is closely related to the structure of the minima ofe our main theorem states that ife is chosen according to the ito measure and t 1 t p according to lebesgue measure on 0 e p the treet p e t 1 t p is distributed according to the uniform measure on the set of trees withp branches the proof of this result yields additional information about the subexcursions ofe corresponding to the different branches of the tree thus generalizing a well known representation theorem of bismut if we replace the ito measure by the law of the normalized excursion a simple conditioning argument leads to another remarkable result originally proved by aldous with a very different method

ray knight representation of flows of branching processes with competition by pruning of levy trees we introduce flows of branching processes with competition which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term this generalizes the logistic branching processes studied by lambert ann appl probab 15 2 1506 1535 2005 following the approach developed by dawson and li ann probab 40 2 813 857 2012 we first construct such processes as the solutions of certain flow of stochastic differential equations we then propose a novel genealogical description for branching processes with competition based on interactive pruning of levy trees and establish a ray knight representation result for these processes in terms of the local times of suitably pruned forests

the contour of splitting trees is a levy process splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution to i i d copies of themselves the width process of a splitting tree is then a binary homogeneous crump mode jagers cmj process and is not markovian unless the lifetime distribution is exponential or a dirac mass at here we allow the birth rate to be infinite that is pairs of birth times and life spans of newboms form a poisson point process along the lifetime of their mother with possibly infinite intensity measure a splitting tree is a random so called chronological tree each element of a chronological tree is a so called existence point of some individual vertex in a discrete tree where is a nonnegative real number called chronological level time we introduce a total order on existence points called linear order and a mapping from the tree into the real line which preserves this order the inverse of is called the exploration process and the projection of this inverse on chronological levels the contour process for splitting trees truncated up to level r we prove that a thus defined contour process is a levy process reflected below and killed upon hitting 0 this allows one to derive properties of i splitting trees conceptual proof of le gall le jan s theorem in the finite variation case exceptional points coalescent point process and age distribution ii cmj processes one dimensional marginals conditionings limit theorems and asymptotic numbers of individuals with infinite versus finite descendances

global divergence of spatial coalescents we study several fundamental properties of a class of stochastic processes called spatial coalescents in these models a number of particles perform independent random walks on some underlying graph g in addition particles on the same vertex merge randomly according to a given coalescing mechanism a remarkable property of mean field coalescent processes is that they may come down from infinity meaning that starting with an infinite number of particles only a finite number remains after any positive amount of time almost surely we show here however that in the spatial setting on any infinite and bounded degree graph the total number of particles will always remain infinite at all times almost surely moreover if g mathbb z d and the coalescing mechanism is kingman s coalescent then starting with n particles at the origin the total number of particles remaining is of order log n d at any fixed positive time where log is the inverse tower function at sufficiently large times the total number of particles is of order log n d 2 when d 2 we provide parallel results in the recurrent case d 2 the spatial beta coalescents behave similarly where log log n is replacing log n

bayesian methods for estimation of the size of a closed population summary a bayesian methodology for estimating the size of a closed population from multiple incomplete administrative lists is proposed the approach allows for a variety of dependence structures between the lists can make use of covariates and explicitly accounts for model uncertainty interval estimates from this approach are compared to frequentist and previously published bayesian approaches several examples are considered

hierarchical exchangeability of pure states in mean field spin glass models the main result in this paper is motivated by the mezard parisi ansatz which predicts a very special structure for the distribution of spins in diluted mean field spin glass models such as the random k sat model at positive temperature using the fact that one can safely assume the validity of the ghirlanda guerra identities in these models we prove hierarchical exchangeability of pure states for the asymptotic gibbs measures which allows us to apply a representation result for hierarchically exchangeable arrays recently proved in austin and panchenko in probab theory relat fields 2013 comparing this representation with the predictions of the mezard parisi ansatz one can see that the key property still missing is that the multi overlaps between pure states depend only on their overlaps

the extremal process of critical points of the pure p spin spherical spin glass model recently sharp results concerning the critical points of the hamiltonian of the p spin spherical spin glass model have been obtained by means of moments computations in particular these moments computations allow for the evaluation of the leading term of the ground state i e of the global minimum in this paper we study the extremal point process of critical points that is the point process associated to all critical values in the vicinity of the ground state we show that the latter converges in distribution to a poisson point process of exponential intensity in particular we identify the correct centering of the ground state and prove the convergence in distribution of the centered minimum to a minus gumbel variable these results are identical to what one obtains for a sequence of i i d variables correctly normalized namely we show that the model is in the universality class of rem

a branching random walk seen from the tip we show that all the time dependent statistical properties of the rightmost points of a branching brownian motion can be extracted from the traveling wave solutions of the fisher kpp equation the distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the fisher kpp traveling waves when the initial condition is modified the limiting measure exhibits the surprising property of superposability the statistical properties of the distances between the rightmost points of the union of two realizations of the branching brownian motion shifted by arbitrary amounts are the same as those of a single realization we discuss the extension of our results to more general branching random walks

on the stationary distribution of the block counting process for population models with mutation and selection abstract we consider two population models subject to the evolutionary forces of selection and mutation the moran model and the wright fisher model in such models the block counting process traces back the number of potential ancestors of a sample of the population at present under some conditions the block counting process is positive recurrent and its stationary distribution is described via a linear system of equations in this work we first characterise the measures leading to a geometric stationary distribution the bolthausen sznitman model being the most prominent example having this feature next we solve the linear system of equations corresponding to the moran model for the wright fisher model we show that the probability generating function associated to the stationary distribution of the block counting process satisfies an integro differential equation we solve the latter for the kingman model and the star shaped model

an invariance principle for branching diffusions in bounded domains we study branching diffusions in a bounded domain d of mathbb r d in which particles are killed upon hitting the boundary partial d it is known that any such process undergoes a phase transition when the branching rate beta exceeds a critical value a multiple of the first eigenvalue of the generator of the diffusion we investigate the system at criticality and show that the associated genealogical tree when the process is conditioned to survive for a long time converges to aldous continuum random tree under appropriate rescaling the result holds under only a mild assumption on the domain and is valid for all branching mechanisms with finite variance and a general class of diffusions

structure of finite rsb asymptotic gibbs measures in the diluted spin glass models we suggest a possible approach to proving the mezard parisi formula for the free energy in the diluted spin glass models such as diluted k spin or random k sat model at any positive temperature in the main contribution of the paper we show that a certain small modification of the hamiltonian in any of these models forces all finite rsb asymptotic gibbs measures in the sense of the overlaps to satisfy the mezard parisi ansatz for the distribution of spins unfortunately what is still missing is a description of the general full rsb asymptotic gibbs measures if one could show that the general case can be approximated by finite rsb case in the right sense then one could a posteriori remove the small modification of the hamiltonian to recover the mezard parisi formula for the original model

maximum likelihood applied to a capture recapture model maximum likelihood is applied to a capture recapture model in order to estimate the population size n the likelihoods to which this model gives rise can be highly skew it is shown that the use of n 3 gives an approximately normal likelihood this results in a function that has an approximately standard normal distribution and is linear in n 3 thence complete sets of approximate confidence intervals can be obtained merely by the use of a pocket calculator

the genealogy of continuous state branching processes with immigration recent works by j f le gall and y le jan 15 have extended the genealogical structure of galton watson processes to continuous state branching processes cb we are here interested in processes with immigration cbi

goodness of fit based confidence intervals for estimates of the size of a closed population methods for estimating the size of a closed population often consist of fitting some model e g a log linear model to data with a missing cell corresponding to the members of the population missed by all reporting sources although the use of the asymptotic standard error is the usual method for forming confidence intervals for the population total the sample sizes are not always large enough to produce valid confidence intervals we propose a method for forming confidence intervals based upon changes in a goodness of fit statistic associated with changes in trial values of the population total

the depth first processes of galton watson trees converge to the same brownian excursion in this paper we show a strong relation between the depth first processes associated to galton watson trees with finite variance conditioned by the total progeny the depth first walk the depth first queue process the height process a consequence is that these processes suitably normalized converge to the same brownian excursion this provides an alternative proof of aldous one of the convergence of the depth first walk to the brownian excursion which does not use the existence of a limit tree the methods that we introduce allow one to compute some functionals of trees or discrete excursions for example we compute the limit law of the process of the height of nodes with a given out degree and the process of the height of nodes root of a given subtree

large deviation properties of the extended moran model the distributions of the times to the first common ancestor t mrca is numerically studied for an ecological population model the extended moran model this model has a fixed population size n the number of descendants is drawn from a beta distribution beta alpha 2 alpha for various choices of alpha this includes also the classical moran model alpha 0 as well as the uniform distribution alpha 1 using a statistical mechanics based large deviation approach the distributions can be studied over an extended range of the support down to probabilities like 10 70 which allowed us to study the change of the tails of the distribution when varying the value of alpha in 0 2 we find exponential distributions p t mrca delta t mrca in all cases with systematically varying values for the base delta only for the cases alpha 0 and alpha 1 analytical results are known i e delta exp 2 n 2 and delta 2 3 respectively we recover these values confirming the validity of our approach finally we also study the correlations between t mrca and the number of descendants

random stable laminations of the disk we study large random dissections of polygons we consider random dissections of a regular polygon with n sides which are chosen according to boltzmann weights in the domain of attraction of a stable law of index 2 1 2 as n goes to infinity we prove that these random dissections converge in distribution towards a random compact set called the random stable lamination if 2 we recover aldous brownian triangulation however if 2 1 2 large faces remain in the limit and a dierent random compact set appears we show that the random stable lamination can be coded by the continuous time height function associated to the normalized excursion of a strictly stable spectrally positive levy process of index using this coding we establish that the hausdor dimension of the stable random lamination is almost surely 2 1

a limit theorem for the contour process of condidtioned galton watson trees in this work we study asymptotics of the genealogy of galton watson processes conditioned on the total progeny we consider a fixed aperiodic and critical offspring distribution such that the rescaled galton watson processes converges to a continuous state branching process csbp with a stable branching mechanism of index 1 2 we code the genealogy by two different processes the contour process and the height process that le gall and le jan recently introduced we show that the rescaled height process of the corresponding galton watson family tree with one ancestor and conditioned on the total progeny converges in a functional sense to a new process the normalized excursion of the continuous height process associated with the stable csbp we deduce from this convergence an analogous limit theorem for the contour process in the brownian case 2 the limiting process is the normalized brownian excursion that codes the continuum random tree the result is due to aldous who used a different method

growth of levy trees we construct random locally compact real trees called levy trees that are the genealogical trees associated with continuous state branching processes more precisely we define a growing family of discrete galton watson trees with i i d exponential branch lengths that is consistent under bernoulli percolation on leaves we define the levy tree as the limit of this growing family with respect to the gromov hausdorff topology on metric spaces this elementary approach notably includes supercritical trees and does not make use of the height process introduced by le gall and le jan to code the genealogy of sub critical continuous state branching processes we construct the mass measure of levy trees and we give a decomposition along the ancestral subtree of a poisson sampling directed by the mass measure

a continuum tree valued markov process we present a construction of a levy continuum random tree crt associated with a super critical continuous state branching process using the so called exploration process and a girsanov theorem we also extend the pruning procedure to this super critical case let be a critical branching mechanism we set let or be the set of values of for which is a conservative branching mechanism the pruning procedure allows to construct a decreasing levy crt valued markov process mathcal t theta theta in theta such that mathcal t theta has branching mechanism it is sub critical if 0 and super critical if 0 we then consider the explosion time a of the crt the smallest negative time for which the continuous state branching process cb associated with mathcal t theta has finite total mass i e the length of the excursion of the exploration process that codes the crt is finite we describe the law of a as well as the distribution of the crt just after this explosion time the crt just after explosion can be seen as a crt conditioned not to be extinct which is pruned with an independent intensity related to a we also study the evolution of the crt valued process after the explosion time this extends results from aldous and pitman on galton watson trees for the particular case of the quadratic branching mechanism we show that after explosion the total mass of the cb behaves like the inverse of a stable subordinator with index 1 2 this result is related to the size of the tagged fragment for the fragmentation of aldous s crt

percolation on random triangulations and stable looptrees we study site percolation on angel and schramm s uniform infinite planar triangulation we compute several critical and near critical exponents and describe the scaling limit of the boundary of large percolation clusters in all regimes subcritical critical and supercritical we prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index 3 2 which was introduced in curien and kortchemski random stable looptrees arxiv 1304 1044 2014 we also give a conjecture linking looptrees of any index alpha in 1 2 with scaling limits of cluster boundaries in random triangulations decorated with o n models

the genealogy of branching brownian motion with absorption 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 logn 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 non rigorous predictions by brunet derrida muller and munier for a closely related model

variational bounds for the generalized random energy model we compute the pressure of the random energy model rem and generalized random energy model grem by establishing variational upper and lower bounds for the upper bound we generalize guerra s broken replica symmetry bounds and identify the random probability cascade as the appropriate random overlap structure for the model for the rem the lower bound is obtained in the high temperature regime using talagrand s concentration of measure inequality and in the low temperature regime using convexity and the high temperature formula the lower bound for the grem follows from the lower bound for the rem by induction while the argument for the lower bound is fairly standard our proof of the upper bound is new

sub exponential tail bounds for conditioned stable bienayme galton watson trees we establish uniform sub exponential tail bounds for the width height and maximal outdegree of critical bienayme galton watson trees conditioned on having a large fixed size whose offspring distribution belongs to the domain of attraction of a stable law this extends results obtained for the height and width by addario berry devroye and janson in the finite variance case

the stable trees are nested we show that we can construct simultaneously all the stable trees as a nested family more precisely if 1 alpha alpha prime le 2 we prove that hidden inside any alpha stable tree we can find a version of an alpha prime stable tree rescaled by an independent mittag leffler type distribution this tree can be explicitly constructed by a pruning procedure of the underlying stable tree or by a modification of the fragmentation associated with it our proofs are based on a recursive construction due to marchal which is proved to converge almost surely towards a stable tree

a finiteness condition on subgroups of large derived length abstract the authors discuss the class s d r of groups in which every finitely generated subgroup is either at most r generated or soluble of derived length at most d such groups need not be of finite rank or soluble of derived length at most d in general a structure theorem is obtained for locally finite and for certain locally nilpotent s d r groups

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion