arXiv subjects

Igor Kortchemski

Publications and source records attributed to Igor Kortchemski.

At least 19 recordsLinked to original sources

arXiv2026

Rare subtree patterns in size-conditioned Bienaym\'e trees: Poisson approximation and declumping

We establish a general Poisson approximation for rare local patterns in critical Bienaym\'e--Galton--Watson trees with offspring distribution $\mu$ in the domain of attraction of a stable law, conditioned to have a large number of vertices. A pattern is specified by a sequence-dependent mark on fringe subtrees. If marked fringe subtrees remain microscopic and nearby marked occurrences have negligible clustering, then their count is asymptotically Poisson in total variation whenever its mean remains bounded; when the mean diverges, the count satisfies a law of large numbers. The main difficulty is the global dependence created by size conditioning. We overcome it by combining the cyclic-shift representation with a refined form of the Chen--Stein bound and a bridge-removal estimate controlling the interaction between a local mark and the remainder of the conditioned random walk. For non-fringe patterns, overlapping occurrences may form clusters and the raw count need not be asymptotically Poisson. We introduce declumped indicators which select boundary witnesses of these clusters and prove a general Poisson approximation for their count. As applications, we obtain sharp asymptotics for the maximum leaf-height, equivalently the maximum protection number, and for the height of the largest complete $r$-ary tree appearing as a non-fringe subtree. Unary-chain maxima, and the maximum leaf-height when $\mu_1>0$, exhibit lattice-modulated Gumbel behavior. Complete $r$-ary patterns for $r\ge2$, and the maximum leaf-height when $\mu_1=0$, are localized on one or two consecutive integers. The results require no exponential moment and include offspring distributions with infinite variance.

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arXiv2025

Does freezing impede the growth of random recursive trees?

Uniform attachment with freezing is an extension of the classical model of random recursive trees, in which trees are recursively built by attaching new vertices to old ones. In the model of uniform attachment with freezing, vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones. We study the impact of removing attachment and/or freezing steps on the height of the trees. We show in particular that removing an attachment step can increase the expected height, and that freezing cannot substantially decrease the height of random recursive trees. Our methods are based on coupling arguments.

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arXiv2025

The height of the infection tree

We are interested in the geometry of the ``infection tree'' in a stochastic SIR (Susceptible-Infectious-Recovered) model, starting with a single infectious individual. This tree is constructed by drawing an edge between two individuals when one infects the other. We focus on the regime where the infectious period before recovery follows an exponential distribution with rate $1$, and infections occur at a rate $\lambda_{n} \sim \frac{\lambda}{n}$ where $n$ is the initial number of healthy individuals with $\lambda>1$. We show that provided that the infection does not quickly die out, the height of the infection tree is asymptotically $\kappa(\lambda) \log n$ as $n \rightarrow \infty$, where $\kappa(\lambda)$ is a continuous function in $\lambda$ that undergoes a second-order phase transition at $\lambda_{c}\simeq 1.8038$. Our main tools include a connection with the model of uniform attachment trees with freezing and the application of martingale techniques to control profiles of random trees.

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arXiv2025

Condensation in subcritical Cauchy Bienaym\'e trees

The goal of this note is to study the geometry of large size-conditioned Bienaym\'e trees whose offspring distribution is subcritical, belongs to the domain of attraction of a stable law of index $\alpha=1$ and satisfies a local regularity assumption. We show that a condensation phenomenon occurs: one unique vertex of macroscopic degree emerges, and its height converges in distribution to a geometric random variable. Furthermore, the height of such trees grows logarithmically in their size. Interestingly, the behavior of subcritical Bienaym\'ee trees with $\alpha=1$ is quite similar to the case $\alpha \in( 1,2]$, in contrast with the critical case. This completes the study of the height of heavy-tailed size-conditioned Bienaym\'e trees. Our approach is to check that a random-walk one-big-jump principle due to Armend\'ariz & Loulakis holds, by using local estimates due to Berger, combined with the previous approach to study subcritical Bienaym\'e trees with $\alpha>1$.

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arXiv2024

Uniform attachment with freezing: Scaling limits

We investigate scaling limits of trees built by uniform attachment with freezing, which is a variant of the classical model of random recursive trees introduced in a companion paper. Here vertices are allowed to freeze, and arriving vertices cannot be attached to already frozen ones. We identify a phase transition when the number of non-frozen vertices roughly evolves as the total number of vertices to a given power. In particular, we observe a critical regime where the scaling limit is a random compact real tree, closely related to a time non-homogenous Kingman coalescent process identified by Aldous. Interestingly, in this critical regime, a condensation phenomenon can occur.

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arXiv2024

Random L\'evy Looptrees and L\'evy Maps

What is the analogue of L\'evy processes for random surfaces? Motivated by scaling limits of random planar maps in random geometry, we introduce and study L\'evy looptrees and L\'evy maps. They are defined using excursions of general L\'evy processes with no negative jump and extend the known stable looptrees and stable maps, associated with stable processes. We compute in particular their fractal dimensions in terms of the upper and lower Blumenthal--Getoor exponents of the coding L\'evy process. The case where the L\'evy process is a stable process with a drift naturally appears in the context of stable-Boltzmann planar maps conditioned on having a fixed number of vertices and edges in a near-critical regime.

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arXiv2023

Coupling Bertoin's and Aldous-Pitman's representations of the additive coalescent

We construct a coupling between two seemingly very different constructions of the standard additive coalescent, which describes the evolution of masses merging pairwise at rates proportional to their sums. The first construction, due to Aldous \& Pitman, involves the components obtained by logging the Brownian Continuum Random Tree (CRT) by a Poissonian rain on its skeleton as time increases. The second one, due to Bertoin, involves the excursions above its running infimum of a linear-drifted standard Brownian excursion as its drift decreases. Our main tool is the use of an exploration algorithm of the so-called cut-tree of the Brownian CRT, which is a tree that encodes the genealogy of the fragmentation of the CRT.

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arXiv2023

Critical trees are neither too short nor too fat

We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaym\'e trees. Our bounds are optimal at this level of generality. We also obtain precise asymptotics for offspring distributions within the domain of attraction of a Cauchy distribution, under a local regularity assumption. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaym\'e trees.

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arXiv2023

Uniform attachment with freezing

In the classical model of random recursive trees, trees are recursively built by attaching new vertices to old ones. What happens if vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones? We are interested in the impact of freezing on the height of such trees.

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arXiv2021

The mesoscopic geometry of sparse random maps

We investigate the structure of large uniform random maps with $n$ edges, $\mathrm{f}_n$ faces, and with genus $\mathrm{g}_n$ in the so-called sparse case, where the ratio between the number vertices and edges tends to $1$. We focus on two regimes: the planar case $(\mathrm{f}_n, 2\mathrm{g}_n) = (\mathrm{s}_n, 0)$ and the unicellular case with moderate genus $(\mathrm{f}_n, 2 \mathrm{g}_n) = (1, \mathrm{s}_n-1)$, both when $1 \ll \mathrm{s}_n \ll n$. Albeit different at first sight, these two models can be treated in a unified way using a probabilistic version of the classical core-kernel decomposition. In particular, we show that the number of edges of the core of such maps, obtained by iteratively removing degree $1$ vertices, is concentrated around $\sqrt{n \mathrm{s}_{n}}$. Further, their kernel, obtained by contracting the vertices of the core with degree $2$, is such that the sum of the degree of its vertices exceeds that of a trivalent map by a term of order $\sqrt{\mathrm{s}_{n}^{3}/n}$; in particular they are trivalent with high probability when $\mathrm{s}_{n} \ll n^{1/3}$. This enables us to identify a mesoscopic scale $\sqrt{n/\mathrm{s}_n}$ at which the scaling limits of these random maps can be seen as the local limit of their kernels, which is the dual of the UIPT in the planar case and the infinite three-regular tree in the unicellular case, where each edge is replaced by an independent (biased) Brownian tree with two marked points.

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arXiv2021

Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps

We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time $n$ at an arbitrary value, encompassing in particular large deviation regimes. This enables us to derive scaling limits of such random walks conditioned by their terminal value at time $n$ in various regimes. We believe both to be of independent interest. We then apply these results to obtain invariance principles for the Lukasiewicz path of Bienaymé-Galton-Watson trees conditioned on having a fixed number of leaves and of vertices at the same time, which constitutes a first step towards understanding their large scale geometry. We finally deduce from this scaling limit theorems for random bipartite planar maps under a new conditioning by fixing their number of vertices, edges, and faces at the same time. In the particular case of the uniform distribution, our results confirm a prediction of Fusy & Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian map.

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arXiv2019

On conditioning a self-similar growth-fragmentation by its intrinsic area

The genealogical structure of self-similar growth-fragmentations can be described in terms of a branching random walk. The so-called intrinsic area $\mathrm{A}$ arises in this setting as the terminal value of a remarkable additive martingale. Motivated by connections with some models of random planar geometry, the purpose of this work is to investigate the effect of conditioning a self-similar growth-fragmentation on its intrinsic area. The distribution of $\mathrm{A}$ satisfies a useful smoothing transform which enables us to establish the existence of a regular density $a$ and to determine the asymptotic behavior of $a(r)$ as $r\to \infty$ (this can be seen as a local version of Kesten-Grincevicius-Goldie theorem's for random affine fixed point equations in a particular setting). In turn, this yields a family of martingales from which the formal conditioning on $\mathrm{A}=r$ can be realized by probability tilting. We point at a limit theorem for the conditional distribution given $\mathrm{A}=r$ as $r\to \infty$, and also observe that such conditioning still makes sense under the so-called canonical measure for which the growth-fragmentation starts from $0$

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arXiv2019

Trajectories in random minimal transposition factorizations

We study random typical minimal factorizations of the $n$-cycle, which are factorizations of $(1, \ldots,n)$ as a product of $n-1$ transpositions, chosen uniformly at random. Our main result is, roughly speaking, a local convergence theorem for the trajectories of finitely many points in the factorization. The main tool is an encoding of the factorization by an edge and vertex-labelled tree, which is shown to converge to Kesten's infinite Bienaymé-Galton-Watson tree with Poisson offspring distribution, uniform i.i.d. edge labels and vertex labels obtained by a local exploration algorithm.

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arXiv2018

The geometry of random minimal factorizations of a long cycle via biconditioned bitype random trees

We study random typical minimal factorizations of the $n$-cycle into transpositions, which are factorizations of $(1, \ldots,n)$ as a product of $n-1$ transpositions. By viewing transpositions as chords of the unit disk and by reading them one after the other, one obtains a sequence of increasing laminations of the unit disk (i.e. compact subsets of the unit disk made of non-intersecting chords). When an order of $\sqrt{n}$ consecutive transpositions have been read, we establish, roughly speaking, that a phase transition occurs and that the associated laminations converge to a new one-parameter family of random laminations, constructed from excursions of specific Lévy processes. Our main tools involve coding random minimal factorizations by conditioned two-type Bienaymé--Galton--Watson trees. We establish in particular limit theorems for two-type BGW trees conditioned on having given numbers of vertices of both types, and with an offspring distribution depending on the conditioning size. We believe that this could be of independent interest.

math.PR↗

arXiv2018

Condensation in critical Cauchy Bienaymé-Galton-Watson trees

We are interested in the structure of large Bienaymé-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index $α=1$. In stark contrast to the case $α\in (1,2]$, we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges. To this end, we establish limit theorems for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. These results are of independent interest. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called non-generic of parameter $3/2$). This supports the conjecture that faces in Le Gall & Miermont's $3/2$-stable maps are self-avoiding.

math.PR↗

arXiv2018

The boundary of random planar maps via looptrees

We study the scaling limits of looptrees associated with Bienaymé--Galton--Watson (BGW) trees, that are obtained by replacing every vertex of the tree by a "cycle" whose size is its degree. First, we consider BGW trees whose offspring distribution is critical and in the domain of attraction of a Gaussian distribution. We prove that the Brownian CRT is the scaling limit of the associated looptrees, thereby confirming a prediction of [CK14b]. Then, we deal with BGW trees whose offspring distribution is critical and heavy-tailed. We show that the scaling limit of the associated looptrees is a multiple of the unit circle. This corresponds to a so-called condensation phenomenon, meaning that the underlying tree exhibits a vertex with macroscopic degree. Here, we rely on an invariance principle for random walks with negative drift, which is of independent interest. Finally, we apply these results to the study of the scaling limits of large faces of Boltzmann planar maps. We complete the results of [Ric17] and establish a phase transition for the topology of these maps in the non-generic critical regime.

math.PR↗

arXiv2017

Martingales in self-similar growth-fragmentations and their connections with random planar maps

The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework. In particular, we establish many-to-one formulas for growth-fragmentations and define the notion of intrinsic area of a growth-fragmentation. Second, we identify a distinguished family of growth-fragmentations closely related to stable Lévy processes, which are then shown to arise as the scaling limit of the perimeter process in Markovian explorations of certain random planar maps with large degrees (which are, roughly speaking, the dual maps of the stable maps of Le Gall & Miermont. As a consequence of this result, we are able to identify the law of the intrinsic area of these distinguished growth-fragmentations. This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in arXiv:1507.02265 .

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arXiv2017

Random planar maps & growth-fragmentations

We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial martingale involving the perimeters of cycles at given heights. We also use a recent result concerning self-similar scaling limits of Markov chains on the nonnegative integers. A motivation for this work is to give a new construction of the Brownian map from a growth-fragmentation process.

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