arXiv subjects

David Clancy Jr

Publications and source records attributed to David Clancy Jr.

13 recordsLinked to original sources

arXiv2025

Multitype L\'evy trees as scaling limits of multitype Bienaym\'e-Galton-Watson trees

We establish sufficient mild conditions for a sequence of multitype Bienaym\'e-Galton-Watson trees, conditioned in some sense to be large, to converge to a limiting compact metric space which we call a \emph{multitype L\'{e}vy tree}. More precisely, we condition on the size of the maximal subtree of vertices of the same type joined by the root to be large. While we employ a different conditioning, our result can be seen as a generalization to the multitype setting of the continuum random trees defined by Aldous, Duquesne and Le Gall in [Ald91a,Ald91b,Ald93,DLG02]. Our main result is an invariance principle for the convergence of such trees, by gluing single-type L\'{e}vy trees together in a method determined by the limiting spectrally positive additive L\'{e}vy field, as constructed by Chaumont and Marolleau [CM21]. Our approach is an improvement of a result about the convergence in the Gromov-Hausdorff-Prohorov topology, of compact marked metric spaces equipped with vector-valued measures, which are then glued via an iterative operation. To analyze the gluing operation, we extend the techniques developed by S\'enizergues [Sen19,Sen22] to the multitype setting. While the single-type case exhibits a more homogeneous structure with simpler dependency patterns, the multitype case introduces interactions between different types, leading to a more intricate dependency structure where functionals must account for type-specific behaviors and inter-type relationships.

math.PR↗

arXiv2025

A central limit theorem for the giant in a stochastic block model

We provide a simple proof for of the central limit theorem for the number of vertices in the giant for super-critical stochastic block model using the breadth-first walk of Konarovskyi, Limic and the author (2024). Our approach follows the recent work of Corujo, Limic and Lemaire (2024) and reduces to the classic central limit theorem for the Erd\H{o}s-R\'{e}nyi model obtained by Stepanov (1970).

math.PR↗

arXiv2025

Fluctuations of the giant of Poisson random graphs

Enriquez, Faraud, and Lemaire (2023) have established process-level fluctuations for the giant of the dynamic Erd\H{o}s-R\'{e}nyi random graph above criticality and show that the limit is a centered Gaussian process with continuous sample paths. A random walk proof was recently obtained by Corujo, Limic and Lemaire (2024). We show that a similar result holds for rank-one inhomogeneous models whenever the empirical weight distribution converges to a limit and its second moment converges as well.

math.PR↗

arXiv2024

Asymptotics for the number of bipartite graphs with fixed surplus

In a recent work on the bipartite Erd\H{o}s-R\'{e}nyi graph, Do et al. (2023) established upper bounds on the number of connected labeled bipartite graphs with a fixed surplus. We use some recent encodings of bipartite random graphs in order to provide a probabilistic formula for the number of bipartite graphs with fixed surplus. Using this, we obtain asymptotics as the number of vertices in each class tend to infinity.

math.CO↗

arXiv2024

Near-critical bipartite configuration models and their associated intersection graphs

Recently, van der Hofstad, Komj\'{a}thy, and Vadon (2022) identified the critical point for the emergence of a giant connected component for the bipartite configuration model (BCM) and used this to analyze its associated random intersection graph (RIG) (2021). We extend some of this analysis to understand the graph at, and near, criticality. In particular, we show that under certain moment conditions on the empirical degree distributions, the number of vertices in each connected component listed in decreasing order of their size converges, after appropriate re-normalization, to the excursion lengths of a certain thinned L\'{e}vy process. Our approach allows us to obtain the asymptotic triangle counts in the RIG built from the BCM. Our limits agree with the limits recently identified by Wang (2023) for the RIG built from the bipartite Erd\H{o}s-R\'{e}nyi random graph.

math.PR↗

arXiv2024

Component sizes of rank-2 multiplicative random graphs

We show that in three different critical regimes, the masses of the connected components of rank-2 multiplicative random graph converge to lengths of excursions of a thinned L\'{e}vy process, perhaps with random coefficients. The three critical regimes are those identified by Bollob\'{a}s, Janson and Riordan (2007), the interacting regime identified by Konarovskyi and Limic (2021), and what we call the nearly bipartite regime which has recently gained interest for its connection to random intersection graphs. Our results are able to extend some of the results by Baslingker et al. (2023) on component sizes of the stochastic blockmodel with two types and those of Federico (2019) and Wang (2023) on the sizes of the connected components of random intersection graphs.

math.PR↗

arXiv2024

Degree corrected stochastic block model: excursion representation

This is the first of two complementary works in which we analyze the connected components of the degree-corrected stochastic block model (DCSBM). Our model is a random graph with an underlying community structure and degree in-homogeneity. It belongs to a class of non-rank one models. The scaling limit of connected component sizes in the near-critical regime, obtained by Konarovskyi and Limic (2021) for a subfamily of DCSBM, is non-trivially different (although related to) the standard eternal multiplicative coalescent of Aldous (1997). The Aldous (1997) excursion representation combined with weak convergence approach to the scaling limits of connected components of random graphs proved to be much more difficult (and therefore rare) for non rank-one models. In this work we show how to build a random field encoding for the connected component structure of DCSBM, in part relying on the theory of Chaumont and Marolleau (2020). We then show how one can, under additional assumptions, reformulate the minimization problem stated in terms of multidimensional first hitting times into an equivalent minimization problem stated for a single real-valued stochastic process. This reformulation relies on a novel composition-like operator on pairs of compatible non-decreasing rcll functions, which might be of independent interest.

math.PR↗

arXiv2024

Inhomogeneous percolation on the hierarchical configuration model with a heavy-tailed degree distribution

We consider inhomogeneous percolation on a hierarchical configuration model with a heavy-tailed degree distribution. This graph is the configuration model where all the half-edges are colored either black or white, and edges are formed by uniformly matching edges of the same color. When only the white half-edges are paired, we provide sufficient conditions for the size and total number of incident black half-edges of the connected components to converge in an $\ell^2$-sense. The limiting vector is described by an $\mathbb{R}^2$-valued thinned L\'{e}vy process. We also establish an $\ell^2$-limit for the number of vertices in connected components when a critical proportion of the black edges are included. A key part of our analysis is establishing a Feller-type property for the multiplicative coalescent with mass and weight recently studied in (Dhara et. al 2017, Dhara et. al 2020).

math.PR↗

arXiv2021

Occupation Times for Time-changed Processes with Applications to Parisian Options

Stochastic processes time-changed by an inverse subordinator have been suggested as a way to model the price of assets in illiquid markets, where the jumps of the subordinator correspond to periods of time where one is unable to sell an asset. We develop an excursion theory for time-changed reflected Brownian motion and use this to express the price of certain European options with Parisian barrier condition in terms of solutions of a time-fractional PDE. We provide a general description of the occupation measures of time-changed processes and use this to prove a Ray-Knight theorem for the occupation measure of a time-changed Brownian motion with negative drift. We also show that the duration of the excursions on finite time intervals obey a Poisson-Dirichlet distribution when a reflected Brownian motion is time-changed by an inverse stable subordinator.

math.PR↗

arXiv2021

Encoding multitype Galton-Watson forests and a multitype Ray-Knight theorem

We provide a simple forest model to encode the genealogical structure of a multitype Galton-Watson process with immigration. We provide two encodings of these forests by stochastic processes. We show, under appropriate conditions, the depth-first encodings of each particular type converge to a solution to a system of stochastic integral equations involving height processes perturbed by functionals of their local times. The forest picture allows us to extend the Ray-Knight theorem and show that local time of the solution to the system of equations form a multitype continuous state branching process with immigration. These assumptions underlying our weak convergence arguments are easily seen to be met in the Brownian setting, and more generally an $\alpha$-stable setting for any $\alpha\in(1,2]$.

math.PR↗

arXiv2021

Epidemics on critical random graphs with heavy-tailed degree distribution

We study the susceptible-infected-recovered (SIR) epidemic on a random graph chosen uniformly over all graphs with certain critical, heavy-tailed degree distributions. For this model, each vertex infects all its susceptible neighbors and recovers the day after it was infected. When a single individual is initially infected, the total proportion of individuals who are eventually infected approaches zero as the size of the graph grows towards infinity. Using different scaling, we prove process level scaling limits for the number of individuals infected on day $h$ on the largest connected components of the graph. The scaling limits are contain non-negative jumps corresponding to some vertices of large degree, that is these vertices are super-spreaders. Using weak convergence techniques, we can describe the height profile of the $\alpha$-stable continuum random graph (Goldschmidt et. al. 2018, Conchon-Kerjan - Goldschmidt 2020), extending results known in the Brownian case (Miermont - Sen 2019). We also prove abstract results that can be used on other critical random graph models.

math.PR↗

arXiv2020

A new relationship between Erd\H{o}s-R\'{e}nyi graphs, epidemic models and Brownian motion with parabolic drift

In the Reed-Frost model, an example of an SIR epidemic model, one can examine a statistic that counts the number of concurrently infected individuals. This statistic can be reformulated as a statistic on the \ER random graph $G(n,p)$. Within the critical window of Aldous and Martin-L\"{o}f, i.e. when $p = p(n) = n^{-1}+\lambda n^{-4/3}$, the cumulative sum of this statistic converges weakly to the integral of a Brownian motion with parabolic drift. This same statistic exhibits a deterministic scaling limit when $p = (1+\lambda \varepsilon_n)/n$ whenever $\varepsilon_n\to 0$ and $n^{1/3}\varepsilon_n\to\infty$.

math.PR↗

arXiv2019

The Gorin-Shkolnikov identity and its random tree generalization

In a recent pair of papers Gorin and Shkolnikov (2018) and Hariya (2016) have shown that the area under normalized Brownian excursion minus one half the integral of the square of its total local time is a centered normal random variable with variance $\frac{1}{12}$. Gaudreau Lamarre and Shkolnikov (2019) generalized this to Brownian bridges, and ask for a combinatorial interpretation. We provide a combinatorial interpretation using random forests on $n$ vertices. In particular, we show that there is a process level generalization for a certain infinite forest model. We also show analogous results for a variety of other related models using stochastic calculus.

math.PR↗