arXiv subjects

Sanchayan Sen

Publications and source records attributed to Sanchayan Sen.

18 recordsLinked to original sources

arXiv2025

Geometry of critical discrete structures: long-range percolation on the hierarchical lattice and the discrete torus

Consider (a) balls $\Lambda_n$ of growing volumes in the $d$-dimensional hierarchical lattice, and (b) the $d$-dimensional discrete torus $\mathbb{T}_n^d$ on $n^d$ vertices. Place edges independently between each pair of vertices $x\neq y\in\Lambda_n$ or $\mathbb{T}_n^d$ with probability $1-\exp(-\beta J(x, y) )$ where $J(x, y) \asymp \| x-y \|^{-\alpha}$ for some $0<\alpha<d$. For both of these models, we prove the following: (i) We obtain tight bounds, up to constants, on the two-point function in the barely subcritical regime. We show that in part of the barely subcritical regime, the two-point function has a plateau [47, 51]. (ii) We identify the critical window when $0<\alpha<5d/6$. Further, using the bound on the two-point function mentioned in (i) together with a universality principle proven in [10, 14], we establish the scaling limit of the maximal components, viewed as metric measure spaces, within the critical window. More precisely, we show that the metric scaling limit of the maximal components is Brownian, and that these models belong to the Erdos-Renyi universality class when $0<\alpha<5d/6$. It was recently conjectured by Hutchcroft [45, Section~7.1] that the model of critical hierarchical percolation with $\alpha\in(d, 4d/3]$ is a member of the Erdos-Renyi universality class, and we believe that this is also true for all $\alpha\in (0, d]$. Similarly, critical long-range percolation on the discrete torus is expected to be in this universality class when the effective dimension is high enough. These results take a first step in that direction. (iii) We show that when $0<\alpha<2d/3$, the girth of each maximal component in the critical window is $\Omega_P(|\Lambda_n|^{1/3})$ and $\Omega_P(n^{d/3})$ respectively for these two models, contrary to the situation when $d<\alpha$ where the girth would equal $3$ .

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arXiv2023

Scaling limits and universality: Critical percolation on weighted graphs converging to an $L^3$ graphon

We develop a general universality technique for establishing metric scaling limits of critical random discrete structures exhibiting mean-field behavior that requires four ingredients: (i) from the barely subcritical regime to the critical window, components merge approximately like the multiplicative coalescent, (ii) asymptotics of the susceptibility functions are the same as that of the Erdos-Renyi random graph, (iii) asymptotic negligibility of the maximal component size and the diameter in the barely subcritical regime, and (iv) macroscopic averaging of distances between vertices in the barely subcritical regime. As an application of the general universality theorem, we establish, under some regularity conditions, the critical percolation scaling limit of graphs that converge, in a suitable topology, to an $L^3$ graphon. In particular, we define a notion of the critical window in this setting. The $L^3$ assumption ensures that the model is in the Erdos-Renyi universality class and that the scaling limit is Brownian. Our results do not assume any specific functional form for the graphon. As a consequence of our results on graphons, we obtain the metric scaling limit for Aldous-Pittel's RGIV model [9] inside the critical window. Our universality principle has applications in a number of other problems including in the study of noise sensitivity of critical random graphs [52]. In [10], we use our universality theorem to establish the metric scaling limit of critical bounded size rules. Our method should yield the critical metric scaling limit of Rucinski and Wormald's random graph process with degree restrictions [56] provided an additional technical condition about the barely subcritical behavior of this model can be proved.

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arXiv2020

Geometry of the minimal spanning tree in the heavy-tailed regime: new universality classes

A well-known open problem on the behavior of optimal paths in random graphs in the strong disorder regime, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [31,32,38,70] is as follows: for a large class of random graph models with degree exponent $\tau\in (3,4)$, the distance between two typical points on the minimal spanning tree (MST) on the giant component in the supercritical regime scales like $n^{(\tau-3)/(\tau-1)}$. The aim of this paper is to make progress towards a proof of this conjecture. We consider a supercritical inhomogeneous random graph model with degree exponent $\tau\in(3, 4)$ that is closely related to Aldous's multiplicative coalescent, and show that the MST constructed by assigning i.i.d. continuous weights to the edges in its giant component, endowed with the tree distance scaled by $n^{-(\tau-3)/(\tau-1)}$, converges in distribution with respect to the Gromov-Hausdorff topology to a random compact real tree. Further, almost surely, every point in this limiting space either has degree one (leaf), or two, or infinity (hub), both the set of leaves and the set of hubs are dense in this space, and the Minkowski dimension of this space equals $(\tau-1)/(\tau-3)$. The multiplicative coalescent, in an asymptotic sense, describes the evolution of the component sizes of various near-critical random graph processes. We expect the limiting spaces in this paper to be the candidates for the scaling limit of the MST constructed for a wide array of other heavy-tailed random graph models.

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arXiv2020

Global lower mass-bound for critical configuration models in the heavy-tailed regime

We establish the global lower mass-bound property for the largest connected components in the critical window for the configuration model when the degree distribution has an infinite third moment. The scaling limit of the critical percolation clusters, viewed as measured metric spaces, was established in [7] with respect to the Gromov-weak topology. Our result extends those scaling limit results to the stronger Gromov-Hausdorff-Prokhorov topology under slightly stronger assumptions on the degree distribution. This implies the distributional convergence of global functionals such as the diameter of the largest critical components. Further, our result gives a sufficient condition for compactness of the random metric spaces that arise as scaling limits of critical clusters in the heavy-tailed regime.

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arXiv2019

On breadth-first constructions of scaling limits of random graphs and random unicellular maps

We give alternate constructions of (i) the scaling limit of the uniform connected graphs with given fixed surplus, and (ii) the continuum random unicellular map (CRUM) of a given genus that start with a suitably tilted Brownian continuum random tree and make `horizontal' point identifications, at random heights, using the local time measures. Consequently, this can be seen as a continuum analogue of the breadth-first construction of a finite connected graph. In particular, this yields a breadth-first construction of the scaling limit of the critical Erd\H{o}s-R\'enyi random graph which answers a question posed in [2]. As a consequence of this breadth-first construction we obtain descriptions of the radii, the distance profiles, and the two point functions of these spaces in terms of functionals of tilted Brownian excursions.

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arXiv2018

Geometry of the minimal spanning tree of a random $3$-regular graph

The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and so far the scaling limit of the MST viewed as a metric measure space has only been identified in the case of the complete graph [5]. In this work, we show that the MST constructed by assigning i.i.d. continuous edge-weights to either the random (simple) $3$-regular graph or the $3$-regular configuration model on $n$ vertices, endowed with the tree distance scaled by $n^{-1/3}$ and the uniform probability measure on the vertices, converges in distribution with respect to Gromov-Hausdorff-Prokhorov topology to a random compact metric measure space. Further, this limiting space has the same law as the scaling limit of the MST of the complete graph identified in [5] up to a scaling factor of $6^{1/3}$. Our proof relies on a novel argument that proceeds via a comparison between a $3$-regular configuration model and the largest component in the critical Erd\H{o}s-R\'enyi random graph. The techniques of this paper can be used to establish the scaling limit of the MST in the setting of general random graphs with given degree sequences provided two additional technical conditions are verified.

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arXiv2017

A probabilistic approach to the leader problem in random graphs

We study the fixation time of the identity of the leader, i.e., the most massive component, in the general setting of Aldous's multiplicative coalescent [4, 5], which in an asymptotic sense describes the evolution of the component sizes of a wide array of near-critical coalescent processes, including the classical Erd\H{o}s-R\'enyi process. We show tightness of the fixation time in the "Brownian" regime, explicitly determining the median value of the fixation time to within an optimal $O(1)$ window. This generalizes {\L}uczak's result [31] for the Erd\H{o}s-R\'enyi random graph using completely different techniques. In the heavy-tailed case, in which the limit of the component sizes can be encoded using a thinned pure-jump L\'{e}vy process, we prove that only one-sided tightness holds. This shows a genuine difference in the possible behavior in the two regimes. The solution to the leader problem in the setting of the Erd\H{o}s-R\'enyi random graph played an important role in the study of the scaling limit of the minimal spanning tree on the complete graph [2]. We believe that analogous results, such as those proved herein, will be useful in establishing universality of the intrinsic geometry of the minimal spanning tree across a large class of models.

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arXiv2017

Universality for critical heavy-tailed network models: Metric structure of maximal components

We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such results were derived by Bhamidi, van der Hofstad, Sen [Probab. Theory Relat. Fields 2018]. We develop general principles under which the identical scaling limits as the rank-one case can be obtained. Of independent interest, we derive refined asymptotics for various susceptibility functions and the maximal diameter in the barely subcritical regime.

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arXiv2017

Critical window for the configuration model: finite third moment degrees

We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erdős-Rényi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.

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arXiv2017

The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $τ\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(τ-3)/(τ-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of Lévy processes "without replacement" [10], yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $τ\in (3,4)$ where edges are rescaled by $n^{-(τ-3)/(τ-1)}$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact "tree-like" random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal $(τ-2)/(τ-3)$ a.s., in stark contrast to the Erdős-Rényi scaling limit whose Minkowski dimension is 2 a.s. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $τ\in (3,4)$.

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arXiv2016

Heavy-tailed configuration models at criticality

We study the critical behavior of the component sizes for the configuration model when the tail of the degree distribution of a randomly chosen vertex is a regularly-varying function with exponent $\tau-1$, where $\tau\in (3,4)$. The component sizes are shown to be of the order $n^{(\tau-2)/(\tau-1)}L(n)^{-1}$ for some slowly-varying function $L(\cdot)$. We show that the re-scaled ordered component sizes converge in distribution to the ordered excursions of a thinned L\'evy process. This proves that the scaling limits for the component sizes for these heavy-tailed configuration models are in a different universality class compared to the Erd\H{o}s-R\'enyi random graphs. Also the joint re-scaled vector of ordered component sizes and their surplus edges is shown to have a distributional limit under a strong topology. Our proof resolves a conjecture by Joseph, Ann. Appl. Probab. (2014) about the scaling limits of uniform simple graphs with i.i.d degrees in the critical window, and sheds light on the relation between the scaling limits obtained by Joseph and in this paper, which appear to be quite different. Further, we use percolation to study the evolution of the component sizes and the surplus edges within the critical scaling window, which is shown to converge in finite dimension to the augmented multiplicative coalescent process introduced by Bhamidi et. al., Probab. Theory Related Fields (2014). The main results of this paper are proved under rather general assumptions on the vertex degrees. We also discuss how these assumptions are satisfied by some of the frameworks that have been studied previously.

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arXiv2016

Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (i) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (ii) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by Cerny and Teixeira by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.

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arXiv2016

Minimal spanning trees and Stein's method

Kesten and Lee [36] proved that the total length of a minimal spanning tree on certain random point configurations in $\mathbb{R}^d$ satisfies a central limit theorem. They also raised the question: how to make these results quantitative? However, techniques employed to tackle the same problem for other functionals studied in geometric probability do not apply directly to the minimal spanning tree. Thus the problem of determining the convergence rate in the central limit theorem for Euclidean minimal spanning trees has remained open. In this work, we establish bounds on the convergence rate for the Poissonized version of this problem by using a variation of Stein's method. We also derive bounds on the convergence rate for the analogous problem in the setup of the lattice $\mathbb{Z}^d$. The contribution of this paper is twofold. First, we develop a general technique to compute convergence rates in central limit theorems satisfied by minimal spanning trees on sequence of weighted graphs which includes minimal spanning trees on Poisson points. Secondly, we present a way of quantifying the Burton-Keane argument for the uniqueness of the infinite open cluster. The latter is interesting in its own right and based on a generalization of our technique, Duminil-Copin, Ioffe and Velenik [28] have recently obtained bounds on probability of two-arm events in a broad class of translation-invariant percolation models.

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arXiv2016

Continuum limit of critical inhomogeneous random graphs

Motivated by applications, the last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models. In this study we analyze the maximal components at criticality of one famous class of such models, the rank-one inhomogeneous random graph model. Viewing these components as measured random metric spaces, under finite moment assumptions for the weight distribution, we show that the components in the critical scaling window with distances scaled by $n^{-1/3}$ converge in the Gromov-Haussdorf-Prokhorov metric to rescaled versions of the limit objects identified for the Erdős-Rényi random graph components at criticality Addario-Berry, Broutin and Goldschmidt (2012). A key step is the construction of connected components of the random graph through an appropriate tilt of a famous class of random trees called $\mathbf{p}$-trees (studied previously by Aldous, Miermont and Pitman (2004) and by Camarri and Pitman (2000)). This is the first step in rigorously understanding the scaling limits of objects such as the Minimal spanning tree and other strong disorder models from statistical physics (see Braunstein et al., 2003) for such graph models. By asymptotic equivalence (Janson, 2010), the same results are true for the Chung-Lu model and the Britton-Deijfen-Lof model. A crucial ingredient of the proof of independent interest is tail bounds for the height of $\mathbf{p}$-trees. The techniques developed in this paper form the main technical bedrock for proving continuum scaling limits in the critical regime for a wide array of other random graph models (Bhamidi, Broutin, Sen and Wang, 2014) including the configuration model and inhomogeneous random graphs with general kernels which were introduced by Bollobas, Janson and Riordan (2007).

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arXiv2015

The Speed of a Biased Walk on a Galton-Watson Tree without Leaves is Monotonic with Respect to Progeny Distributions for High Values of Bias

Consider biased random walks on two Galton-Watson trees without leaves having progeny distributions $P_1$ and $P_2$ (GW$(P_1)$ and GW$(P_2)$) where $P_1$ and $P_2$ are supported on positive integers and $P_1$ dominates $P_2$ stochastically. We prove that the speed of the walk on GW$(P_1)$ is bigger than the same on GW$(P_2)$ when the bias is larger than a threshold depending on $P_1$ and $P_2$. This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius.

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arXiv2014

Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erd\H{o}s-R\'enyi random graph

A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large class of models, the nature of this emergence is similar to that of the Erd\H{o}s-R\'enyi random graph, in the sense that (a) the sizes of the maximal components in the critical regime scale like $n^{2/3}$, and (b) the structure of the maximal components at criticality (rescaled by $n^{-1/3}$) converges to random fractals. To date, (a) has been proven for a number of models using different techniques. This paper develops a general program for proving (b) that requires three ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent, (ii) scaling exponents of susceptibility functions are the same as that of the Erd\H{o}s-R\'enyi random graph, and (iii) macroscopic averaging of distances between vertices in the barely subcritical regime. We show that these apply to two fundamental random graph models: the configuration model and inhomogeneous random graphs with a finite ground space. For these models, we also obtain new results for component sizes at criticality and structural properties in the barely subcritical regime.

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arXiv2014

Limiting spectral distribution of sample autocovariance matrices

We show that the empirical spectral distribution (ESD) of the sample autocovariance matrix (ACVM) converges as the dimension increases, when the time series is a linear process with reasonable restriction on the coefficients. The limit does not depend on the distribution of the underlying driving i.i.d. sequence and its support is unbounded. This limit does not coincide with the spectral distribution of the theoretical ACVM. However, it does so if we consider a suitably tapered version of the sample ACVM. For banded sample ACVM the limit has unbounded support as long as the number of non-zero diagonals in proportion to the dimension of the matrix is bounded away from zero. If this ratio tends to zero, then the limit exists and again coincides with the spectral distribution of the theoretical ACVM. Finally, we also study the LSD of a naturally modified version of the ACVM which is not non-negative definite.

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arXiv2013

On the largest component in the subcritical regime of the Bohman-Frieze process

Kang, Perkins and Spencer showed that the size of the largest component of the Bohman-Frieze process at a fixed time $t$ smaller than $t_c$, the critical time for the process is $L_1(t)=Ω(\log n/(t_c-t)^2)$ with high probability. They also conjectured that this is the correct order, that is $L_1(t)=O(\log n/(t_c-t)^2)$ with high probability for fixed $t$ smaller than $t_c$. Using a different approach, Bhamidi, Budhiraja and Wang showed that $L_1(t_n)=O((\log n)^4/(t_c-t_n)^2)$ with high probability for $t_n\leq t_c-n^{-γ}$ where $γ\in(0,1/4)$. In this paper, we improve their result by showing that for any fixed $λ>0$, $L_1(t_n)=O(\log n/(t_c-t_n)^2)$ with high probability for $t_n\leq t_c-λn^{-1/3}$. In particular, this settles the conjecture of Kang, Perkins and Spencer. We also prove some generalizations for general bounded size rules.

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