arXiv subjects

Sourav Chatterjee

Publications and source records attributed to Sourav Chatterjee.

At least 91 records · Page 5Linked to original sources

arXiv2021

Fluctuation lower bounds in planar random growth models

We prove $\sqrt{\log n}$ lower bounds on the order of growth fluctuations in three planar growth models (first-passage percolation, last-passage percolation, and directed polymers) under no assumptions on the distribution of vertex or edge weights other than the minimum conditions required for avoiding pathologies. Such bounds were previously known only for certain restrictive classes of distributions.In addition, the first-passage shape fluctuation exponent is shown to be at least $1/8$, extending previous results to more general distributions.

math.PR↗

arXiv2021

Localization in Gaussian disordered systems at low temperature

For a broad class of Gaussian disordered systems at low temperature, we show that the Gibbs measure is asymptotically localized in small neighborhoods of a small number of states. From a single argument, we obtain (i) a version of "complete" path localization for directed polymers that is not available even for exactly solvable models; and (ii) a result about the exhaustiveness of Gibbs states in spin glasses not requiring the Ghirlanda-Guerra identities.

math.PR↗

arXiv2021

The endpoint distribution of directed polymers

Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called "partitioned subprobability measure", to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on large deviations of the occupation measure of Brownian motion, we define a suitable topology on the space of partitioned subprobability measures and prove that this topology is compact. Then using a variant of the cavity method from the theory of spin glasses, we show that any limit law of a sequence of endpoint distributions must satisfy a fixed point equation on this abstract space, and that the limiting free energy of the model can be expressed as the solution of a variational problem over the set of fixed points. As a first application of the theory, we prove that in an environment with finite exponential moment, the endpoint distribution is asymptotically purely atomic if and only if the system is in the low temperature phase. The analogous result for a heavy-tailed environment was proved by Vargas in 2007. As a second application, we prove a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. All our results hold in arbitrary dimensions, and make no use of integrability.

math.PR↗

arXiv2021

White Dwarf Subsystems in Core-Collapsed Globular Clusters

Numerical and observational evidence suggests that massive white dwarfs dominate the innermost regions of core-collapsed globular clusters by both number and total mass. Using NGC 6397 as a test case, we constrain the features of white dwarf populations in core-collapsed clusters, both at present day and throughout their lifetimes. The dynamics of these white dwarf subsystems have a number of astrophysical implications. We demonstrate that the collapse of globular cluster cores is ultimately halted by the dynamical burning of white dwarf binaries. We predict core-collapsed clusters in the local universe yield a white dwarf merger rate of $\mathcal{O}(10\rm{)\,Gpc}^{-3}\,\rm{yr}^{-1}$, roughly $0.1-1\%$ of the observed Type Ia supernova rate. We show that prior to merger, inspiraling white dwarf binaries will be observable as gravitational wave sources at milli- and decihertz frequencies. Over $90\%$ of these mergers have a total mass greater than the Chandrasekhar limit. If the merger/collision remnants are not destroyed completely in an explosive transient, we argue the remnants may be observed in core-collapsed clusters as either young neutron stars/pulsars/magnetars (in the event of accretion-induced collapse) or as young massive white dwarfs offset from the standard white dwarf cooling sequence. Finally, we show collisions between white dwarfs and main sequence stars, which may be detectable as bright transients, occur at a rate of $\mathcal{O}(100\rm{)\,Gpc}^{-3}\,\rm{yr}^{-1}$ in the local universe. We find that these collisions lead to depletion of blue straggler stars and main sequence star binaries in the centers of core-collapsed clusters.

astro-ph.GA↗

arXiv2021

A deterministic theory of low rank matrix completion

The problem of completing a large low rank matrix using a subset of revealed entries has received much attention in the last ten years. The main result of this paper gives a necessary and sufficient condition, stated in the language of graph limit theory, for a sequence of matrix completion problems with arbitrary missing patterns to be asymptotically solvable. It is then shown that a small modification of the Candès-Recht nuclear norm minimization algorithm provides the required asymptotic solution whenever the sequence of problems is asymptotically solvable. The theory is fully deterministic, with no assumption of randomness. A number of open questions are listed.

math.ST↗

arXiv2021

Micro-Estimates of Wealth for all Low- and Middle-Income Countries

Many critical policy decisions, from strategic investments to the allocation of humanitarian aid, rely on data about the geographic distribution of wealth and poverty. Yet many poverty maps are out of date or exist only at very coarse levels of granularity. Here we develop the first micro-estimates of wealth and poverty that cover the populated surface of all 135 low and middle-income countries (LMICs) at 2.4km resolution. The estimates are built by applying machine learning algorithms to vast and heterogeneous data from satellites, mobile phone networks, topographic maps, as well as aggregated and de-identified connectivity data from Facebook. We train and calibrate the estimates using nationally-representative household survey data from 56 LMICs, then validate their accuracy using four independent sources of household survey data from 18 countries. We also provide confidence intervals for each micro-estimate to facilitate responsible downstream use. These estimates are provided free for public use in the hope that they enable targeted policy response to the COVID-19 pandemic, provide the foundation for new insights into the causes and consequences of economic development and growth, and promote responsible policymaking in support of the Sustainable Development Goals.

econ.GN↗

arXiv2021

A simple measure of conditional dependence

We propose a coefficient of conditional dependence between two random variables $Y$ and $Z$ given a set of other variables $X_1,\ldots,X_p$, based on an i.i.d. sample. The coefficient has a long list of desirable properties, the most important of which is that under absolutely no distributional assumptions, it converges to a limit in $[0,1]$, where the limit is $0$ if and only if $Y$ and $Z$ are conditionally independent given $X_1,\ldots,X_p$, and is $1$ if and only if $Y$ is equal to a measurable function of $Z$ given $X_1,\ldots,X_p$. Moreover, it has a natural interpretation as a nonlinear generalization of the familiar partial $R^2$ statistic for measuring conditional dependence by regression. Using this statistic, we devise a new variable selection algorithm, called Feature Ordering by Conditional Independence (FOCI), which is model-free, has no tuning parameters, and is provably consistent under sparsity assumptions. A number of applications to synthetic and real datasets are worked out.

math.ST↗

arXiv2021

A phase transition for repeated averages

Let $x_1,\ldots,x_n$ be a fixed sequence of real numbers. At each stage, pick two indices $I$ and $J$ uniformly at random and replace $x_I$, $x_J$ by $(x_I+x_J)/2$, $(x_I+x_J)/2$. Clearly all the coordinates converge to $(x_1+\cdots+x_n)/n$. We determine the rate of convergence, establishing a sharp "cutoff" transition, answering a question of Jean Bourgain.

math.PR↗

arXiv2021

A probabilistic mechanism for quark confinement

The confinement of quarks is one of the enduring mysteries of modern physics. There is a longstanding physics heuristic that confinement is a consequence of `unbroken center symmetry'. This article gives mathematical confirmation of this heuristic, by rigorously defining of center symmetry in lattice gauge theories and proving that a theory is confining when center symmetry is unbroken. Furthermore, a sufficient condition for unbroken center symmetry is given: It is shown that if the center of the gauge group is nontrivial, and correlations decay exponentially under arbitrary boundary conditions, then center symmetry does not break.

math.PR↗

arXiv2021

Fast Optical Transients from Stellar-Mass Black Hole Tidal Disruption Events in Young Star Clusters

Observational evidence suggests that the majority of stars may have been born in stellar clusters or associations. Within these dense environments, dynamical interactions lead to high rates of close stellar encounters. A variety of recent observational and theoretical indications suggest stellar-mass black holes may be present and play an active dynamical role in stellar clusters of all masses. In this study, we explore the tidal disruption of main sequence stars by stellar-mass black holes in young star clusters. We compute a suite of over 3000 independent $N$-body simulations that cover a range in cluster mass, metallicity, and half-mass radii. We find stellar-mass black hole tidal disruption events (TDEs) occur at an overall rate of up to roughly $200\,\rm{Gpc}^{-3}\,\rm{yr}^{-1}$ in young stellar clusters in the local universe. These TDEs are expected to have several characteristic features, namely fast rise times of order a day, peak X-ray luminosities of at least $10^{44}\,\rm{erg\,s}^{-1}$, and bright optical luminosities (roughly $10^{41}-10^{44}\,\rm{erg\,s}^{-1}$) associated with reprocessing by a disk wind. In particular, we show these events share many features in common with the emerging class of Fast Blue Optical Transients.

astro-ph.HE↗

arXiv2021

No Black Holes in NGC 6397

Recently, \citet{vitral2021does} detected a central concentration of dark objects in the core-collapsed globular cluster NGC 6397, which could be interpreted as a subcluster of stellar-mass black holes. However, it is well established theoretically that any significant number of black holes in the cluster would provide strong dynamical heating and is fundamentally inconsistent with this cluster's core-collapsed profile. Claims of intermediate-mass black holes in core-collapsed clusters should similarly be treated with suspicion, for reasons that have been understood theoretically for many decades. Instead, the central dark population in NGC 6397 is exactly accounted for by a compact subsystem of white dwarfs, as we demonstrate here by inspection of a previously published model that provides a good fit to this cluster. These central subclusters of heavy white dwarfs are in fact a generic feature of core-collapsed clusters, while central black hole subclusters are present in all {\em non\/}-collapsed clusters.

astro-ph.GA↗

arXiv2021

Matching Globular Cluster Models to Observations

As ancient, gravitationally bound stellar populations, globular clusters are abundant, vibrant laboratories characterized by high frequencies of dynamical interactions coupled to complex stellar evolution. Using surface brightness and velocity dispersion profiles from the literature, we fit $59$ Milky Way globular clusters to dynamical models from the \texttt{CMC Cluster Catalog}. Without doing any interpolation, and without any directed effort to fit any particular cluster, $26$ globular clusters are well-matched by at least one of our models. We discuss in particular the core-collapsed clusters NGC 6293, NGC 6397, NGC 6681, and NGC 6624, and the non-core-collapsed clusters NGC 288, NGC 4372, and NGC 5897. As NGC 6624 lacks well-fitting snapshots on the main \texttt{CMC Cluster Catalog}, we run six additional models in order to refine the fit. We calculate metrics for mass segregation, explore the production of compact object sources such as millisecond pulsars, cataclysmic variables, low-mass X-ray binaries, and stellar-mass black holes, finding reasonable agreement with observations. Additionally, closely mimicking observational cuts, we extract the binary fraction from our models, finding good agreement except in the dense core regions of core-collapsed clusters. Accompanying this paper are a number of \textsf{python} methods for examining the publicly accessible \texttt{CMC Cluster Catalog}, as well as any other models generated using \texttt{CMC}.

astro-ph.GA↗

arXiv2021

The Observed Rate of Binary Black Hole Mergers can be Entirely Explained by Globular Clusters

Since the first signal in 2015, the gravitational-wave detections of merging binary black holes (BBHs) by the LIGO and Virgo collaborations (LVC) have completely transformed our understanding of the lives and deaths of compact object binaries, and have motivated an enormous amount of theoretical work on the astrophysical origin of these objects. We show that the phenomenological fit to the redshift-dependent merger rate of BBHs from Abbott et al. (2020) is consistent with a purely dynamical origin for these objects, and that the current merger rate of BBHs from the LVC could be explained entirely with globular clusters alone. While this does not prove that globular clusters are the dominant formation channel, we emphasize that many formation scenarios could contribute a significant fraction of the current LVC rate, and that any analysis that assumes a single (or dominant) mechanism for producing BBH mergers is implicitly using a specious astrophysical prior.

astro-ph.HE↗

arXiv2021

Intermediate-mass Black Holes from High Massive-star Binary Fractions in Young Star Clusters

Black holes formed in dense star clusters, where dynamical interactions are frequent, may have fundamentally different properties than those formed through isolated stellar evolution. Theoretical models for single star evolution predict a gap in the black hole mass spectrum from roughly $40-120\,M_{\odot}$ caused by (pulsational) pair-instability supernovae. Motivated by the recent LIGO/Virgo event GW190521, we investigate whether black holes with masses within or in excess of this "upper-mass gap" can be formed dynamically in young star clusters through strong interactions of massive stars in binaries. We perform a set of $N$-body simulations using the CMC cluster-dynamics code to study the effects of the high-mass binary fraction on the formation and collision histories of the most massive stars and their remnants. We find that typical young star clusters with low metallicities and high binary fractions in massive stars can form several black holes in the upper-mass gap and often form at least one intermediate-mass black hole. These results provide strong evidence that dynamical interactions in young star clusters naturally lead to the formation of more massive black hole remnants.

astro-ph.HE↗

arXiv2021

Black Hole Mergers from Star Clusters with Top-Heavy Initial Mass Functions

Recent observations of globular clusters (GCs) provide evidence that the stellar initial mass function (IMF) may not be universal, suggesting specifically that the IMF grows increasingly top-heavy with decreasing metallicity and increasing gas density. Non-canonical IMFs can greatly affect the evolution of GCs, mainly because the high end determines how many black holes (BHs) form. Here we compute a new set of GC models, varying the IMF within observational uncertainties. We find that GCs with top-heavy IMFs lose most of their mass within a few Gyr through stellar winds and tidal stripping. Heating of the cluster through BH mass segregation greatly enhances this process. We show that, as they approach complete dissolution, GCs with top-heavy IMFs can evolve into 'dark clusters' consisting of mostly BHs by mass. In addition to producing more BHs, GCs with top-heavy IMFs also produce many more binary BH (BBH) mergers. Even though these clusters are short-lived, mergers of ejected BBHs continue at a rate comparable to, or greater than, what is found for long-lived GCs with canonical IMFs. Therefore these clusters, although they are no longer visible today, could still contribute significantly to the local BBH merger rate detectable by LIGO/Virgo, especially for sources with higher component masses well into the BH mass gap. We also report that one of our GC models with a top-heavy IMF produces dozens of intermediate-mass black holes (IMBHs) with masses $M>100\,{\rm M_\odot}$, including one with $M>500\,{\rm M_\odot}$. Ultimately, additional gravitational wave observations will provide strong constraints on the stellar IMF in old GCs and the formation of IMBHs at high redshift.

astro-ph.GA↗

arXiv2020

Average Gromov hyperbolicity and the Parisi ansatz

Gromov hyperbolicity of a metric space measures the distance of the space from a perfect tree-like structure. The measure has a "worst-case" aspect to it, in the sense that it detects a region in the space which sees the maximum deviation from tree-like structure. In this article we introduce an "average-case" version of Gromov hyperbolicity, which detects whether the "most of the space", with respect to a given probability measure, looks like a tree. The main result of the paper is that if this average hyperbolicity is small, then the space can be approximately embedded in a tree. The proof uses a weighted version of Szemeredi's regularity lemma from graph theory. The result applies to Gromov hyperbolic spaces as well, since average hyperbolicity is bounded above by Gromov hyperbolicity. As an application, we give a construction of hierarchically organized pure states in any model of a spin glass that satisfies the Parisi ultrametricity ansatz.

math.PR↗

arXiv2020

Black Hole Mergers from Hierarchical Triples in Dense Star Clusters

Hierarchical triples are expected to be produced by the frequent binary-mediated interactions in the cores of globular clusters. In some of these triples, the tertiary companion can drive the inner binary to merger following large eccentricity oscillations, as a result of the eccentric Kozai-Lidov mechanism. In this paper, we study the dynamics and merger rates of black hole (BH) hierarchical triples, formed via binary--binary encounters in the CMC Cluster Catalog, a suite of cluster simulations with present-day properties representative of the Milky Way's globular clusters. We compare the properties of the mergers from triples to the other merger channels in dense star clusters, and show that triple systems do not produce significant differences in terms of mass and effective spin distribution. However, they represent an important pathway for forming eccentric mergers, which could be detected by LIGO--Virgo/KAGRA (LVK), and future missions such as LISA and DECIGO. We derive a conservative lower limit for the merger rate from this channel of $0.35$ Gpc$^{-3}$yr$^{-1}$ in the local Universe and up to $\sim9\%$ of these events may have a detectable eccentricity at LVK design sensitivity. Additionally, we find that triple systems could play an important role in retaining second-generation BHs, which can later merge again in the core of the host cluster.

astro-ph.GA↗

arXiv2020

Populating the upper black hole mass gap through stellar collisions in young star clusters

Theoretical modeling of massive stars predicts a gap in the black hole (BH) mass function above $\sim 40-50\,M_{\odot}$ for BHs formed through single star evolution, arising from (pulsational) pair-instability supernovae. However, in dense star clusters, dynamical channels may exist that allow construction of BHs with masses in excess of those allowed from single star evolution. The detection of BHs in this so-called "upper-mass gap" would provide strong evidence for the dynamical processing of BHs prior to their eventual merger. Here, we explore in detail the formation of BHs with masses within or above the pair-instability gap through collisions of young massive stars in dense star clusters. We run a suite of 68 independent cluster simulations, exploring a variety of physical assumptions pertaining to growth through stellar collisions, including primordial cluster mass segregation and the efficiency of envelope stripping during collisions. We find that as many as $\sim20\%$ of all BH progenitors undergo one or more collisions prior to stellar collapse and up to $\sim1\%$ of all BHs reside within or above the pair-instability gap through the effects of these collisions. We show that these BHs readily go on to merge with other BHs in the cluster, creating a population of massive BH mergers at a rate that may compete with the "multiple-generation" merger channel described in other analyses. This has clear relevance for the formation of very massive BH binaries as recently detected by LIGO/Virgo in GW190521. Finally, we describe how stellar collisions in clusters may provide a unique pathway to pair-instability supernovae and briefly discuss the expected rate of these events and other electromagnetic transients.

astro-ph.HE↗