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Sourav Chatterjee

Publications and source records attributed to Sourav Chatterjee.

At least 199 records · Page 11Linked to original sources
arXiv2012

Planets in Open Clusters Detectable by Kepler

While hundreds of planets have been discovered around field stars, only a few are known in star clusters. To explain the lack of short-period giant planets in globular clusters (GC), such as 47 Tucane and ω Centauri, it has been suggested that their low metallicities may have prevented planet formation. Alternatively, the high rates of close stellar encounters in these clusters may have influenced the formation and subsequent evolution of planetary systems. How common are planets in clusters around normal main-sequence stars? Here we consider whether this question can be addressed using data from the Kepler mission. The Kepler field of view contains 4 low-density (relative to GCs) open clusters where the metallicities are about solar (or even higher) and stellar encounters are much less frequent than in typical GCs. We provide detailed $N$-body models and show that most planets in Kepler-detectable orbits are not significantly perturbed by stellar encounters in these open clusters. We focus on the most massive cluster, NGC 6791, which has super-solar metallicity, and find that if planets formed in this cluster at the same frequency as observed in the field, Kepler could detect 1 -- 20 transiting planets depending on the planet-size distribution and the duration of data collection. Due to the large distance to NGC 6791 Kepler will have to search relatively faint ($K_p<20$) stars for the full extended mission to achieve such a yield.

astro-ph.EP↗
arXiv2012

The universal relation between scaling exponents in first-passage percolation

It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent $χ$ and the wandering exponent $ξ$ are related through the universal relation $χ=2ξ-1$, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. This article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense.

math.PR↗
arXiv2012

Gravitational Slingshot of Young Massive Stars in Orion

The Orion Nebula Cluster (ONC) is the nearest region of massive star formation and thus a crucial testing ground for theoretical models. Of particular interest amongst the ONC's ~1000 members are: θ^1 Ori C, the most massive binary in the cluster with stars of masses 38 and 9 MSun (Kraus et al. 2009); the Becklin-Neugebauer (BN) object, a 30 km/s runaway star of ~8 MSun (Tan 2004); and the Kleinmann-Low (KL) nebula protostar, a highly-obscured, ~15 MSun object still accreting gas while also driving a powerful, apparently "explosive" outflow (Allen & Burton 1993). The unusual behavior of BN and KL is much debated: How did BN acquire its high velocity? How is this related to massive star formation in the KL nebula? Here we report the results of a systematic survey using ~ 10^7 numerical experiments of gravitational interactions of the θ^1C and BN stars. We show that dynamical ejection of BN from this triple system at its observed velocity leaves behind a binary with total energy and eccentricity matching those observed for θ^1C. Five other observed properties of θ^C are also consistent with it having ejected BN and altogether we estimate there is only a <~ 10^{-5} probability that θ^1C has these properties by chance. We conclude that BN was dynamically ejected from the θ^1C system about 4,500 years ago. BN has then plowed through the KL massive-star-forming core within the last 1,000 years causing its recently-enhanced accretion and outflow activity.

astro-ph.SR↗
arXiv2012

Central limit theorem for first-passage percolation time across thin cylinders

We prove that first-passage percolation times across thin cylinders of the form $[0,n]\times [-h_n,h_n]^{d-1}$ obey Gaussian central limit theorems as long as $h_n$ grows slower than $n^{1/(d+1)}$. It is an open question as to what is the fastest that $h_n$ can grow so that a Gaussian CLT still holds. Under the natural but unproven assumption about existence of fluctuation and transversal exponents, and strict convexity of the limiting shape in the direction of $(1,0,...,0)$, we prove that in dimensions 2 and 3 the CLT holds all the way up to the height of the unrestricted geodesic. We also provide some numerical evidence in support of the conjecture in dimension 2.

math.PR↗
arXiv2012

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.

math.PR↗
arXiv2012

Monte Carlo Simulations of Globular Cluster Evolution. VI. The Influence of an Intermediate Mass Black Hole

We present results of a series of Monte Carlo simulations investigating the imprint of a central intermediate-mass black hole (IMBH) on the structure of a globular cluster. We investigate the three-dimensional and projected density profiles, and stellar disruption rates for idealized as well as realistic cluster models, taking into account a stellar mass spectrum and stellar evolution, and allowing for a larger, more realistic, number of stars than was previously possible with direct N-body methods. We compare our results to other N-body and Fokker-Planck simulations published previously. We find, in general, very good agreement for the overall cluster structure and dynamical evolution between direct N-body simulations and our Monte Carlo simulations. Significant differences exist in the number of stars that are tidally disrupted by the IMBH, which is most likely an effect of the wandering motion of the IMBH, not included in the Monte Carlo scheme. These differences, however, are negligible for the final IMBH masses in realistic cluster models as the disruption rates are generally much lower than for single-mass clusters. As a direct comparison to observations we construct a detailed model for the cluster NGC 5694, which is known to possess a central surface brightness cusp consistent with the presence of an IMBH. We find that not only the inner slope but also the outer part of the surface brightness profile agree well with observations. However, there is only a slight preference for models harboring an IMBH compared to models without.

astro-ph.GA↗
arXiv2011

Spectral clustering and the high-dimensional stochastic blockmodel

Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social networks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasible method to discover these communities. The stochastic blockmodel [Social Networks 5 (1983) 109--137] is a social network model with well-defined communities; each node is a member of one community. For a network generated from the Stochastic Blockmodel, we bound the number of nodes "misclustered" by spectral clustering. The asymptotic results in this paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes, hence the name high-dimensional. In order to study spectral clustering under the stochastic blockmodel, we first show that under the more general latent space model, the eigenvectors of the normalized graph Laplacian asymptotically converge to the eigenvectors of a "population" normalized graph Laplacian. Aside from the implication for spectral clustering, this provides insight into a graph visualization technique. Our method of studying the eigenvectors of random matrices is original.

stat.ML↗
arXiv2011

Random graphs with a given degree sequence

Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have graph limits in the sense of Lovász and Szegedy with identifiable limits. This allows simple determination of other features such as the number of triangles. The argument proceeds by studying a natural exponential model having the degree sequence as a sufficient statistic. The maximum likelihood estimate (MLE) of the parameters is shown to be unique and consistent with high probability. Thus $n$ parameters can be consistently estimated based on a sample of size one. A fast, provably convergent, algorithm for the MLE is derived. These ingredients combine to prove the graph limit theorem. Along the way, a continuous version of the Erdős--Gallai characterization of degree sequences is derived.

math.PR↗
arXiv2011

Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie--Weiss model

Let $(W,W')$ be an exchangeable pair. Assume that \[E(W-W'|W)=g(W)+r(W),\] where $g(W)$ is a dominated term and $r(W)$ is negligible. Let $G(t)=\int_0^tg(s)\,ds$ and define $p(t)=c_1e^{-c_0G(t)}$, where $c_0$ is a properly chosen constant and $c_1=1/\int_{-\infty}^{\infty}e^{-c_0G(t)}\,dt$. Let $Y$ be a random variable with the probability density function $p$. It is proved that $W$ converges to $Y$ in distribution when the conditional second moment of $(W-W')$ given $W$ satisfies a law of large numbers. A Berry-Esseen type bound is also given. We use this technique to obtain a Berry-Esseen error bound of order $1/\sqrt{n}$ in the noncentral limit theorem for the magnetization in the Curie-Weiss ferromagnet at the critical temperature. Exponential approximation with application to the spectrum of the Bernoulli-Laplace Markov chain is also discussed.

math.PR↗
arXiv2011

The missing log in large deviations for triangle counts

This paper solves the problem of sharp large deviation estimates for the upper tail of the number of triangles in an Erdos-Renyi random graph, by establishing a logarithmic factor in the exponent that was missing till now. It is possible that the method of proof may extend to general subgraph counts.

math.PR↗
arXiv2011

The large deviation principle for the Erdős-Rényi random graph

What does an Erdos-Renyi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovasz and coauthors and Szemeredi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition.

math.PR↗
arXiv2011

Random multiplicative functions in short intervals

We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.

math.NT↗
arXiv2010

How planet-planet scattering can create high-inclination as well as long-period orbits

Recent observations have revealed two new classes of planetary orbits. Rossiter- Mclaughlin (RM) measurements have revealed hot Jupiters in high-obliquity orbits. In addition, direct-imaging has discovered giant planets at large (~ 100 AU) separations via direct-imaging technique. Simple-minded disk-migration scenarios are inconsistent with the high-inclination (and even retrograde) orbits as seen in recent RM measurements. Furthermore, forming giant planets at large semi-major axis (a) may be challenging in the core-accretion paradigm. We perform many N-body simulations to explore the two above-mentioned orbital architectures. Planet-planet scattering in a multi-planet system can naturally excite orbital inclinations. Planets can also get scattered to large distances. Large-a planetary orbits created from planet-planet scattering are expected to have high eccentricities (e). Theoretical models predict that the observed long-period planets, such as Fomalhaut-b have moderate e \approx 0.3. Interestingly, these are also in systems with disks. We find that if a massive-enough outer disk is present, a scattered planet may be circularized at large a via dynamical friction from the disk and repeated scattering of the disk particles.

astro-ph.EP↗
arXiv2010

Applications of Stein's method for concentration inequalities

Stein's method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie--Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erdős--Rényi random graph $G(n,p)$ when $p\ge0.31$. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.

math.PR↗
arXiv2010

Properties of Uniform Doubly Stochastic Matrices

We investigate the properties of uniform doubly stochastic random matrices, that is non-negative matrices conditioned to have their rows and columns sum to 1. The rescaled marginal distributions are shown to converge to exponential distributions and indeed even large sub-matrices of side-length $o(n^{1/2-ε})$ behave like independent exponentials. We determine the limiting empirical distribution of the singular values the the matrix. Finally the mixing time of the associated Markov chains is shown to be exactly 2 with high probability.

math.PR↗
arXiv2010

A new approach to strong embeddings

We revisit strong approximation theory from a new perspective, culminating in a proof of the Komlós-Major-Tusnády embedding theorem for the simple random walk. The proof is almost entirely based on a series of soft arguments and easy inequalities. The new technique, inspired by Stein's method of normal approximation, is applicable to any setting where Stein's method works. In particular, one can hope to take it beyond sums of independent random variables.

math.PR↗
arXiv2010

Unstable Planetary Systems Emerging Out Of Gas Disks

The discovery of over 400 extrasolar planets allows us to statistically test our understanding of formation and dynamics of planetary systems via numerical simulations. Traditional N-body simulations of multiple-planet systems without gas disks have successfully reproduced the eccentricity (e) distribution of the observed systems, by assuming that the planetary systems are relatively closely packed when the gas disk dissipates, so that they become dynamically unstable within the stellar lifetime. However, such studies cannot explain the small semi-major axes (a) of extrasolar planetary systems, if planets are formed, as the standard planet formation theory suggests, beyond the ice line. In this paper, we numerically study the evolution of three-planet systems in dissipating gas disks, and constrain the initial conditions that reproduce the observed semi-major axis and eccentricity distributions simultaneously. We adopt the initial conditions that are motivated by the standard planet formation theory, and self-consistently simulate the disk evolution, and planet migration by using a hybrid N-body and 1D gas disk code. We also take account of eccentricity damping, and investigate the effect of saturation of corotation resonances on the evolution of planetary systems. We find that the semi-major axis distribution is largely determined in a gas disk, while the eccentricity distribution is determined after the disk dissipation. We also find that there may be an optimum disk mass which leads to the observed a-e distribution. Our simulations generate a larger fraction of planetary systems trapped in mean-motion resonances (MMRs) than the observations, indicating that the disk's perturbation to the planetary orbits may be important to explain the observed rate of MMRs. We also find much lower occurrence of planets on retrograde orbits than the current observations of close-in planets suggest.

astro-ph.EP↗