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Sourav Chatterjee

Publications and source records attributed to Sourav Chatterjee.

At least 181 records · Page 10Linked to original sources
arXiv2015

Planetesimal Interactions Can Explain the Mysterious Period Ratios of Small Near-Resonant Planets

An intriguing trend among \kepler's multi-planet systems is an overabundance of planet pairs with period ratios just wide of a mean motion resonance (MMR) and a dearth of systems just narrow of them. Traditional planet formation models are at odds with these observations. They are also in contrast with the period ratios of radial-velocity-discovered multi-planet systems which tend to pile up at 2:1 MMR. We propose that gas-disk migration traps planets in a MMR. After gas dispersal, orbits of these trapped planets are altered through interaction with a residual planetesimal disk. We study the effects of planetesimal disk interactions on planet pairs trapped in 2:1 MMR using planets of mass typical of the Kepler planet candidates (KPC) and explore large ranges for the mass, and density profile of the planetesimal disk. We find that planet-planetesimal disk interactions naturally create the observed asymmetry in period-ratio distribution for large ranges of planetesimal disk and planet properties. If the planetesimal disk mass is above a threshold of ~0.2x the planet mass, these interactions typically disrupt MMR. Afterwards, the planets migrate in such a way that the final period-ratio is slightly higher than the integer ratio corresponding to the initial MMR. Below this threshold these interactions typically cannot disrupt the resonance and the period ratio stays close to the integer ratio. The threshold explains why the more massive planet pairs found by RV surveys are still in resonance. We encourage future research to explore how significantly the associated accretion would change the planets' atmospheric and surface properties.

astro-ph.EP↗
arXiv2014

Matrix estimation by Universal Singular Value Thresholding

Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Candès and collaborators. This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has "a little bit of structure." Surprisingly, this simple estimator achieves the minimax error rate up to a constant factor. The method is applied to solve problems related to low rank matrix estimation, blockmodels, distance matrix completion, latent space models, positive definite matrix completion, graphon estimation and generalized Bradley--Terry models for pairwise comparison.

math.ST↗
arXiv2014

Vulcan Planets: Inside-Out Formation of the Innermost Super-Earths

The compact multi-transiting systems discovered by Kepler challenge traditional planet formation theories. These fall into two broad classes: (1) formation further out followed by migration; (2) formation in situ from a disk of gas and planetesimals. In the former, an abundance of resonant chains is expected, which the Kepler data do not support. In the latter, required disk mass surface densities may be too high. A recently proposed mechanism hypothesizes that planets form in situ at the pressure trap associated with the dead-zone inner boundary (DZIB) where radially drifting "pebbles" accumulate. This scenario predicts planet masses ($M_p$) are set by the gap-opening process that then leads to DZIB retreat, followed by sequential, inside-out planet formation (IOPF). For typical disk accretion rates, IOPF predictions for $M_p$, $M_p$ versus orbital radius $r$, and planet-planet separations are consistent with observed systems. Here we investigate the IOPF prediction for how the masses, $M_{p,1}$, of the innermost ("Vulcan") planets vary with $r$. We show that for fiducial parameters, $M_{p,1}\simeq5.0(r/{\rm{0.1\:AU}})\:M_\oplus$, independent of the disk's accretion rate at time of planet formation. Then, using Monte Carlo sampling of a population of these innermost planets, we test this predicted scaling against observed planet properties, allowing for intrinsic dispersions in planetary densities and Kepler's observational biases. These effects lead to a slightly shallower relation $M_{p,1}\propto{r}^{0.9\pm0.2}$, which is consistent with $M_{p,1}\propto{r}^{0.7\pm0.2}$ of the observed Vulcans. The normalization of the relation constrains the gap-opening process, favoring relatively low viscosities in the inner dead zone.

astro-ph.EP↗
arXiv2014

A new perspective on least squares under convex constraint

Consider the problem of estimating the mean of a Gaussian random vector when the mean vector is assumed to be in a given convex set. The most natural solution is to take the Euclidean projection of the data vector on to this convex set; in other words, performing "least squares under a convex constraint." Many problems in modern statistics and statistical signal processing theory are special cases of this general situation. Examples include the lasso and other high-dimensional regression techniques, function estimation problems, matrix estimation and completion, shape-restricted regression, constrained denoising, linear inverse problems, etc. This paper presents three general results about this problem, namely, (a) an exact computation of the main term in the estimation error by relating it to expected maxima of Gaussian processes (existing results only give upper bounds), (b) a theorem showing that the least squares estimator is always admissible up to a universal constant in any problem of the above kind and (c) a counterexample showing that least squares estimator may not always be minimax rate-optimal. The result from part (a) is then used to compute the error of the least squares estimator in two examples of contemporary interest.

math.ST↗
arXiv2014

Pebble Delivery for Inside-Out Planet Formation

Inside-Out Planet Formation (IOPF; Chatterjee & Tan 2014, hereafter CT14) is a scenario for sequential in situ planet formation at the pressure traps of retreating dead zone inner boundaries (DZIBs) motivated to explain the many systems with tightly packed inner planets (STIPs) discovered by Kepler. The scenario involves build-up of a pebble-dominated protoplanetary ring, supplied by radial drift of pebbles from the outer disk. It may also involve further build-up of planetary masses to gap-opening scales via continued pebble accretion. Here we study radial drift & growth of pebbles delivered to the DZIB in fiducial IOPF disk models.

astro-ph.EP↗
arXiv2014

Assumptionless consistency of the Lasso

The Lasso is a popular statistical tool invented by Robert Tibshirani for linear regression when the number of covariates is greater than or comparable to the number of observations. The purpose of this note is to highlight the simple fact (noted in a number of earlier papers in various guises) that for the loss function considered in Tibshirani's original paper, the Lasso is consistent under almost no assumptions at all.

math.ST↗
arXiv2014

A short survey of Stein's method

Stein's method is a powerful technique for proving central limit theorems in probability theory when more straightforward approaches cannot be implemented easily. This article begins with a survey of the historical development of Stein's method and some recent advances. This is followed by a description of a "general purpose" variant of Stein's method that may be called the generalized perturbative approach, and an application of this method to minimal spanning trees. The article concludes with the descriptions of some well known open problems that may possibly be solved by the perturbative approach or some other variant of Stein's method.

math.PR↗
arXiv2014

Fluctuations of the Bose-Einstein condensate

This article gives a rigorous analysis of the fluctuations of the Bose-Einstein condensate for a system of non-interacting bosons in an arbitrary potential, assuming that the system is governed by the canonical ensemble. As a result of the analysis, we are able to tell the order of fluctuations of the condensate fraction as well as its limiting distribution upon proper centering and scaling. This yields interesting results. For example, for a system of $n$ bosons in a 3D harmonic trap near the transition temperature, the order of fluctuations of the condensate fraction is $n^{-1/2}$ and the limiting distribution is normal, whereas for the 3D uniform Bose gas, the order of fluctuations is $n^{-1/3}$ and the limiting distribution is an explicit non-normal distribution. For a 2D harmonic trap, the order of fluctuations is $n^{-1/2}(\log n)^{1/2}$, which is larger than $n^{-1/2}$ but the limiting distribution is still normal. All of these results come as easy consequences of a general theorem.

math.PR↗
arXiv2013

Estimating and understanding exponential random graph models

We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large-deviations approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and Varadhan [European J. Combin. 32 (2011) 1000-1017]. The theory explains a host of difficulties encountered by applied workers: many distinct models have essentially the same MLE, rendering the problems ``practically'' ill-posed. We give the first rigorous proofs of ``degeneracy'' observed in these models. Here, almost all graphs have essentially no edges or are essentially complete. We supplement recent work of Bhamidi, Bresler and Sly [2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2008) 803-812 IEEE] showing that for many models, the extra sufficient statistics are useless: most realizations look like the results of a simple Erdős-Rényi model. We also find classes of models where the limiting graphs differ from Erdős-Rényi graphs. A limitation of our approach, inherited from the limitation of graph limit theory, is that it works only for dense graphs.

math.PR↗
arXiv2013

Inside-Out Planet Formation

The compact multi-transiting planet systems discovered by Kepler challenge planet formation theories. Formation in situ from disks with radial mass surface density, $Σ$, profiles similar to the minimum mass solar nebula (MMSN) but boosted in normalization by factors $\gtrsim 10$ has been suggested. We propose that a more natural way to create these planets in the inner disk is formation sequentially from the inside-out via creation of successive gravitationally unstable rings fed from a continuous stream of small (~cm--m size) "pebbles", drifting inwards via gas drag. Pebbles collect at the pressure maximum associated with the transition from a magneto-rotational instability (MRI)-inactive ("dead zone") region to an inner MRI-active zone. A pebble ring builds up until it either becomes gravitationally unstable to form an $\sim 1\ M_\oplus$ planet directly or induces gradual planet formation via core accretion. The planet may undergo Type I migration into the active region, allowing a new pebble ring and planet to form behind it. Alternatively if migration is inefficient, the planet may continue to accrete from the disk until it becomes massive enough to isolate itself from the accretion flow. A variety of densities may result depending on the relative importance of residual gas accretion as the planet approaches its isolation mass. The process can repeat with a new pebble ring gathering at the new pressure maximum associated with the retreating dead zone boundary. Our simple analytical model for this scenario of inside-out planet formation yields planetary masses, relative mass scalings with orbital radius, and minimum orbital separations consistent with those seen by Kepler. It provides an explanation of how massive planets can form with tightly-packed and well-aligned system architectures, starting from typical protoplanetary disk properties.

astro-ph.EP↗
arXiv2013

On level sets of Gaussian fields

In this short note, we present a theorem concerning certain "additive structure" for the level sets of non-degenerate Gaussian fields, which yields the multiple valley phenomenon for extremal fields with exponentially many valleys.

math.PR↗
arXiv2013

Stellar Collisions and Blue Straggler Stars in Dense Globular Clusters

Blue straggler stars (BSS) are abundantly observed in all Galactic globular clusters (GGC) where data exist. However, observations alone cannot reveal the relative importance of various formation channels or the typical formation times for this well studied population of anomalous stars. Using a state-of-the-art Hénon-type Monte Carlo code that includes all relevant physical processes, we create 128 models with properties typical of the observed GGCs. These models include realistic numbers of single and binary stars, use observationally motivated initial conditions, and span large ranges in central density, concentration, binary fraction, and mass. Their properties can be directly compared with those of observed GGCs. We can easily identify the BSSs in our models and determine their formation channels and birth times. We find that for central densities above ~10^3 Msun/pc^3 the dominant formation channel is stellar collisions while for lower density clusters, mass transfer in binaries provides a significant contribution (up to ~ 60% in our models). The majority of these collisions are binary-mediated, occurring during 3-body and 4-body interactions. As a result a strong correlation between the specific frequency of BSSs and the binary fraction in a cluster can be seen in our models. We find that the number of BSSs in the core shows only a weak correlation with the collision rate estimator Γtraditionally used by observers, in agreement with the latest Hubble Space Telescope (ACS) data. Using an idealized "full mixing" prescription for collision products, our models indicate that the BSSs observed today may have formed several Gyrs ago. However, denser clusters tend to have younger (~1 Gyr) BSSs.

astro-ph.SR↗
arXiv2013

Mock Observations of Blue Stragglers in Globular Cluster Models

We created artificial color-magnitude diagrams of Monte Carlo dynamical models of globular clusters, and then used observational methods to determine the number of blue stragglers in those clusters. We compared these blue stragglers to various cluster properties, mimicking work that has been done for blue stragglers in Milky Way globular clusters to determine the dominant formation mechanism(s) of this unusual stellar population. We find that a mass-based prescription for selecting blue stragglers will choose approximately twice as many blue stragglers than a selection criterion that was developed for observations of real clusters. However, the two numbers of blue stragglers are well-correlated, so either selection criterion can be used to characterize the blue straggler population of a cluster. We confirm previous results that the simplified prescription for the evolution of a collision or merger product in the BSE code overestimates their lifetimes. We show that our model blue stragglers follow similar trends with cluster properties (core mass, binary fraction, total mass, collision rate) as the true Milky Way blue stragglers, as long as we restrict ourselves to model clusters with an initial binary fraction higher than 5%. We also show that, in contrast to earlier work, the number of blue stragglers in the cluster core does have a weak dependence on the collisional parameter Gamma in both our models and in Milky Way globular clusters.

astro-ph.SR↗
arXiv2013

Stochastic solutions of the wave equation

Unlike the heat equation or the Laplace equation, solutions of the wave equation on general domains have no known stochastic representation. This short note gives a simple solution to this well known problem in arbitrary dimensions. The proposed representation has several shortcomings, one of which is that it does not cover all solutions. Still, it is proof that a large class of nontrivial solutions of the wave equation in general dimensions and domains may indeed be represented stochastically.

math.PR↗
arXiv2013

Large Deviations for Random Matrices

We prove a large deviation result for a random symmetric n x n matrix with independent identically distributed entries to have a few eigenvalues of size n. If the spectrum S survives when the matrix is rescaled by a factor of n, it can only be the eigenvalues of a Hilbert-Schmidt kernel k(x,y) on [0,1] x [0,1]. The rate function for k is $I(k)=1/2\int h(k(x,y) dxdy$ where h is the Cramer rate function for the common distribution of the entries that is assumed to have a tail decaying faster than any Gaussian. The large deviation for S is then obtained by contraction.

math.PR↗
arXiv2013

Invariant measures and the soliton resolution conjecture

The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. This paper proves a "statistical version" of this conjecture at mass-subcritical nonlinearity, in the following sense. The uniform probability distribution on the set of all functions with a given mass and energy, if such a thing existed, would be a natural invariant measure for the NLS flow and would reflect the long-term behavior for "generic initial data" with that mass and energy. Unfortunately, such a probability measure does not exist. We circumvent this problem by constructing a sequence of discrete measures that, in principle, approximate this fictitious probability distribution as the grid size goes to zero. We then show that a continuum limit of this sequence of probability measures does exist in a certain sense, and in agreement with the soliton resolution conjecture, the limit measure concentrates on the unique ground state soliton. Combining this with results from ergodic theory, we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting.

math.AP↗
arXiv2012

Understanding the Dynamical State of Globular Clusters: Core-Collapsed vs Non Core-Collapsed

We study the dynamical evolution of globular clusters using our Hénon-type Monte Carlo code for stellar dynamics including all relevant physics such as two-body relaxation, single and binary stellar evolution, Galactic tidal stripping, and strong interactions such as physical collisions and binary mediated scattering. We compute a large database of several hundred models starting from broad ranges of initial conditions guided by observations of young and massive star clusters. We show that these initial conditions very naturally lead to present day clusters with properties including the central density, core radius, half-light radius, half-mass relaxation time, and cluster mass, that match well with those of the old Galactic globular clusters. In particular, we can naturally reproduce the bimodal distribution in observed core radii separating the "core-collapsed" vs the "non core-collapsed" clusters. We see that the core-collapsed clusters are those that have reached or are about to reach the equilibrium "binary burning" phase. The non core-collapsed clusters are still undergoing gravo-thermal contraction.

astro-ph.GA↗