Mathematical Physics Pk Chattopadhyay Pdf
Chapin Ratte
Phone: +1-510-225-5152 --> Selected publications I am interested in probability theory, statistics, and mathematical physics. The following are some of my favorite works, and some recent preprints. (This list is subject to change from time to time.) For the complete list of publications and preprints, click here or visit my page in Google Scholar.
We define the spectral gap of a Markov chain on a finite state space as the second-smallest singular value of the generator of the chain, generalizing the usual definition of spectral gap for reversible chains. We then define the relaxation time of the chain as the inverse of this spectral gap, and show that this relaxation time can be characterized, for any Markov chain, as the time required for convergence of empirical averages. This relaxation time is related to the Cheeger constant and the mixing time of the chain through inequalities that are similar to the reversible case, and the path argument can be used to get upper bounds. Several examples are worked out. An interesting finding from the examples is that the time for convergence of empirical averages in nonreversible chains can often be substantially smaller than the mixing time.
While the analysis of mean-field spin glass models has seen tremendous progress in the last twenty years, lattice spin glasses have remained largely intractable. This article presents the solutions to a number of questions about the Edwards-Anderson model of short-range spin glasses (in all dimensions) that were raised in the physics literature many years ago. First, it is shown that the ground state is sensitive to small perturbations of the disorder, in the sense that a small amount of noise gives rise to a new ground state that is nearly orthogonal to the old one with respect to the site overlap inner product. Second, it is shown that one can overturn a macroscopic fraction of the spins in the ground state with an energy cost that is negligible compared to the size of the boundary of the overturned region - a feature that is believed to be typical of spin glasses but clearly absent in ferromagnets. The third result is that the boundary of the overturned region in dimension d has fractal dimension strictly greater than d - 1, confirming a prediction from physics. The fourth result is that the correlations between bonds in the ground state can decay at most like the inverse of the distance. This contrasts with the random field Ising model, where it has been shown recently that the correlation decays exponentially in distance in dimension two. The fifth result is that the expected size of the critical droplet of a bond grows at least like a power of the volume. Taken together, these results comprise the first mathematical proof of glassy behavior in a short-range spin glass model.
Is it possible to define a coefficient of correlation which is (a) as simple as the classical coefficients like Pearson's correlation or Spearman's correlation, and yet (b) consistently estimates some simple and interpretable measure of the degree of dependence between the variables, which is 0 if and only if the variables are independent and 1 if and only if one is a measurable function of the other, and (c) has a simple asymptotic theory under the hypothesis of independence, like the classical coefficients? This article answers this question in the affirmative, by producing such a coefficient.
This paper develops techniques for understanding large deviations of sparse random graphs. The techniques for dense graphs, based on Szemerdi's regularity lemma, are not applicable in the sparse regime. The technology developed here applies more broadly to large deviations for nonlinear functions of independent random variables, going beyond classical methods which cater mostly to linear functions.
This monograph studies three features of Gaussian random fields, called superconcentration, chaos, and multiple valleys, and explores the relations between them. It is shown that superconcentration is equivalent to chaos, and chaos implies multiple valleys. Superconcentration has been a known feature in probability theory for a while (under different names). This book connects it to chaos and multiple valleys. Two main results in the book are proofs of the disorder chaos and multiple valley conjectures for mean-field spin glasses.
It is a longstanding conjecture that in the model of first-passage percolation on a lattice, two important numbers, known as the fluctuation exponent and the wandering exponent, are related through a universal relation that does not depend on the dimension. This is sometimes called the KPZ relation. This paper gives a rigorous proof of the KPZ relation assuming that the exponents exist in a certain sense.
This paper introduces a new version of Stein's method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod embeddings, the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily. As an application, a general CLT is proved for functions that are obtained as combinations of many local contributions, where the definition of "local" itself depends on the data. Several examples are given, including the solution to a nearest-neighbor CLT problem posed by P. Bickel.
Consider a deterministically growing surface of any dimension, where the growth at a point is an arbitrary nonlinear function of the heights at that point and its neighboring points. Assuming that this nonlinear function is monotone, invariant under the symmetries of the lattice, equivariant under constant shifts, and twice continuously differentiable, it is shown that any such growing surface approaches a solution of the deterministic KPZ equation in a suitable space-time scaling limit.
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Sourav Chatterjee received the Bachelor of Statistics and Master of Statistics degrees from the Indian Statistical Institute, Calcutta, in 2000 and 2002. He received the PhD degree in statistics from Stanford University in 2005. He joined UC Berkeley as a Neyman Assistant Professor of statistics in 2005, and started as a tenure-track assistant professor in the UC Berkeley statistics department in 2006. He moved to New York University as an associate professor of mathematics in the Courant Institute of Mathematical Sciences in 2009. Since 2013, he has been a professor of mathematics and statistics at Stanford University, Stanford, California. His areas of interest are probability theory, statistics, and mathematical physics.
Dr Chatterjee was awarded a Sloan Research Fellowship in mathematics in 2007, the 2008 Tweedie New Researcher Award from the Institute of Mathematical Statistics, the 2010 Rollo Davidson Prize for work in probability theory, the 2012 Doeblin Prize from the Bernoulli Society, the 2012 Young Researcher Award from the International Indian Statistical Association, and the 2013 Line and Michel Loeve Prize in probability from UC Berkeley. He gave a Medallion Lecture of the Institute of Mathematical Statistics in 2012 and was an invited speaker at the International Congress of Mathematicians in 2014. He was elected a Fellow of the Institute of Mathematical Statistics in 2018, and received the 2020 Infosys Prize in Mathematical Sciences.
Joining us from academia and industry, they bring a wide range of subject matter expertise, including deep learning, robotics, precision health, informatics, user design, computer security, environmental engineering, and more. They also bring an unwavering commitment to teaching and mentoring the next generation of engineers.
Ashesh Chattopadhyay joins Baskin Engineering from industry, most recently working as a staff scientist at the SRI Palo Alto Research Center. He received his Ph.D. in mechanical engineering from Rice University. His research lies at the intersection of theoretical deep learning, dynamical systems, and computational physics aimed at understanding atmospheric dynamics and climate extremes.
Alexander Ioannidis joins Baskin Engineering from Stanford University, where he received his Ph.D. and taught for the past two years as an adjunct professor in computational and mathematical engineering and an instructor in biomedical data science at the medical school. His research centers on the design of algorithms and application of computational methods for problems in genomics, biomedical data science, and precision health. Ioannidis also conducts work in population genetics with a focus on diverse and underrepresented populations, particularly in Oceania and Latin America, working closely with these communities to help reclaim ancestral stories and dispersed community connections.
Hao Ye joined the Baskin School of Engineering in spring 2023. He received his Ph.D. in electrical and computer engineering from Georgia Institute of Technology. Before joining UC Santa Cruz, he worked as a machine learning researcher at Qualcomm AI Research. His research focuses on leveraging machine learning techniques to advance communication networks, especially in the context of AI applications that involve a multitude of edge devices, such as autonomous vehicles.
Prior to joining Baskin Engineering as an associate teaching professor, Tela Favaloro taught as a continuing lecturer and adjunct professor in capstone programming in the Electrical and Computer Engineering Department. Her Ph.D is in electrical engineering with an emphasis in the design and fabrication of laboratory apparatus and techniques for electro-thermal characterization, as well as the design of learner-centered experiential curriculum. As a program designer for the sustainability studies minor and as the director of the UCSC Sustainability Lab (S-Lab), Favaloro has worked to establish holistic interdisciplinary programming centered in experiential learning.
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