We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

5分で分かる!? 有名論文ナナメ読み：Jacob Devlin et al. : BERT : Pre-training of Deep Bidirectional Transformers for Language Understanding

The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

/pdf/an-introduction-to-random-matrices-1gerxro6r6.pdf

An Introduction to Random Matrices

The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.

/pdf/topics-in-random-matrix-theory-3dl0sp9tcn.pdf

Topics in Random Matrix Theory

Starting with a “relativistic” Schrodinger Hamiltonian for neutral gravitating particles, we prove that as the particle number N → ∞ and the gravitation constant G → 0 we obtain the well known semiclassical theory for the ground state of stars. For fermions; the correct limit is to fix GN 2/3 and the Chandrasekhar formula is obtained. For bosons the correct limit is to fix GN and a Hartree type equation is obtained. In the fermion case we also prove that the semiclassical equation has a unique solution — a fact which had not been established previously.

/pdf/the-chandrasekhar-theory-of-stellar-collapse-as-the-limit-of-46u9a1pew4.pdf

The Chandrasekhar Theory of Stellar Collapse as the Limit of Quantum Mechanics

We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schrodinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k.

Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

https://dash.harvard.edu/bitstream/handle/1/25426536/1007.4652v7.pdf?sequence=1

Rigidity of eigenvalues of generalized Wigner matrices

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V (N(xi − xj)), where x = (x1, . . ., xN) denotes the positions of the particles. Let HN denote the Hamiltonian of the system and let ψN,t be the solution to the Schrodinger equation. Suppose that the initial data ψN,0 satisfies the energy condition h ψN,0, H k N ψN,0i ≤ C k N k for k = 1,2, . . .. We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density matrices of ψN,t are also asymptotically factorized and the one particle orbital wave function solves the Gross- Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant given by the scattering length of the potential V. We also prove the same conclusion if the energy condition holds only for k = 1 but the factorization of ψN,0 is assumed in a stronger sense. AMS Subject Classification Number: 81V70, 81T18, 35Q55

https://annals.math.princeton.edu/wp-content/uploads/annals-v172-n1-p06-s.pdf

Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/2001/0005/0006/ATMP-2001-0005-0006-a006.pdf

Horng-Tzer Yau

Papers

The Chandrasekhar Theory of Stellar Collapse as the Limit of Quantum Mechanics

Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

Rigidity of eigenvalues of generalized Wigner matrices

Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

Derivation of the nonlinear Schrödinger equation from a many body Coulomb system