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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

This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

/pdf/many-body-physics-with-ultracold-gases-552s10zfdn.pdf

Many-Body Physics with Ultracold Gases

Poly(N-isopropylacrylamide): experiment, theory and application

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

The work presents a short proof of localization under the conditions of either strong disorder (λ > λ0) or extreme energies for a wide class of self adjoint operators with random matrix elements, acting inl
2 spaces. A prototypical example is the discrete Schrodinger operatorH=−Δ+U
0(x)+λV

 x
 onZ

 d
 ,d≧1, withU
0(x) a specified background potential and {V

 x
 } generated as random variables. The general results apply to operators with −Δ replaced by a non-local self adjoint operatorT whose matrix elements satisfy: ∑
 y
 |T

 x,y
 |
 S
 ≦Const., uniformly inx, for somes<1. Localization means here that within a specified energy range the spectrum ofH is of the pure-point type, or equivalently — the wave functions do not spread indefinitely under the unitary time evolution generated byH. The effect is produced by strong disorder in either the potential or in the off-diagonal matrix elementsT

 x, y
 . Under rapid decay ofT

 x, y
 , the corresponding eigenfunctions are also proven to decay exponentially. The method is based on resolvent techniques. The central technical ideas include the use of low moments of the resolvent kernel, i.e. <|G

 E
 
(x, y)|
 s
 > withs small enough (<1) to avoid the divergence caused by the distribution's Cauchy tails, and an effective use of the simple form of the dependence ofG

 E
 
(x, y) on the individual matrix elements ofH in elucidating the implications of the fundamental equation (H−E)G

 E
 (x,x
0)=δ
 x,x0
 . This approach simplifies previous derivations of localization results, avoiding the small denominator difficulties which have been hitherto encountered in the subject. It also yields some new results which include localization under the following sets of conditions: i) potentials with an inhomogeneous non-random partU
0
(x), ii) the Bethe lattice, iii) operators with very slow decay in the off-diagonal terms (T
x,y≈1/|x−y|(d+e)), and iv) localization produced by disordered boundary conditions.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104253939

Localization at large disorder and at extreme energies: an elementary derivation

Brownian motion and harnack inequality for Schrödinger operators

The equality of two critical points - the percolation threshold pH and the point pτ where the cluster size distribution ceases to decay exponentially - is proven for all translation invariant independent percolation models on homogeneous d-dimensional lattices (d^ 1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter M(β, h\ which for h = Q reduces to the percolation density P^ - at the bond density p = l—e~β in the single parameter case. These are: (1) M^hdM/dh + M2 + βMdM/dβ, and (2) dM/dβ^\J\MdM/dh. Inequality (1) is intriguing in that its derivation provides yet another hint of a "φ3 structure" in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents β and δ. One of these resembles an Ising model inequality of Frόhlich and Sokal and yields the mean field bound (5^2, and the other implies the result of Chayes and Chayes that β^ί. An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation /?((5 —1)^1 and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.

/pdf/sharpness-of-the-phase-transition-in-percolation-models-1yvssm2m8y.pdf

Sharpness of the phase transition in percolation models

It is shown, by a general argument, that in 2D quenched randomness results in the elimination of discontinuities in the density of the thermodynamic variable conjugate to the fluctuating parameter. Analogous results for continuous symmetry breaking extend to d\ensuremath{\le}4. In particular, for random-field models we rigorously prove uniqueness of the Gibbs state 2D Ising systems, and absence of continuous symmetry breaking in the Heisenberg model in d\ensuremath{\le}4, as predicted by Imry and Ma. Another manifestation of the general statement is found in 2D random-bond Potts models where a phase transition persists, but ceases to be first order.

Rounding of First-Order Phase Transitions in Systems with Quenched Disorder

We provide here the details of the proof, announced in [1], that ind>4 dimensions the (even) φ4 Euclidean field theory, with a lattice cut-off, is inevitably free in the continuum limit (in the single phase regime). The analysis is nonperturbative, and is based on a representation of the field variables (or spins in Ising systems) as source/sink creation operators in a system of random currents — which may be viewed as the mediators of correlations. In this dual representation, the onset of long-range-order is attributed to percolation in an ensemble of sourceless currents, and the physical interaction in the φ4 field — and other aspects of the critical behavior in Ising models — are directly related to the intersection properties of long current clusters. An insight into the criticality of the dimensiond=4 is derived from an analogy (foreseen by K. Symanzik) with the intersection properties of paths of Brownian motion. Other results include the proof that in certain respect, the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensionsd>4, but not in the low dimensiond=2 — for which we establish the “universality” of hyperscaling.

/pdf/geometric-analysis-of-ph4-fields-and-ising-models-parts-i-2tit2ekl7j.pdf

Michael Aizenman

Papers

Localization at large disorder and at extreme energies: an elementary derivation

Brownian motion and harnack inequality for Schrödinger operators

Sharpness of the phase transition in percolation models

Rounding of First-Order Phase Transitions in Systems with Quenched Disorder

Geometric analysis of φ4 fields and Ising models. Parts I and II