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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

Convergence of Probability Measures. By Patrick Billingsley. Pp. xii, 253. 117s. 1968. (John Wiley & Sons, Inc.)

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

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

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

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

Frozen-in disorder in an otherwise homogeneous system, is modeled by interaction terms with random coefficients, given by independent random variables with a translation-invariant distribution. For such systems, it is proven that ind=2 dimensions there can be no first-order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter. Discontinuities which would amount to a continuous symmetry breaking, in systems which are (stochastically) invariant under the action of a continuous subgroup ofO(N), are suppressed by the randomness in dimensionsd≦4. Specific implications are found in the Random-Field Ising Model, for which we conclude that ind=2 dimensions at all (β,h) the Gibbs state is unique for almost all field configurations, and in the Random-Bond Potts Model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point. The results are explained by the argument of Imry and Ma [1]. The proofs involve the analysis of fluctuations of free energy differences, which are shown (using martingale techniques) to be Gaussian on the suitable scale.

/pdf/rounding-effects-of-quenched-randomness-on-first-order-phase-24mud5yi00.pdf

Rounding effects of quenched randomness on first-order phase transitions

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