Random Walks and Polymers in the Presence of Quenched Disorder

After a general introduction to the field, we describe some recent results concerning disorder effects on both ‘random walk models’, where the random walk is a dynamical process generated by local transition rules, and on ‘polymer models’, where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points: thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures T _c(i,L) over the ensemble of samples (i) of size L. We describe the results of this analysis for the bidimensional wetting and for the Poland–Scheraga model of DNA denaturation.Conference Proceedings “Mathematics and Physics”, I.H.E.S., France, November 2005     

Short-Range Spin Glasses and Random Overlap Structures

Properties of Random Overlap Structures (ROSt)’s constructed from the Edwards-Anderson (EA) Spin Glass model on ℤ ^d with periodic boundary conditions are studied. ROSt’s are ℕ×ℕ random matrices whose entries are the overlaps of spin configurations sampled from the Gibbs measure. Since the ROSt construction is the same for mean-field models (like the Sherrington-Kirkpatrick model) as for short-range ones (like the EA model), the setup is a good common ground to study the effect of dimensionality on the properties of the Gibbs measure. In this spirit, it is shown, using translation invariance, that the ROSt of the EA model possesses a local stability that is stronger than stochastic stability, a property known to hold at almost all temperatures in many spin glass models with Gaussian couplings. This fact is used to prove stochastic stability for the EA spin glass at all temperatures and for a wide range of coupling distributions. On the way, a theorem of Newman and Stein about the pure state decomposition of the EA model is recovered and extended.     

Superbosonization of Invariant Random Matrix Ensembles

‘Superbosonization’ is a new variant of the method of commuting and anti-commuting variables as used in studying random matrix models of disordered and chaotic quantum systems. We here give a concise mathematical exposition of the key formulas of superbosonization. Conceived by analogy with the bosonization technique for Dirac fermions, the new method differs from the traditional one in that the superbosonization field is dual to the usual Hubbard-Stratonovich field. The present paper addresses invariant random matrix ensembles with symmetry group U _n , O _n , or USp _n , giving precise definitions and conditions of validity in each case. The method is illustrated at the example of Wegner’s n-orbital model. Superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non-Gaussian and/or non-invariant type.     

Quasi-long-range order in the random anisotropy Heisenberg model

The random field and random anisotropy N-vector models are studied with the functional renormalization group in 4−ε dimensions. The random anisotropy Heisenberg (N=3) model has a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law 〈m(r ₁)m(r ₂)〉∼|r ₁−r ₂|^{− 0.62ε}. The magnetic susceptibility diverges at low fields as χ∼H ^−1+0.15ε. In the random field N-vector model the correlation length is finite at arbitrarily weak disorder for any N>3. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 2, 130–135 (25 July 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Spherical Model in a Random Field

We investigate the properties of the Gibbs states and thermodynamic observables of the spherical model in a random field. We show that on the low-temperature critical line the magnetization of the model is not a self-averaging observable, but it self-averages conditionally. We also show that an arbitrarily weak homogeneous boundary field dominates over fluctuations of the random field once the model transits into a ferromagnetic phase. As a result, a homogeneous boundary field restores the conventional self-averaging of thermodynamic observables, like the magnetization and the susceptibility. We also investigate the effective field created at the sites of the lattice by the random field, and show that at the critical temperature of the spherical model the effective field undergoes a transition into a phase with long-range correlations ∼r ^4−d .     

Exact Asymptotics of the Freezing Transition of a Logarithmically Correlated Random Energy Model

We consider a logarithmically correlated random energy model, namely a model for directed polymers on a Cayley tree, which was introduced by Derrida and Spohn. We prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the KPP-equation—thus translating Bramson’s work on the KPP-equation into a discrete time case. We also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point.     

Variational Bounds for the Generalized Random Energy Model

We compute the pressure of the random energy model (REM) and generalized random energy model (GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra’s “broken replica symmetry bounds,” and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand’s concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new.     

Spin density wave in oxypnictide superconductors in a three-band model

The spin density wave and its temperature dependence in oxypnictide are studied in a three-band model. The spin susceptibilities with various interactions are calculated in the random phase approximation (PPA). It is found that the spin susceptibility peaks around the M point show a spin density wave (SDW) with momentum (0, π) and a clear stripe-like spin configuration. The intra-band Coulomb repulsion enhances remarkably the SDW but the Hund’s coupling weakens it. It is shown that a new resonance appears at higher temperatures at the Γ point indicating the formation of a paramagnetic phase. There is a clear transition from the SDW phase to the paramagnetic phase.     

Random Walk of Second Class Particles in Product Shock Measures

We consider shock measures in a class of conserving stochastic particle systems on ℤ. These shock measures have a product structure with a step-like density profile and include a second class particle at the shock position. We show for the asymmetric simple exclusion process, for the exponential bricklayers’ process, and for a generalized zero range process, that under certain conditions these shocks, and therefore the second class particles, perform a simple random walk. Some previous results, including random walks of product shock measures and stationary shock measures seen from a second class particle, are direct consequences of our more general theorem. Multiple shocks can also be handled easily in this framework. Similar shock structure is also found in a nonconserving model, the branching coalescing random walk, where the role of the second class particle is played by the rightmost (or leftmost) particle.     

A mean-field theory for strongly disordered non-frustrated antiferromagnets

Motivated by impurity-induced magnetic ordering phenomena in spin-gap materials like TlCuCl₃, we develop a mean-field theory for strongly disordered antiferromagnets, designed to capture the broad distribution of coupling constants in the effective model for the impurity degrees of freedom. Based on our results, we argue that in the presence of random magnetic couplings the conventional first-order spin-flop transition of an anisotropic antiferromagnet is split into two transitions at low temperatures, associated with separate order parameters along and perpendicular to the field axis. We demonstrate the existence of either a bicritcal point or a critical endpoint in the temperature–field phase diagram, with the consequence that signatures of the spin flop are more pronounced at elevated temperature.     

Stability of interfaces in a random environment. A rigorous renormalization group analysis of a hierarchical model

We study a hierarchical model of domain walls in aD-dimensional bond disordered Ising model at low temperatures. Using a renormalization group method inspired by the work of Bricmont and Kupiainen for the random field Ising model, we prove the existence of rigid interfaces at low enough temperatures in dimensionsD>3.     

Quark confinement in 2+1 dimensional pure Yang-Mills theory

We report some progress on the quark confinement problem in 2 + 1 dim. pure Yang-Mills theory, using Euclidean instanton methods. The instantons are regularized Wu-Yang ‘monopoles’, whose long range Coulomb field is screened by collective effects. Such configurations are stable to small perturbations unlike the case of singular, undressed monopoles. Using exact non-perturbative results for the 3-dim. Coulomb gas, where Debye screening holds for arbitrarily low temperatures, we show in a self-consistent way that a mass gap is dynamically generated in the gauge theory. The mass gap also determines the size of the monopoles. We also identify the disorder operator of the model in terms of the Sine-Gordon field of the Coulomb gas and hence obtain a dual representation whose symmetry is the centre ofSU(2).     

Nonlinear theory of mirror instability near its threshold

An asymptotic model based on a reductive perturbative expansion of the drift kinetic and Maxwell’s equations is used to demonstrate that, near the instability threshold, the nonlinear dynamics of mirror modes in a magnetized plasma with anisotropic ion temperatures involves a subcritical bifurcation, leading to the formation of small-scale structures with amplitudes comparable with the ambient magnetic field.     

Destruction of localized electron pairs above the magnetic-field-driven superconductor-insulator transition in amorphous In-O films

The field-induced superconductivity-destroying quantum transition in amorphous indium oxide films are investigated at low temperatures down to 30 mK. It is found that, on the high-field side of the transition, the magnetoresistance reaches a maximum and the phase can be insulating as well as metallic. With further increase of the magnetic field the resistance of the film drops and in the high-field limit approaches the resistance value at the transition point, so that at high fields the metallic phase occurs for both cases. We give a qualitative account of this behavior in terms of field-induced destruction of localized electron pairs. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 4, 337–342 (25 August 1998)     

A Random Surface Wave Equation for Bosonic QCD(<Emphasis Type="Italic">SU</Emphasis>(∞))

We propose a quantum surface wave functional describing the interaction between a colored SU(N _c ) membrane and a quantized Yang-Mills field. Additionally, we deduce its associated wave equation in the t’Hooft N _c →∞ limit. We show that its reproduces the Yang-Mills Field Theory at a large rigid random surface scale.     

Modal Quantum Theory

We present a discrete model theory similar in structure to ordinary quantum mechanics, but based on a finite field instead of complex amplitudes. The interpretation of this theory involves only the “modal” concepts of possibility and necessity rather than quantitative probability measures. Despite its simplicity, our model theory includes entangled states and has versions of both Bell’s theorem and the no cloning theorem.     

Some mathematical aspects of the scaling limit of critical two-dimensional systems

It has been observed long ago that many systems from statistical physics behave randomly on macroscopic level at their critical temperature. In two dimensions, these phenomena have been classified by theoretical physicists thanks to conformal field theory, that led to the derivation of the exact value of various critical exponents that describe their behavior near the critical temperature. In the last couple of years, combining ideas of complex analysis and probability theory, mathematicians have constructed and studied a family of random fractals (called ‘Schramm-Loewner evolutions’ or SLE) that describe the only possible conformally invariant limits of the interfaces for these models. This gives a concrete construction of these random systems, puts various predictions on a rigorous footing, and leads to further understanding of their behavior. The goal of this paper is to survey some of these recent mathematical developments, and to describe a couple of basic underlying ideas. We will also briefly describe some very recent and ongoing developments relating SLE, Brownian loop soups and conformal field theory.     

Cartan–Weyl Dirac and Laplacian Operators,Brownian Motions: The Quantum Potential and Scalar Curvature,Maxwell’s and Dirac-Hestenes Equations,and Supersymmetric Systems

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q