Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝ^d, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.     

Random Walk on the Incipient Infinite Cluster for Oriented Percolation in High Dimensions

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on . In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is , and thereby prove a version of the Alexander–Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resistance for the random graph. A second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d > 6, by extending results about critical oriented percolation obtained previously via the lace expansion.     

Random Walk Weakly Attracted to a Wall

We consider a random walk X _n in ℤ₊, starting at X ₀=x≥0, with transition probabilities

Torelli Theorem for the Deligne–Hitchin Moduli Space

Fix integers g ≥ 3 and r ≥ 2, with r ≥ 3 if g = 3. Given a compact connected Riemann surface X of genus g, let denote the corresponding Deligne–Hitchin moduli space. We prove that the complex analytic space determines (up to an isomorphism) the unordered pair , where is the Riemann surface defined by the opposite almost complex structure on X.     

Random Graph Asymptotics on High-Dimensional Tori

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when d > 6 for sufficiently spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V ^2/3 and below by a small constant times , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on under which the lower bound can be improved to small constant times , i.e. we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by [1], apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. Our method is crucially based on the results in [11, 12], where the scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on . We also strongly rely on mean-field results for percolation on proved in [17–20].     

Spectral Measure of Heavy Tailed Band and Covariance Random Matrices

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix whose (i, j) entry is , where (x _ij , 1 ≤ i ≤ j < ∞) is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an α-stable law, , and σ is a deterministic function. For random diagonal D _N independent of and with appropriate rescaling a _N , we prove that converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries. Supported in part by a Discovery grant from the Natural Sciences and Engineering Research Council of Canada and a University of Saskatchewan start-up grant. Research partially supported by NSF grant #DMS-0806211.     

On Almost Randomizing Channels with a Short Kraus Decomposition

For large d, we study quantum channels on C ^d obtained by selecting randomly N independent Kraus operators according to a probability measure μ on the unitary group . When μ is the Haar measure, we show that for , such a channel is ε-randomizing with high probability, which means that it maps every state within distance ε/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discretization argument. Moreover, for general μ, we obtain an ε-randomizing channel provided . For d = 2 ^k (k qubits), this includes Kraus operators obtained by tensoring k random Pauli matrices. This leads to more efficient constructions of almost randomizing channels. The proof uses recent results on empirical processes in Banach spaces.     

Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation

We consider an aggregation equation in , n ≥ 2 with fractional dissipation, namely, , where 0 ≤ γ < 1 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e ^−|x|. We prove that for a class of smooth initial data, the solutions develop blow-up in finite time.     

Surface Tension in the Dilute Ising Model. The Wulff Construction

We study the surface tension and the phenomenon of phase coexistence for the Ising model on with ferromagnetic but random couplings. We prove the convergence in probability (with respect to random couplings) of surface tension and analyze its large deviations: upper deviations occur at volume order while lower deviations occur at surface order. We study the asymptotics of surface tension at low temperatures and relate the quenched value τ ^q of surface tension to maximal flows (first passage times if d =  2). For a broad class of distributions of the couplings we show that the inequality –where τ ^a is the surface tension under the averaged Gibbs measure – is strict at low temperatures. We also describe the phenomenon of phase coexistence in the dilute Ising model and discuss some of the consequences of the media randomness. All of our results hold as well for the dilute Potts and random cluster models.     

Computer Simulations for Some One-Dimensional Models of Random Walks in Fluctuating Random Environment

We report some results of computer simulations for two models of random walks in random environment (rwre) on the one-dimensional lattice for fixed space–time configuration of the environment (“quenched rwre”): a “Markov model” with Markov dependence in time, and a “quasi stationary” model with long range space–time correlations. We compare with the corresponding results for a model with i.i.d. (in space time) environment. In the range of times available to us the quenched distributions of the random walk displacement are far from gaussian, but as the behavior is similar for all three models one cannot exclude asymptotic gaussianity, which is proved for the model with i.i.d. environment. We also report results on the random drift and on some time correlations which show a clear power decay     

Phase Transition in the 1d Random Field Ising Model with Long Range Interaction

We study one–dimensional Ising spin systems with ferromagnetic, long–range interaction decaying as n ^−2+α , , in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, gaussian or subgaussian. We show, for temperature and strength of the randomness (variance) small enough, with IP = 1 with respect to the random fields, that there are at least two distinct extremal Gibbs measures. Supported by: GDRE 224 GREFI-MEFI, CNRS-INdAM. P.P was also partially supported by INdAM program Professori Visitatori 2007; M.C and E.O were partially supported by Prin07: 20078XYHYS.     

Single-Site Approximation for Reaction-Diffusion Processes

We consider the branching and annihilating random walk and with reaction rates σ and λ, respectively, and hopping rate D, and study the phase diagram in the λ/D,σ/D) plane. According to standard mean-field theory, this system is in an active state for all σ/D≥0, and perturbative renormalization suggests that this mean-field result is valid for d>2; however, nonperturbative renormalization predicts that for all d there is a phase transition line to an absorbing state in the λ/D,σ/D) plane. We show here that a simple single-site approximationreproduces with minimal effort the nonperturbative phase diagram both qualitatively and quantitatively for all dimensions d>2. We expect the approach to be useful for other reaction-diffusion processes involving absorbing state transitions.     

Local Semicircle Law and Complete Delocalization for Wigner Random Matrices

We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales . Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the ℓ ^∞-norm of the corresponding eigenvectors is of order O(N ^−1/2), modulo logarithmic corrections. The upper bound O(N ^−1/2) implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements. Supported by Sofja-Kovalevskaya Award of the Humboldt Foundation. On leave from Cambridge University, UK. Partially supported by NSF grant DMS-0602038.     

Interfaces for Random Cluster Models

A random cluster measure on that is not translationally invariant is constructed for d3, the critical density p _c , and sufficiently large q. The resulting measure is proven to be a Gibbs state satisfying cluster model DLR- equations.     

Harmonic Bilocal Fields Generated by Globally Conformal Invariant Scalar Fields

The twist two contribution in the operator product expansion of for a pair of globally conformal invariant, scalar fields of equal scaling dimension d in four space–time dimensions is a field V ₁ (x₁, x₂) which is harmonic in both variables. It is demonstrated that the Huygens bilocality of V ₁ can be equivalently characterized by a “single–pole property” concerning the pole structure of the (rational) correlation functions involving the product . This property is established for the dimension d = 2 of . As an application we prove that any system of GCI scalar fields of conformal dimension 2 (in four space–time dimensions) can be presented as a (possibly infinite) superposition of products of free massless fields.     

One-and-a-Half Quantum de Finetti Theorems

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation of the unitary group U(d), for highest weights μ, ν and μ + ν. Let ξ_μ be the state obtained by tracing out U _ν. Then ξ_μ is close to a convex combination of the coherent states , where and is the highest weight vector in U _μ. For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.     

The Spectrum of Heavy Tailed Random Matrices

Let X _N be an N → N random symmetric matrix with independent equidistributed entries. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of X _N , once renormalized by , converges almost surely and in expectation to the so-called semicircular distribution as N goes to infinity. In this paper we study the same question when P is in the domain of attraction of an α-stable law. We prove that if we renormalize the eigenvalues by a constant a _N of order , the corresponding spectral distribution converges in expectation towards a law which only depends on α. We characterize and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero. This work was partially supported by Miller institute for Basic Research in Science, University of California Berkeley.     

Reconstruction of Random Colourings

Reconstruction problems have been studied in a number of contexts including biology, information theory and statistical physics. We consider the reconstruction problem for random k-colourings on the Δ-ary tree for large k. Bhatnagar et al. [2] showed non-reconstruction when . We tighten this result and show non-reconstruction when , which is very close to the best known bound establishing reconstruction which is . Supported by NSF grants DMS-0528488 and DMS-0548249.     

Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤ ^d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.     

Compound Basis Arising from the Basic {A^{(1)}_{1}}-Module

Given a conditionally completely positive map on a unital *-algebra , we find an interesting connection between the second Hochschild cohomology of with coefficients in the bimodule of adjointable maps, where M is the GNS bimodule of , and the possibility of constructing a quantum random walk [in the sense of (Attal et al. in Ann Henri Poincar 7(1):59–104, 2006; Lindsay and Parthasarathy in Sankhya Ser A 50(2):151–170, 1988; Sahu in Quantum stochastic Dilation of a class of Quantum dynamical Semigroups and Quantum random walks. Indian Statistical Institute, 2005; Sinha in Banach Center Publ 73:377–390, 2006)] corresponding to . D. Goswami was supported by a project funded by the Indian National Academy of Sciences. L. Sahu had research support from the National Board of Higher Mathematics, DAE (India) is gratefully acknowledged.