Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model

We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry.     

A Hardy’s Uncertainty Principle Lemma in Weak Commutation Relations of Heisenberg-Lie Algebra

In this article we consider linear operators satisfying a generalized commutation relation of a type of the Heisenberg-Lie algebra. It is proven that a generalized inequality of the Hardy’s uncertainty principle lemma follows. Its applications to time operators and abstract Dirac operators are also investigated.     

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.     

Efficient Algorithm for Generating Maxwell Random Variables

A fast, easily implemented and high efficiency algorithm is derived for sampling from the Maxwell distribution. The algorithm is derived from the rejection-acceptance sampling method using the simple exponential decay function as an envelope function for the Maxwell distribution. The derived algorithm requires less number of random numbers per iteration, consumes less number of random numbers per sample and requires less expensive computation functions than the direct and Johnk’s algorithms. The speed of the proposed algorithm is about 1.6 times that of the direct algorithm and is about 1.5 times that of Johnk’s algorithm. Since the proposed algorithm for sampling from Maxwell distribution verified high efficiency and speed, Watt random variables can be generated by transforming Maxwell random variables generated by the proposed algorithm. The speed of generating Watt random variables using the proposed algorithm is about 1.1 times that generated from Kalos’s algorithm.     

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.     

Typical Support and Sanov Large Deviations of Correlated States

Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov’s theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states.     

Wegner Bounds for a Two-Particle Tight Binding Model

We consider a quantum two-particle system on a lattice with interaction and in presence of an IID external potential. We establish Wegner-type estimates for such a model. The main tool used is Stollmann’s lemma.     

Directional quantum random walk induced by coherence

Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker's position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker's position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker's position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker's position distribution.     

It?’s Lemma with Quantum Calculus (q-Calculus): Some Implications

q-derivatives are part of so called quantum calculus. In this paper we investigate how such derivatives can possibly be used in It?’s lemma. This leads us to consider how such derivatives can be used in a social science setting. We conclude that in a It? Lemma setting we cannot use a macroscopic version of the Heisenberg uncertainty principle with q-derivatives.     

Nonlinear surface magnetic polaritons in a nonuniform magnetic field

High-frequency surface magnetic polaritons of finite amplitude propagating along the interface between a ferrite and a nonlinear insulator in a weakly nonuniform, shaft-shaped external magnetic field are investigated theoretically. The analysis is based on employment of the variational method together with bilinear relations having the form of Lorentz’s lemma. It is shown that the wave dispersion and the transverse profile of a wave along the field nonuniformity depend significantly on the amplitude of the wave. Zh. Tekh. Fiz. 68, 92–95 (September 1998)     

A Possible Reinterpretation of Einstein’s Equations

In this paper, we first review Huei’s formulation in which it is shown that the linearized Einstein equations can be written in the same form as the Maxwell equations. We eliminate some imperfections like the scalar potential which is ill linked to the electric-type field, the Lorentz-type force which is obtained with a time independence restriction and the undesired factor 4 which appears in the magnetic-type part. Second, from these results and in the light of a recent work by C.C. Barros, we propose an extension of the equivalence principle and we suggest a new interpretation for Einstein’s equations by showing that the electromagnetic Maxwell equations can be derived from a new version of Einstein’s ones.     

Chaos Control for a Single Pendulum Damping System with Proper Random Phase Under Forced Oscillation

This paper deals with the effect of phase of a Gaussian white noise for a single pendulum system with damping. Based on the Khasminskii’s formulation of spherical coordinate and the extension of Wedig’s algorithm for linear stochastic system, the largest Lyapunov exponent is computed. Due to the change of the sign for the largest Lyapunov exponent, the chaotic behavior of the system is suppressed. Also Poincaré surface of section, phase portrait and the time evolution are analyzed to confirm the stability of the system, which demonstrates the above control methods are effective.     

A Monte Carlo and continuum study of mechanical properties of nanoparticle based films

A combination Monte Carlo and equivalent-continuum simulation approach was used to investigate the structure-mechanical property relationships of titania nanoparticle deposits. Films of titania composed of nanoparticle aggregates were simulated using a Monte Carlo approach with diffusion-limited aggregation. Each aggregate in the simulation is fractal-like and random in structure. In the film structure, it is assumed that bond strength is a function of distance with two limiting values for the bond strengths: one representing the strong chemical bond between the particles at closest proximity in the aggregate and the other representing the weak van der Waals bond between particles from different aggregates. The Young’s modulus of the film is estimated using an equivalent-continuum modeling approach, and the influences of particle diameter (5–100 nm) and aggregate size (3–400 particles per aggregate) on predicted Young’s modulus are investigated. The Young’s modulus is observed to increase with a decrease in primary particle size and is independent of the size of the aggregates deposited. Decreasing porosity resulted in an increase in Young’s modulus as expected from results reported previously in the literature.     

Strict Convexity of the Free Energy for a Class of Non-Convex Gradient Models

We consider a gradient interface model on the lattice with interaction potential which is a non-convex perturbation of a convex potential. We show using a one-step multiple scale analysis the strict convexity of the surface tension at high temperature. This is an extension of Funaki and Spohn’s result [8], where the strict convexity of potential was crucial in their proof. Supported by the DFG-Forschergruppe 718 ‘Analysis and stochastics in complex physical systems’.     

Stein’s Method and Characters of Compact Lie Groups

Stein’s method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson’s circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold. The author received funding from NSF grant DMS-0503901.     

Generalized Jarzynski’s Equality of Inhomogeneous Multidimensional Diffusion Processes

Applying the well-known Feynman-Kac formula of inhomogeneous case, an interesting and rigorous mathematical proof of generalized Jarzynski’s equality of inhomogeneous multidimensional diffusion processes is presented, followed by an extension of the second law of thermodynamics. Then, we explain its physical meaning and applications, extending Hummer and Szabo’s work (Proc. Natl. Acad. Sci. USA 98(7):3658–3661, [2001]) and Hatano-Sasa equality of steady state thermodynamics (Phys. Rev. Lett. 86:3463–3466, [2001]) to the general multidimensional case.     

Symmetry in stochasticity: Random walk models of large-scale structure

This paper describes the insights gained from the excursion set approach, in which various questions about the phenomenology of large-scale structure formation can be mapped to problems associated with the first crossing distribution of appropriately defined barriers by random walks. Much of this is summarized in R K Sheth, AIP Conf. Proc. 1132, 158 (2009). So only a summary is given here, and instead a few new excursion set related ideas and results which are not published elsewhere are presented. One is a generalization of the formation time distribution to the case in which formation corresponds to the time when half the mass was first assembled in pieces, each of which was at least 1/n times the final mass, and where n ≥ 2; another is an analysis of the first crossing distribution of the Ornstein–Uhlenbeck process. The first derives from the mirror-image symmetry argument for random walks which Chandrasekhar described so elegantly in 1943; the second corrects a misuse of this argument. Finally, some discussion of the correlated steps and correlated walks assumptions associated with the excursion set approach, and the relation between these and peaks theory are also included. These are problems in which Chandra’s mirror-image symmetry is broken.     

A Noncommutative de Finetti Theorem: Invariance under Quantum Permutations is Equivalent to Freeness with Amalgamation

We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen “exchangeability” (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables , we prove that invariance of the joint distribution of the x _i ’s under quantum permutations is equivalent to the fact that the x _i ’s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the x _i ’s. Research supported by Discovery and LSI grants from NSERC (Canada) and by a Killam Fellowship from the Canada Council for the Arts.