Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles

This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral intervals ofO(n ^–1) is determined by the type of matrix (real symmetric, Hermitian, or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions.     

Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States

We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product states in \({(\mathbb {C}^d)^{\otimes k}}\), where k and p/d ^k are fixed while d → ∞. When k = 1, the Mar?enko-Pastur law determines (up to small corrections) not only the largest eigenvalue (\({(1+\sqrt{p/d^k})^2}\)) but the smallest eigenvalue \({(\min(0,1-\sqrt{p/d^k})^2)}\) and the spectral density in between. We use the method of moments to show that for k > 1 the largest eigenvalue is still approximately \({(1+\sqrt{p/d^k})^2}\) and the spectral density approaches that of the Mar?enko-Pastur law, generalizing the random matrix theory result to the random tensor case. Our bound on the largest eigenvalue has implications both for sampling from a particular heavy-tailed distribution and for a recently proposed quantum data-hiding and correlation-locking scheme due to Leung and Winter.Since the matrices we consider have neither independent entries nor unitary invariance, we need to develop new techniques for their analysis. The main contribution of this paper is to give three different methods for analyzing mixtures of random product states: a diagrammatic approach based on Gaussian integrals, a combinatorial method that looks at the cycle decompositions of permutations and a recursive method that uses a variant of the Schwinger-Dyson equations.     

The Largest Eigenvalue of Real Symmetric,Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source,Part I

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β=1), Hermitian (β=2), and Hermitian self-dual (β=4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “one-band” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β>0.     

A Perron–Frobenius Type of Theorem for Quantum Operations

We define a special class of quantum operations we call Markovian and show that it has the same spectral properties as a corresponding Markov chain. We then consider a convex combination of a quantum operation and a Markovian quantum operation and show that under a norm condition its spectrum has the same properties as in the conclusion of the Perron–Frobenius theorem if its Markovian part does. Moreover, under a compatibility condition of the two operations, we show that its limiting distribution is the same as the corresponding Markov chain. We apply our general results to partially decoherent quantum random walks with decoherence strength \(0 \le p \le 1\). We obtain a quantum ergodic theorem for partially decoherent processes. We show that for \(0 < p \le 1\), the limiting distribution of a partially decoherent quantum random walk is the same as the limiting distribution for the classical random walk.     

A New Family of Solvable Pearson-Dirichlet Random Walks

An n-step Pearson-Gamma random walk in ? ^d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained random walks of n steps in ? ^d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value. Simple closed-form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d≥d ₀ and any n≥2 when q is either \(q = \frac{d}{2} - 1 \) (d ₀=3) or q=d?1 (d ₀=2) (Le Caër in J. Stat. Phys. 140:728–751, 2010). When the total walk length is chosen, without loss of generality, to be equal to 1, then the constrained step lengths have a Dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a Pearson-Dirichlet random walk. The density of the endpoint position of a n-step planar walk of this type (n≥2), with q=d=2, was shown recently to be a weighted mixture of 1+floor(n/2) endpoint densities of planar Pearson-Dirichlet walks with q=1 (Beghin and Orsingher in Stochastics 82:201–229, 2010). The previous result is generalized to any walk space dimension and any number of steps n≥2 when the parameter of the Pearson-Dirichlet random walk is q=d>1. We rely on the connection between an unconstrained random walk and a constrained one, which have both the same n and the same q=d, to obtain a closed-form expression of the endpoint density. The latter is a weighted mixture of 1+floor(n/2) densities with simple forms, equivalently expressed as a product of a power and a Gauss hypergeometric function. The weights are products of factors which depends both on d and n and Bessel numbers independent of d.     

A Pearson Random Walk with Steps of Uniform Orientation and Dirichlet Distributed Lengths

A constrained diffusive random walk of n steps in ℝ ^d and a random flight in ℝ ^d , which are equivalent, were investigated independently in recent papers (J. Stat. Phys. 127:813, 2007; J. Theor. Probab. 20:769, 2007, and J. Stat. Phys. 131:1039, 2008). The n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1,2,4. Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D.     

Trapping in the Random Conductance Model

We consider random walks on ? ^d among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time 2n. We show that in the situations when the heat kernel exhibits subdiffusive decay—which is known to occur in dimensions d≥4—the walk gets trapped for a time of order n in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.     

Limit Theorems for Open Quantum Random Walks

We consider the limit distributions of open quantum random walks on one-dimensional lattice space. We introduce a dual process to the original quantum walk process, which is quite similar to the relation of Schrödinger-Heisenberg representation in quantum mechanics. By this, we can compute the distribution of the open quantum random walks concretely for many examples and thereby we can also obtain the limit distributions of them. In particular, it is possible to get rid of the initial state when we consider the evolution of the walk, it appears only in the last step of the computation.     

One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion,Phase-Separating Point

We consider the phase separation problem for the one-dimensional ferromagnetic Ising model with long-range two-body interaction, J(n) = n ^?2+α , where ${n\in {\rm {I\!N}}}$ denotes the distance of the two spins and ${\alpha \in [0,\alpha_+[}$ with α ₊ = (log 3)/(log 2) ?1. We prove that when α = 0 the localization of the phase separation fluctuates macroscopically with a non-uniform explicit limiting law, while when 0 < α < α ₊ the macroscopic fluctuations disappear and mesoscopic ones appear with a gaussian behavior when conveniently scaled. The mean magnetization profile is also given.     

Investigation of two resonances in the18O(p,n)18F reaction

Excitation functions for the¹⁸O(p,n)¹⁸F reaction were measured at bombarding energiesE _p=4.6 to 6.6 MeV. In and near two resonances of the yield curves atE _p=5.622 and 6.061 MeV, angular distributions were measured with neutron time-of-flight techniques. The strong neutron decay to theT=1 state in¹⁸F and the similarity of the¹⁸O(p,n) and¹⁸O(n,n) yield curves give good evidence that the structures in the¹⁸O(p,n) yield curve arise from the formation ofT=3/2 states in¹⁹F. A two-level-analysis does not give satisfactory fits to the strongly asymmetric angular distributions.     

The Random Average Process and Random Walk in a Space-Time Random Environment in One Dimension

We study space-time fluctuations around a characteristic line for a one-dimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n ^1/4, where n is the ratio of macroscopic and microscopic scales in the system. The limits of the fluctuations are described by a family of Gaussian processes. In cases of known product-form invariant distributions, this limit is a two-parameter process whose time marginals are fractional Brownian motions with Hurst parameter 1/4. Along the way we study the limits of quenched mean processes for a random walk in a space-time random environment. These limits also happen at scale n ^1/4 and are described by certain Gaussian processes that we identify. In particular, when we look at a backward quenched mean process, the limit process is the solution of a stochastic heat equation.     

The Markov branching random walk and systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov) equations

A general model of a branching random walk inR ¹ is considered, with several types of particles, where the branching occurs with probabilities determined by the type of a parent particle. Each new particle starts moving from the place where it was born, independently of other particles. The distribution of the displacement of a particle, before it splits, depends on its type. A necessary and sufficient condition is given for the random variable $$X^0 = \mathop {\sup max}\limits_{ n \geqq 0 1 \leqq k \leqq N_n } X_{n,k} $$ to be finite. Here,X _{n, k} is the position of thek ^th particle in then ^th generation,N _n is the number of particles in then ^th generation (regardless of their type). It turns out that the distribution ofX ⁰ gives a minimal solution to a natural system of stochastic equations which has a linearly ordered continuum of other solutions. The last fact is used for proving the existence of a monotone travelling-wave solution to systems of coupled non-linear parabolic PDE's.     

Electrical Resistance of the Low Dimensional Critical Branching Random Walk

We show that the electrical resistance between the origin and generation n of the incipient infinite oriented branching random walk in dimensions d < 6 is O(n ^1-α ) for some universal constant α > 0. This answers a question of Barlow et al. (Commun Math Phys 278:385–431, 2008).     

Alzheimer random walk

Using the Monte Carlo simulation, we investigate a memory-impaired self-avoiding walk on a square lattice in which a random walker marks each of sites visited with a given probability p and makes a random walk avoiding the marked sites. Namely, p = 0 and p = 1 correspond to the simple random walk and the self-avoiding walk, respectively. When p> 0, there is a finite probability that the walker is trapped. We show that the trap time distribution can well be fitted by Stacy’s Weibull distribution \(b{\left( {\tfrac{a}{b}} \right)^{\tfrac{{a + 1}}{b}}}{\left[ {\Gamma \left( {\tfrac{{a + 1}}{b}} \right)} \right]^{ - 1}}{x^a}\exp \left( { - \tfrac{a}{b}{x^b}} \right)\) where a and b are fitting parameters depending on p. We also find that the mean trap time diverges at p = 0 as ~p ^{? α} with α = 1.89. In order to produce sufficient number of long walks, we exploit the pivot algorithm and obtain the mean square displacement and its Flory exponent ν(p) as functions of p. We find that the exponent determined for 1000 step walks interpolates both limits ν(0) for the simple random walk and ν(1) for the self-avoiding walk as [ ν(p) ? ν(0) ] / [ ν(1) ? ν(0) ] = p ^β with β = 0.388 when p ? 0.1 and β = 0.0822 when p ? 0.1.     

Weak Multiplicativity for Random Quantum Channels

It is known that random quantum channels exhibit significant violations of multiplicativity of maximum output p-norms for any p > 1. In this work, we show that a weaker variant of multiplicativity nevertheless holds for these channels. For any constant p > 1, given a random quantum channel ${\mathcal{N}}$ (i.e. a channel whose Stinespring representation corresponds to a random subspace S), we show that with high probability the maximum output p-norm of ${\mathcal{N}^{\otimes n}}$ decays exponentially with n. The proof is based on relaxing the maximum output ∞-norm of ${\mathcal{N}}$ to the operator norm of the partial transpose of the projector onto S, then calculating upper bounds on this quantity using ideas from random matrix theory.     

Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

We consider the ensemble of adjacency matrices of Erd?s-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption \({p N \gg N^{2/3}}\), we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erd?s-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erd?s-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.     

An Approach to Wegner’s Estimate Using Subharmonicity

We develop a strategy to establish a Wegner estimate and localization in random lattice Schrödinger operators on $\Bbb{Z}^{d}$ , which does not rely on the usual eigenvalue variation argument. Our assumption is that the potential V(ω) depends real analytically on ω and we use a distributional property of analytic functions in many variables. An application is given to models where V _n is a self-adjoint matrix obtained by random unitary conjugation V _n =U _n AU _n ^* of a fixed matrix A.     

Investigation of the 6p 2(3 P 0)n p Rydberg series of bismuth by multiphoton excitation

Rydberg states of the odd-parity series 6p ²(³ p ₀)n p of BiI are excited by a three-photon process. A two-photon dissociation of Bi₂ into excited atomic states followed by a one-photon absorption leads to highly excited atomic Rydberg states up ton = 32. States of the even-parity Rydberg series 6p ²(³ p ₀)nsJ=1/2,ndJ=3/2 andndJ=5/2 are also observed. In order to avoid the background caused by ionization of the bismuth molecules we performed a two-color excitation with pulsed dye lasers. With this experiment the 6p ²(³ p ₀)npJ=3/2 Rydberg series could be resolved up ton=75. The increasing quantum defect of this series is due to a perturbing state close to the first ionization limit. By a MQDT analysis we obtain the energy of the perturbing state and a value of 58,761.68±0.1 cm^?1 for the first ionization limit of atomic bismuth.     

Simulation of euclidean quantum field theories by a random walk process

A new Monte Carlo method for euclidean lattice field theory is introduced by writing the Boltzmann distribution e^?s as a solution of a diffusion type equation and constructing the associated random walk process. It is practically tested for a quantum mechanical model and a non-compact version of lattice QCD. It is explained where the main interest in this algorithm lies: the diffusion process coming from an action that can be generalized to include non-conservative forces. This possibility is exploited in our QCD version to implement gauge fixing without Faddeev-Popov ghosts.     

Quantum efficiency and responsivity of InSb photodiodes utilizing the Moss-Burstein effect

In this work a detailed analysis of the quantum efficiency of InSb n⁺-p photodetectors produced by liquid phase epitaxy is given, in the case when the n⁺ region is doped to such a level that the Moss-Burstein effect plays an important role. Our starting point was the theoretically determined coefficient of intrinsic absorption and derived expressions for the generated photocurrent in the n⁺ region, depletion layer and p-phase of the photodetector. The results are presented in the form of graphical dependence of the quantum efficiency on the wavelength, with the electron concentration in the n⁺ layer as a parameter.