scispace - formally typeset
Search or ask a question

Showing papers in "Communications in Mathematical Physics in 2008"


Journal ArticleDOI
TL;DR: In this paper, it was shown that the volume function of a Sasaki-Einstein manifold is a function on the space of Reeb vector fields, and that it can be computed in terms of topological fixed point data.
Abstract: We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat–Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki–Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler–Einstein metric.

461 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider the problem of identifying sharp criteria under which radial H1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrodinger equation (NLS) i∂tu + Δu + |u|2u = 0 scatter, i.e., approach the solution to a linear Schroffinger equation as t → ±∞.
Abstract: We consider the problem of identifying sharp criteria under which radial H1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrodinger equation (NLS) i∂tu + Δu + |u|2u = 0 scatter, i.e., approach the solution to a linear Schrodinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities \({\|u_0\|_{L^2}\| abla u_0\|_{L^2}}\) and M[u]E[u], where u0 denotes the initial data, and M[u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution eitQ(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if M[u]E[u] \|Q\|_{L^2}\| abla Q\|_{L^2}}\), then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle [17] in their study of the energy-critical NLS.

326 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps.
Abstract: We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.

289 citations


Journal ArticleDOI
TL;DR: In this article, the authors obtained general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor hopping rates p to the right and q = 1−p to the left.
Abstract: In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice $${\mathbb{Z}}$$ with nearest neighbor hopping rates p to the right and q = 1−p to the left. For the most part we consider an N-particle system but for certain of these formulas we can take the $$N\to\infty$$ limit. First we obtain, for the N-particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schutz for the case N = 2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m th left-most particle is at x at time t. In one of these formulas we can take the $$N\to\infty$$ limit, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones.

275 citations


Journal ArticleDOI
TL;DR: The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X as mentioned in this paper.
Abstract: The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP2 and IP1 × IP1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold \({{\mathbb {C}^3} / {\mathbb {Z}_3}}\).

260 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the equations of the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows in a bounded domain, and constructed a solution to the initial-boundary value problem through an approximation scheme and a weak convergence method.
Abstract: The equations of the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic flows are considered in a bounded domain. The viscosity coefficients and heat conductivity can depend on the temperature. A solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global variational weak solution to the three-dimensional full magnetohydrodynamic equations with large data is established.

235 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero.
Abstract: We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing. The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support. The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.

230 citations


Journal ArticleDOI
TL;DR: In this paper, the initial-boundary-value problems for the Navier-Stokes system for compressible fluids with density-dependent viscosities are investigated. But the authors focus on the initial boundary value problems for both bounded spatial domains or periodic domains.
Abstract: The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously.

228 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that two stationary, asymptotically flat vacuum black holes in 5 dimensions with two commuting axial symmetries are identical if and only if their masses, angular momenta, and their "interval structures" coincide.
Abstract: We show that two stationary, asymptotically flat vacuum black holes in 5 dimensions with two commuting axial symmetries are identical if and only if their masses, angular momenta, and their “interval structures” coincide. We also show that the horizon must be topologically either a 3-sphere, a ring, or a Lens-space. Our argument is a generalization of constructions of Morisawa and Ida (based in turn on key work of Maison) who considered the spherical case, combined with basic arguments concerning the nature of the factor manifold of symmetry orbits.

195 citations


Journal ArticleDOI
TL;DR: In this paper, the authors gave a new proof of uniqueness of the Gross-Pitaevskii hierarchy, first established in [1], in a different space, based on space-time estimates similar in spirit to those of [2].
Abstract: The purpose of this note is to give a new proof of uniqueness of the Gross-Pitaevskii hierarchy, first established in [1], in a different space, based on space-time estimates similar in spirit to those of [2].

193 citations


Journal ArticleDOI
TL;DR: In this article, the authors express the GUE Tracy-Widom distribution functions in terms of integrals starting from minus infinity, and show that these integrals can be expressed as the total integral of the Hastings-McLeod solution of the Painleve II equation.
Abstract: The Tracy-Widom distribution functions involve integrals of a Painleve II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painleve II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Widom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom distribution functions are new.

Journal ArticleDOI
Gandalf Lechner1
TL;DR: In this article, the modular nuclearity condition for wedge algebras is introduced, which implies the existence of local observables, and it is shown under which conditions an algebra of observables localized in a wedge-shaped region of spacetime can be used to construct model theories.
Abstract: The subject of this thesis is a novel construction method for interacting relativistic quantum field theories on two-dimensional Minkowski space. Employing the algebraic framework of quantum field theory, it is shown under which conditions an algebra of observables localized in a wedge-shaped region of spacetime can be used to construct model theories. A crucial input in this context is the modular nuclearity condition for wedge algebras, which implies the existence of local observables.As an application of the new method, a rigorous construction of a large family of models with factorizing S-matrices is obtained. In an inverse scattering approach, a given factorizing scattering operator is used to define certain semi-localized Wightman fields associated to it. With the help of these fields, a wedge algebra can be defined, which determines the local observable content of a well-defined quantum field theory. In this approach, the modular nuclearity condition translates to certain analyticity and boundedness conditions on the formfactors of wedge-local observables. These conditions are shown to hold for a large class of underlying S-matrices, including the scattering operators of the Sinh-Gordon model and the scaling Ising model as special examples.The so constructed models are investigated with respect to their scattering properties. They are shown to solve the inverse scattering problem for the underlying S-matrices, and a proof of asymptotic completeness for these models is given.

Journal ArticleDOI
TL;DR: In this article, the asymptotic analysis of a system of coupled kinetic and fluid equations, namely the Vlasov-Fokker-Planck equation and a compressible Navier-Stokes equation, is studied.
Abstract: This article is devoted to the asymptotic analysis of a system of coupled kinetic and fluid equations, namely the Vlasov-Fokker-Planck equation and a compressible Navier-Stokes equation Such a system is used, for example, to model fluid-particle interactions arising in sprays, aerosols or sedimentation problems The asymptotic regime corresponding to a strong drag force and a strong Brownian motion is studied and the convergence toward a two phase macroscopic model is proved The proof relies on a relative entropy method

Journal ArticleDOI
TL;DR: In this article, the Fourier localization technique and Bony's para-product decomposition were used to improve the regularity criterion of the weak solution for the 3D viscous magneto-hydrodynamics equations.
Abstract: We improve and extend some known regularity criterion of the weak solution for the 3D viscous Magneto-hydrodynamics equations by means of the Fourier localization technique and Bony’s para-product decomposition.

Journal ArticleDOI
TL;DR: For all p > 1, the existence of quantum channels with non-multiplicative maximal output p-norms has been shown in this paper, where the violations are large; in all cases, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum entropy of its individual factors.
Abstract: For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p > 1, the minimum output Renyi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p = 1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.

Journal ArticleDOI
TL;DR: In this article, the existence of bound states for Schrodinger systems has been studied in the context of mathematical physics, and different approaches have been proposed depending upon the sizes of the interaction parameters in the system.
Abstract: This paper is concerned with existence of bound states for Schrodinger systems which have appeared as several models from mathematical physics. We establish multiplicity results of bound states for both small and large interactions. This is done by different approaches depending upon the sizes of the interaction parameters in the systems. For small interactions we give a new approach to deal with multiple bound states. The novelty of our approach lies in establishing a certain type of invariant sets of the associated gradient flows. For large interactions we use a minimax procedure to distinguish solutions by analyzing their Morse indices.

Journal ArticleDOI
TL;DR: In this article, a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes is presented, which compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-reversed process.
Abstract: The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-reversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations.

Journal ArticleDOI
TL;DR: In this paper, an expansion of the wave equation on the De Sitter-Schwarzschild metric in terms of resonances is described, and the principal term in the expansion is due to a resonance at 0.
Abstract: We describe an expansion of the solution of the wave equation on the De Sitter–Schwarzschild metric in terms of resonances. The principal term in the expansion is due to a resonance at 0. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an \({\varepsilon}\) derivative loss in the angular directions.

Journal ArticleDOI
TL;DR: In this paper, the impact of the anti-symmetry of the multi-electron wave function on the one electron density matrix was studied, using Berenstein and Sjamaar's theorem on the restriction of an adjoint orbit onto a subgroup.
Abstract: By the Pauli exclusion principle, no quantum state can be occupied by more than one electron. One can state this as a constraint on the one electron density matrix that bounds its eigenvalues by 1. Shortly after its discovery, the Pauli principle was replaced by anti-symmetry of the multi-electron wave function. In this paper we solve a longstanding problem about the impact of this replacement on the one electron density matrix, that goes far beyond the original Pauli principle. Our approach uses Berenstein and Sjamaar’s theorem on the restriction of an adjoint orbit onto a subgroup, and allows us to treat any type of permutational symmetry.

Journal ArticleDOI
TL;DR: In this article, it was shown that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables, and that these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model.
Abstract: We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck’s constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized Greenberger-Horne-Zeilinger (GHZ) states is always bounded so that, in contrast to many other contexts, GHZ states do not lead to extremal quantum correlations in this case. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. Finally, we will relate the existence of diagonal states leading to unbounded violations with a long-standing open problem in the context of Banach algebras.

Journal ArticleDOI
Richard Kenyon1
TL;DR: In this article, the authors studied a model of random surfaces arising in the dimer model on the honeycomb lattice and showed that the large-scale surface fluctuations (height fluctuations) about Σ0 converge as ϵ → 0 to a Gaussian free field for the above conformal structure.
Abstract: We study a model of random surfaces arising in the dimer model on the honeycomb lattice. For a fixed “wire frame” boundary condition, as the lattice spacing ϵ → 0, Cohn, Kenyon and Propp [3] showed the almost sure convergence of a random surface to a non-random limit shape Σ0. In [12], Okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions, in particular constructing a natural conformal structure on them. We show here that when Σ0 has no facets, for a family of boundary conditions approximating the wire frame, the large-scale surface fluctuations (height fluctuations) about Σ0 converge as ϵ → 0 to a Gaussian free field for the above conformal structure. We also show that the local statistics of the fluctuations near a given point x are, as conjectured in [3], given by the unique ergodic Gibbs measure (on plane configurations) whose slope is the slope of the tangent plane of Σ0 at x.

Journal ArticleDOI
TL;DR: In this article, the authors used Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.
Abstract: The authors use Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.

Journal ArticleDOI
TL;DR: Gromov-Witten theory was used to define an enumerative geometry of curves in Calabi-Yau 4-folds as mentioned in this paper, where the main technique is to find exact solutions to moving multiple cover integrals.
Abstract: Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds The main technique is to find exact solutions to moving multiple cover integrals The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation

Journal ArticleDOI
TL;DR: In this paper, a mean-field model for the description of interacting electrons in crystals with local defects was proposed, and the ground state of the self-consistent Fermi sea in the presence of a defect was defined.
Abstract: This article is concerned with the derivation and the mathematical study of a new mean-field model for the description of interacting electrons in crystals with local defects. We work with a reduced Hartree-Fock model, obtained from the usual Hartree-Fock model by neglecting the exchange term. First, we recall the definition of the self-consistent Fermi sea of the perfect crystal, which is obtained as a minimizer of some periodic problem, as was shown by Catto, Le Bris and Lions. We also prove some of its properties which were not mentioned before. Then, we define and study in detail a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. The latter was recently studied by Hainzl, Lewin, Sere and Solovej, based on ideas from Chaix and Iracane. This enables us to define the ground state of the self-consistent Fermi sea in the presence of a defect. We end the paper by proving that our model is in fact the thermodynamic limit of the so-called supercell model, widely used in numerical simulations.

Journal ArticleDOI
TL;DR: In this article, the authors define a Sasaki-Futaki invariant of the polarization of a Reeb vector field and show that it obstructs the existence of constant scalar curvature representatives.
Abstract: Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric.

Journal ArticleDOI
TL;DR: In this paper, the authors consider a polymer with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0.
Abstract: We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length n is given by $$n^{-c}\varphi(n)$$ for some 1 < c < 2 and slowly varying $$\varphi$$ . Disorder is introduced by having the interaction vary from one monomer to another, as a constant u plus i.i.d. mean-0 randomness. There is a critical value of u above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with high probability. To see the effect of disorder on the depinning transition, we compare the contact fraction and free energy (as functions of u) to the corresponding annealed system. We show that for c > 3/2, at high temperature, the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point—the size of this neighborhood scales as $$\beta^{1/(2c-3)}$$ , where β is the inverse temperature. For c < 3/2, given $$\epsilon > 0$$ , for sufficiently high temperature the quenched and annealed curves are within a factor of $$1-\epsilon$$ for all u near the critical point; in particular the quenched and annealed critical points are equal. For c = 3/2 the regime depends on the slowly varying function $$\varphi$$ .

Journal ArticleDOI
TL;DR: In this article, the authors investigated the Doi model for suspensions of rod-like molecules in the dilute regime and proved that the velocity gradient vs. stress relation defined by the stationary and homogeneous flow is not rank-one monotone.
Abstract: We investigate the Doi model for suspensions of rod–like molecules in the dilute regime. For certain parameter values, the velocity gradient vs. stress relation defined by the stationary and homogeneous flow is not rank–one monotone. We then consider the evolution of possibly large perturbations of stationary flows. We prove that, even in the absence of a microscopic cut–off, discontinuities in the velocity gradient cannot occur in finite time. The proof relies on a novel type of estimate for the Smoluchowski equation.

Journal ArticleDOI
TL;DR: In this article, the authors studied the spectrum of the Fibonacci Hamiltonian and proved upper and lower bounds for its fractal dimension in the large coupling regime and showed that it converges to an explicit constant.
Abstract: We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $$\lambda \to \infty, {\rm dim} (\sigma(H_\lambda)) \cdot {\rm log} \lambda$$ converges to an explicit constant, $${\rm log}(1+\sqrt{2})\approx 0.88137$$ . We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrodinger dynamics generated by the Fibonacci Hamiltonian.

Journal ArticleDOI
TL;DR: In this article, it was shown that the spectral distribution of the eigenvalues of a random symmetric matrix with independent equidistributed entries converges almost surely and in expectation to the semicircular distribution as the matrix goes to infinity.
Abstract: 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 $$\sqrt{N}$$ , 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 $$N^{\frac{1}{\alpha}}$$ , the corresponding spectral distribution converges in expectation towards a law $$\mu_\alpha$$ which only depends on α. We characterize $$\mu_\alpha$$ 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.

Journal ArticleDOI
TL;DR: In this paper, simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation was considered and bounds on exit times, transition probabilities, and the range of the random walk were obtained.
Abstract: We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on $${\mathbb{Z}}^{d} \times {\mathbb{Z}}_+$$ . 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 $$\frac {4}{3}$$ , 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.