Abstract: Abstract We calculate, for a branching random walk \(X_n(l)\) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable \(\frac{1}{2^{n}}\sum _{l=1}^{2^n}e^{2\beta X_n(l)}\) , for \(\beta \in {\mathbb {R}}\) . We obtain explicit formulae for the first few moments for finite n. In the limit \(n\rightarrow \infty \) , our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other. PubDate: 2021-01-12
Abstract: Abstract In this paper, we first introduce some new notions of ‘periodic-like’ points, such as almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points, of free semigroup actions. We find that the corresponding sets and gap-sets of these points carry full upper capacity topological entropy of free semigroup actions under certain conditions. Furthermore, \(\phi \) -irregular set acting on free semigroup actions is introduced and it also carries full upper capacity topological entropy in the system with specification property. Finally, we introduce the level set for local recurrence of free semigroup actions and analyze its connections with upper capacity topological entropy. Our analysis generalizes the results obtained by Tian (Different asymptotic behavior versus same dynamical complexity: recurrence & (ir)regularity. Adv. Math. 288:464–526, 2016), Chen et al. (Topological entropy for divergence points. Ergodic Theory Dynam Syst. 25:1173–1208, 2005) and Lau and Shu (The spectrum of Poincaré recurrence. Ergodic Theory Dynam Syst 28:1917–1943, 2007) etc. PubDate: 2021-01-09
Abstract: Abstract We study the dynamics of a Duffing oscillator excited by correlated random perturbations for both fixed and periodically modulated stiffness. In the case of fixed stiffness we see that Poincaré map gets distorted due to the random excitation and, the distortion increases with the increase of correlation of the field. In a strongly correlated field, however, the map becomes purely random. We analyse the maximum value of the Lyapunov exponent and see that the random response competes with the chaotic motion to increase the stability of the system. In the case of periodically modulated stiffness, the periodic parametric excitation causes the Duffing system to execute dynamics of two fixed-point attractors. These attractors remain non-chaotic even in the presence of random field but can get merged due to induced fluctuation in the trajectory. It is seen that the random field can change the status of the system from transit to stable state. PubDate: 2021-01-09
Abstract: Abstract We study a concrete model of a confined particle in form of a Schrödinger operator with a compactly supported smooth potential coupled to a bosonic field at positive temperature. We show, that the model exhibits thermal ionization for any positive temperature, provided the coupling is sufficiently small. Mathematically, one has to rule out that zero is an eigenvalue of the self-adjoint generator of time evolution—the Liouvillian. This will be done by using positive commutator methods with dilations in the space of scattering functions. Our proof relies on a spatial cutoff in the coupling but does otherwise not require any unnatural restrictions. PubDate: 2021-01-08
Abstract: Abstract In this paper we define infinite weakly hyperbolic iterated function systems associated with uniformly Dini continuous weight functions. We study the Ruelle operator theorem for the infinite weakly hyperbolic iterated function systems associated with uniformly Dini continuous weight functions. We prove the existence and uniqueness of the invariant measure for these type systems. PubDate: 2021-01-07
Abstract: Abstract In this paper, we confirm the conjecture of Wang and Zhang (J Stat Phys 134 (5-6):953–968, 2009) in a long time scale, i.e., the displacement of the wavefront for 1D nonlinear random Schrödinger equation is of logarithmic order in time t . PubDate: 2021-01-07
Abstract: Abstract In this article, we introduce a new approach towards the statistical learning problem \(\mathrm{argmin}_{\rho (\theta ) \in {\mathcal {P}}_{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta ))\) to approximate a target quantum state \(\rho _{\star }\) by a set of parametrized quantum states \(\rho (\theta )\) in a quantum \(L^2\) -Wasserstein metric. We solve this estimation problem by considering Wasserstein natural gradient flows for density operators on finite-dimensional \(C^*\) algebras. For continuous parametric models of density operators, we pull back the quantum Wasserstein metric such that the parameter space becomes a Riemannian manifold with quantum Wasserstein information matrix. Using a quantum analogue of the Benamou–Brenier formula, we derive a natural gradient flow on the parameter space. We also discuss certain continuous-variable quantum states by studying the transport of the associated Wigner probability distributions. PubDate: 2021-01-07
Abstract: Abstract We present rigorous results for the SU(n) Fermi–Hubbard models with finite-range hopping in d ( \(\ge 2\) ) dimensions. The models are defined on a class of decorated lattices. We first study the models with flat bands at the bottom of the single-particle spectrum and prove that the ground states exhibit SU(n) ferromagnetism when the number of particles is equal to the number of unit cells. We then perturb the models by adding particular hopping terms and make the bottom bands dispersive. Under the same filling condition, it is proved that the ground states remain SU(n) ferromagnetic when the bottom bands are sufficiently flat and the Coulomb repulsion is sufficiently large. PubDate: 2021-01-07
Abstract: Abstract We consider polygonal billiards with collisions contracting the reflection angle towards the normal to the boundary of the table. In previous work, we proved that such billiards have a finite number of ergodic SRB measures supported on hyperbolic generalized attractors. Here we study the relation of these measures with the ergodic absolutely continuous invariant probabilities (acips) of the slap map, the 1-dimensional map obtained from the billiard map when the angle of reflection is set equal to zero. We prove that if a convex polygon satisfies a generic condition called (*), and the reflection law has a Lipschitz constant sufficiently small, then there exists a one-to-one correspondence between the ergodic SRB measures of the billiard map and the ergodic acips of the corresponding slap map, and moreover that the number of Bernoulli components of each ergodic SRB measure equals the number of the exact components of the corresponding ergodic acip. The case of billiards in regular polygons and triangles is studied in detail. PubDate: 2021-01-07
Abstract: Abstract We obtain the exact large deviation functions of the density profile and of the current, in the non-equilibrium steady state of a one dimensional symmetric simple exclusion process coupled to boundary reservoirs with slow rates. Compared to earlier results, where rates at the boundaries are comparable to the bulk ones, we show how macroscopic fluctuations are modified when the boundary rates are slower by an order of inverse of the system length. PubDate: 2021-01-07
Abstract: Abstract We consider the \(d\ (\ge 3)\) - dimensional lattice Gaussian free field on \(\varLambda _N :=[-N, N]^d\cap \mathbb {Z}^d\) in the presence of a self-potential of the form \(U(r)= -b I( r \le a)\) , \(a>0, b\in \mathbb {R}\) . When \(b>0\) , the potential attracts the field to the level around zero and is called square-well pinning. It is known that the field turns to be localized and massive for every \(a>0\) and \(b>0\) . In this paper, we consider the situation that the parameter \(b<0\) and self-potentials are imposed on \(\varLambda _{\alpha N},\ \alpha \in (0, 1)\) . We prove that once we impose this weak repulsive potential from the level \([-a, a]\) , most sites are located on the same side and the field is pushed to the same level when the original Gaussian field is conditioned to be positive everywhere, or negative everywhere with probability \(\frac{1}{2}\) , respectively. This result can be applied to show the similar path behavior for the disordered pinning model in the delocalized regime. PubDate: 2021-01-07
Abstract: Abstract We introduce quantum Markov states (QMS) in a general tree graph \(G= (V, E)\) , extending the Cayley tree’s case. We investigate the Markov property w.r.t. the finer structure of the considered tree. The main result of the present paper concerns the diagonalizability of a locally faithful QMS \(\varphi \) on a UHF-algebra \({\mathcal {A}}_V\) over the considered tree by means of a suitable conditional expectation into a maximal abelian subalgebra. Namely, we prove the existence of a Umegaki conditional expectation \({\mathfrak {E}} : {\mathcal {A}}_V \rightarrow {\mathcal {D}}_V\) such that \(\varphi =\varphi _{\lceil {\mathcal {D}}_V}\circ {\mathfrak {E}}\) . Moreover, it is clarified the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra \({\mathcal {D}}_V\) . PubDate: 2021-01-07
Abstract: Abstract We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered in the literature and usually dressed up in other clothing. We give a stripped-down account of the subject in the language of sums of independent random variables in order to free ourselves of the consideration of particular models and highlight the essentials. Besides giving a unified presentation of the theory, this work investigates facets so far unexplored in previous studies. Specifically, we show how the study of the class of random allocation models and zero-range processes can serve as a backdrop for the study of the two other classes of processes central to the present work—tied-down and free renewal processes. We then present new insights on the extreme value statistics of these three classes of processes which allow a deeper understanding of the mechanism of condensation and the quantitative analysis of the fluctuations of the condensate. PubDate: 2021-01-07
Abstract: Abstract We introduce the notion of transmission time to study the dynamics of disordered quantum spin chains and prove results relating its behavior to many-body localization properties. We also study two versions of the so-called Local Integrals of Motion (LIOM) representation of spin chain Hamiltonians and their relation to dynamical many-body localization. We prove that uniform-in-time dynamical localization expressed by a zero-velocity Lieb–Robinson bound implies the existence of a LIOM representation of the dynamics as well as a weak converse of this statement. We also prove that for a class of spin chains satisfying a form of exponential dynamical localization, sparse perturbations result in a dynamics in which transmission times diverge at least as a power law of distance, with a power for which we provide lower bound that diverges with increasing sparseness of the perturbation. PubDate: 2021-01-07
Abstract: Abstract We study the vertex classification problem on a graph whose vertices are in \(k\ (k\ge 2)\) different communities, edges are only allowed between distinct communities, and the number of vertices in different communities are not necessarily equal. The observation is a weighted adjacency matrix, perturbed by a scalar multiple of the Gaussian Orthogonal Ensemble (GOE), or Gaussian Unitary Ensemble (GUE) matrix. For the exact recovery of the maximum likelihood estimation (MLE) with various weighted adjacency matrices, we prove sharp thresholds of the intensity \(\sigma \) of the Gaussian perturbation. Roughly speaking, when \(\sigma \) is below (resp. above) the threshold, exact recovery of MLE occurs with probability tending to 1 (resp. 0) as the size of the graph goes to infinity. These weighted adjacency matrices may be considered as natural models for the electric network. Surprisingly, these thresholds of \(\sigma \) do not depend on whether the sample space for MLE is restricted to such classifications that the number of vertices in each group is equal to the true value. In contrast to the \({{\mathbb {Z}}}_2\) -synchronization, a new complex version of the semi-definite programming (SDP) is designed to efficiently implement the community detection problem when the number of communities k is greater than 2, and a common region (independent of k) for \(\sigma \) such that SDP exactly recovers the true classification is obtained. PubDate: 2021-01-04
Abstract: Abstract This study firstly proposes a simple recursive method for deriving the macroscale equations from lattice Boltzmann equations. Similar to the Maxwell iteration based on the convective scaling, this method is used to expand the lattice Boltzmann (LB) equations with the time step \(\delta _{t}\) . It is characterised by the incorporation of a nonequilibrium distribution function not appearing in the Maxwell iteration to considerably reduce the mathematical manipulations required. Next, we define the kinetic equations of a multicomponent (i.e. N-component) system based on a model using the Maxwell velocity distribution law for the equilibrium distribution function appearing in the cross-collision terms. Then, using this simple recursive method, we derive the generalized Stefan–Maxwell equation, which is the macroscale governing equation of a multicomponent system while ensuring the mass conservation. In short, our objective is to firstly define the kinetic equations of a multi-component system having a clear physical interpretation and then formulate the LB equations of any N-component system deductively. PubDate: 2021-01-02
Abstract: Abstract Several theoretical models based on totally asymmetric simple exclusion process (TASEP) have been extensively utilized to study various non-equilibrium transport phenomena. Inspired by the the role of microtubule-transported vesicles in intracellular transport, we propose a generalized TASEP model, where two distinct particles are directed to hop stochastically in opposite directions on a flexible lattice immersed in a three dimensional pool of diffusing particles. We investigate the interplay between lattice conformation and bidirectional transport by obtaining the stationary phase diagrams and density profiles within the framework of mean field theory. For the case when configuration of flexible lattice is independent of particle density on lattice, the phase diagram only differs quantitatively in comparison to that obtained for bidirectional transport on rigid lattice. However, if the lattice occupancy governs the global conformation of lattice, in addition to the pre-existing phases for bidirectional transport a new asymmetric shock-low density phase originates in the system. We identified that this phase is sensitive to finite size effect and vanishes in the thermodynamic limit. PubDate: 2021-01-02
Abstract: Abstract The distribution function of particles over clusters is proposed for a system of identical intersecting spheres, the centers of which are uniformly distributed in space. Consideration is based on the concept of the rank number of clusters, where the rank is assigned to clusters according to the cluster sizes. The distribution function does not depend on boundary conditions and is valid for infinite medium. The form of the distribution is determined by only one parameter, equal to the ratio of the sphere radius (‘interaction radius’) to the average distance between the centers of the spheres. This parameter plays also a role of the order parameter. It is revealed under what conditions the distribution behaves like well known log-normal distribution. Applications of the proposed distribution to some realistic physical situations, which are close to the conditions of the gas condensation to liquid, are considered. PubDate: 2021-01-02
Abstract: Abstract In this paper we study a multi-species disordered model on the Nishimori line. The typical properties of this line, a set of identities and inequalities among correlation functions, allow us to prove the replica symmetry i.e. the concentration of the order parameter. When the interaction structure is elliptic we rigorously compute the exact solution of the model in terms of a finite-dimensional variational principle and we study its properties. PubDate: 2021-01-02
Abstract: We consider multi-dimensional Schrödinger operators with a weak random perturbation distributed in the cells of some periodic lattice. In every cell the perturbation is described by the translate of a fixed abstract operator depending on a random variable. The random variables, indexed by the lattice, are assumed to be independent and identically distributed according to an absolutely continuous probability density. A small global coupling constant tunes the strength of the perturbation. We treat analogous random Hamiltonians defined on multi-dimensional layers, as well. For such models we determine the location of the almost sure spectrum and its dependence on the global coupling constant. In this paper we concentrate on the case that the spectrum expands when the perturbation is switched on. Furthermore, we derive a Wegner estimate and an initial length scale estimate, which together with Combes–Thomas estimate allow to invoke the multi-scale analysis proof of localization. We specify an energy region, including the bottom of the almost sure spectrum, which exhibits spectral and dynamical localization. Due to our treatment of general, abstract perturbations our results apply at once to many interesting examples both known and new. PubDate: 2021-01-01
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