An effective medium approach to the asymptotics of the statistical moments of the parabolic Anderson model and Lifshitz tails

Originally introduced in solid state physics to model amorphous materials and alloys exhibiting disorder induced metal-insulator transitions, the Anderson model $H_{\omega}= -\Delta + V_{\omega}$ on l^2(\bZ^d) has become in mathematical physics as well as in probability theory a paradigmatic example for the relevance of disorder effects. Here $\Delta$ is the discrete Laplacian and V_{\omega} = \{V_{\omega}(x): x \in \bZ^d\} is an i.i.d. random field taking values in \bR. A popular model in probability theory is the parabolic Anderson model (PAM), i.e. the discrete diffusion equation $\partial_t u(x,t) =-H_{\omega} u(x,t)$ on \bZ^d \times \bR_+, $u(x,0)=1$, where random sources and sinks are modelled by the Anderson Hamiltonian. A characteristic property of the solutions of (PAM) is the occurrence of intermittency peaks in the large time limit. These intermittency peaks determine the thermodynamic observables extensively studied in the probabilistic literature using path integral methods and the theory of large deviations. The rigorous study of the relation between the probabilistic approach to the parabolic Anderson model and the spectral theory of Anderson localization is at least mathematically less developed. We see our publication as a step in this direction. In particular we will prove an unified approach to the transition of the statistical moments $$ and the integrated density of states from classical to quantum regime using an effective medium approach. As a by-product we will obtain a logarithmic correction in the traditional Lifshitz tail setting when $V_{\omega}$ satisfies a fat tail condition

Random Block Operators

We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random couplings.Comment: 22 pages, 3 figure

Large deviations for cluster size distributions in a continuous classical many-body system

An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $\beta\in (0,\infty)$ and particle density $\rho\in(0,\rho_{\mathrm{cp}})$ in the thermodynamic limit. Here $\rho_{\mathrm{cp}}>0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $\Gamma$-convergence of the rate function toward an explicit limiting rate function in the low-temperature dilute limit $\beta\to \infty$, $\rho\downarrow0$ such that $-\beta^{-1}\log\rho\to\nu$ for some $\nu\in(0,\infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the decoupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $\nu$. Under additional assumptions on the potential, the $\Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1014 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Gross-Pitaevskii functional with a random background potential and condensation in the single particle ground state

This version corrects a mistake in the proof of Lemma 5.For discrete and continuous Gross-Pitaevskii energy functionals with a random background potential, we study the Gross-Pitaevskii ground state. We characterize a regime of interaction coupling when the Gross-Pitaevskii ground state and the ground state of the random background Hamiltonian asymptotically coincide

Random block operators

We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random coupling

Large deviations for cluster size distributions in a continuous classical many-body system

An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $\beta\in(0,\infty)$ and particle density $\rho\in(0,\rho_{\rm{cp}})$ in the thermodynamic limit. Here $\rho_{\rm{cp}} >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $\Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $\beta\to\infty$, $\rho \downarrow 0$ such that $-\beta^{-1}\log\rho\to \nu$ for some $\nu\in(0,\infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $\nu$. Under additional assumptions on the potential, the Γ-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle

Second-Order Hyperproperties

We introduce Hyper$^2$LTL, a temporal logic for the specification of hyperproperties that allows for second-order quantification over sets of traces. Unlike first-order temporal logics for hyperproperties, such as HyperLTL, Hyper$^2$LTL can express complex epistemic properties like common knowledge, Mazurkiewicz trace theory, and asynchronous hyperproperties. The model checking problem of Hyper$^2$LTL is, in general, undecidable. For the expressive fragment where second-order quantification is restricted to smallest and largest sets, we present an approximate model-checking algorithm that computes increasingly precise under- and overapproximations of the quantified sets, based on fixpoint iteration and automata learning. We report on encouraging experimental results with our model-checking algorithm, which we implemented in the tool~\texttt{HySO}

Live synthesis

Synthesis automatically constructs an implementation that satisfies a given logical specification. In this paper, we study the live synthesis problem, where the synthesized implementation replaces an already running system. In addition to satisfying its own specification, the synthesized implementation must guarantee a sound transition from the previous implementation. This version of the synthesis problem is highly relevant in “always-on” applications, where updates happen while the system is running. To specify the correct handover between the old and new implementation, we introduce an extension of linear-time temporal logic (LTL) called LiveLTL. A LiveLTL specification defines separate requirements on the two implementations and ensures that the new implementation satisfies, in addition to its own requirements, any obligations left unfinished by the old implementation. For specifications in LiveLTL, we show that the live synthesis problem can be solved within the same complexity bound as standard reactive synthesis, i.e., in 2EXPTIME. Our experiments show the necessity of live synthesis for LiveLTL specifications created from benchmarks of SYNTCOMP and robot control

Information Flow Guided Synthesis

Compositional synthesis relies on the discovery of assumptions, i.e., restrictions on the behavior of the remainder of the system that allow a component to realize its specification. In order to avoid losing valid solutions, these assumptions should be necessary conditions for realizability. However, because there are typically many different behaviors that realize the same specification, necessary behavioral restrictions often do not exist. In this paper, we introduce a new class of assumptions for compositional synthesis, which we call information flow assumptions. Such assumptions capture an essential aspect of distributed computing, because components often need to act upon information that is available only in other components. The presence of a certain flow of information is therefore often a necessary requirement, while the actual behavior that establishes the information flow is unconstrained. In contrast to behavioral assumptions, which are properties of individual computation traces, information flow assumptions are hyperproperties, i.e., properties of sets of traces. We present a method for the automatic derivation of information-flow assumptions from a temporal logic specification of the system. We then provide a technique for the automatic synthesis of component implementations based on information flow assumptions. This provides a new compositional approach to the synthesis of distributed systems. We report on encouraging first experiments with the approach, carried out with the BoSyHyper synthesis tool