arXiv subjects
Publications and source records attributed to Michael Aizenman.
We consider radial tree extensions of one-dimensional quasi-periodic Schroedinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The sufficiency criterion for that is the existence of Bloch-Floquet states for the one dimensional operator corresponding to the radial problem.
We present general results for the contact process by a method which applies to all transitive graphs of bounded degree, including graphs of exponential growth. The model's infection rates are varied through a control parameter, for which two natural transition points are defined as: i. $λ_T$, the value up to which the infection dies out exponentially fast if introduced at a single site, and ii. $λ_H$, the threshold for the existence of an invariant measure with a non-vanishing density of infected sites. It is shown here that for all transitive graphs the two thresholds coincide. The method, which proceeds through partial differential inequalities for the infection density, yields also generally valid bounds on two related critical exponents. The main results discussed here were established by Bezuidenhout and Grimmett \cite{BG2} in an extension to the continuous-time process of the discrete-time analysis of Aizenman and Barsky \cite{AB}, and of the partially similar results of Menshikov \cite{M}. The main novelty here is in the direct derivation of the partial differential inequalities by an argument which is formulated for the continuum.
We study localization effects of disorder on the spectral and dynamical properties of Schroedinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.
We summarize recent works on the stability under disorder of the absolutely continuous spectra of random operators on tree graphs. The cases covered include: Schrödinger operators with random potential, quantum graph operators for trees with randomized edge lengths, and radial quasi-periodic operators perturbed by random potentials.
A recent result presented the expansion for the entropy rate of a Hidden Markov Process (HMP) as a power series in the noise variable $\eps$. The coefficients of the expansion around the noiseless ($\eps = 0$) limit were calculated up to 11th order, using a conjecture that relates the entropy rate of a HMP to the entropy of a process of finite length (which is calculated analytically). In this communication we generalize and prove the validity of the conjecture, and discuss the theoretical and practical consequences of our new theorem.
Finding the entropy rate of Hidden Markov Processes is an active research topic, of both theoretical and practical importance. A recently used approach is studying the asymptotic behavior of the entropy rate in various regimes. In this paper we generalize and prove a previous conjecture relating the entropy rate to entropies of finite systems. Building on our new theorems, we establish series expansions for the entropy rate in two different regimes. We also study the radius of convergence of the two series expansions.
The subject of this work are random Schroedinger operators on regular rooted tree graphs $\T$ with stochastically homogeneous disorder. The operators are of the form $H_λ(ω) = T + U + λV(ω)$ acting in $\ell^2(\T)$, with $T $ the adjacency matrix, $U$ a radially periodic potential, and $V(ω)$ a random potential. This includes the only class of homogeneously random operators for which it was proven that the spectrum of $H_λ(ω)$ exhibits an absolutely continuous (ac) component; a results established by A. Klein for weak disorder, in case U=0 and $V(ω)$ given by iid random variables on $\T$. Our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder. The method yields the continuity in the disorder parameter of the ac spectral density of $H_λ(ω)$ at $λ= 0$. The latter is shown to converge in the $L^1$ sense over closed intervals in which $H_0$ has no singular spectrum. The analysis extends to random potentials whose values at different sites need not be independent, assuming only that their joint distribution is weakly correlated across different tree branches.
We consider the Laplacian on a rooted metric tree graph with branching number $ K \geq 2 $ and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the Weyl-Titchmarsh function to trees. The proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder.
We study systems of particles on a line which have a maximum, are locally finite and evolve with independent increments. ``Quasi-stationary states'' are defined as probability measures, on the σ-algebra generated by the gap variables, for which joint distribution of gaps between particles is invariant under the time evolution. Examples are provided by Poisson processes with densities of the form ρ(dx)=e^{-sx}s dx, with s>0, and linear superpositions of such measures. We show that, conversely, any quasi-stationary state for the independent dynamics, with an exponentially bounded integrated density of particles, corresponds to a superposition of Poisson processes with densities ρ(dx)=e^{-sx}s dx with s>0, restricted to the relevant σ-algebra. Among the systems for which this question is of some relevance are spin-glass models of statistical mechanics, where the point process represents the collection of the free energies of distinct ``pure states,'' the time evolution corresponds to the addition of a spin variable and the Poisson measures described above correspond to the so-called REM states.
Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice. We present a model of this phenomenon which we are able to analyze rigorously. The system is a hard core lattice gas at half-filling and the optical lattice is modeled by a periodic potential of strength $λ$. For small $λ$ and temperature, BEC is proved to occur, while at large $λ$ or temperature there is no BEC. At large $λ$ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase. The interparticle interaction is essential for this transition, which occurs even in the ground state. Surprisingly, the condensation is always into the $p=0$ mode in this model, although the density itself has the periodicity of the imposed potential.
The fractional moment method, which was initially developed in the discrete context for the analysis of the localization properties of lattice random operators, is extended to apply to random Schrödinger operators in the continuum. One of the new results for continuum operators are exponentially decaying bounds for the mean value of transition amplitudes, for energies throughout the localization regime. An obstacle which up to now prevented an extension of this method to the continuum is the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. This difficulty is resolved through an analysis of the resonance-diffusing effects of the disorder.
The recent proof by F. Guerra that the Parisi ansatz provides a lower bound on the free energy of the SK spin-glass model could have been taken as offering some support to the validity of the purported solution. In this work we present a broader variational principle, in which the lower bound, as well as the actual value, are obtained through an optimization procedure for which ultrametic/hierarchal structures form only a subset of the variational class. The validity of Parisi's ansatz for the SK model is still in question. The new variational principle may be of help in critical review of the issue.
This paper has been withdrawn by the authors, due to an error in the proof of Lemma 3.1.
We present a mechanism for the creation of gaps in the spectra of self-adjoint operators defined over a Hilbert space of functions on a graph, which is based on the process of graph decoration. The resulting Hamiltonians can be viewed as associated with discrete models exhibiting a repeated local structure and a certain bottleneck in the hopping amplitudes.
A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient `Lifshitz tail estimates' on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the ``multiscale analysis''. In the converse direction, the analysis rules out fast power-law decay of the Green functions at mobility edges.
2D Percolation path exponents $x^{\cal P}_{\ell}$ describe probabilities for traversals of annuli by $\ell$ non-overlapping paths, each on either occupied or vacant clusters, with at least one of each type. We relate the probabilities rigorously to amplitudes of $O(N=1)$ models whose exponents, believed to be exact, yield $x^{\cal P}_{\ell}=({\ell}^2-1)/12$. This extends to half-integers the Saleur--Duplantier exponents for $k=\ell/2$ clusters, yields the exact fractal dimension of the external cluster perimeter, $D_{EP}=2-x^{\cal P}_3=4/3$, and also explains the absence of narrow gate fjords, as originally found by Grossman and Aharony.
A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in $\R^d$. Tightness of the distribution, as $δ\to 0$, is established for the following two-dimensional examples: the uniformly random spanning tree on $δ\Z^2$, the minimal spanning tree on $δ\Z^2$ (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in $\R^2$ with density $δ^{-2}$. In each case, sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for $δ> 0$), ii) the tree branches are given by curves which are regular in the sense of Hölder continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of $\R^2$, of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in $\R^2$, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scale-invariant power bounds on the probabilities of repeated crossings of annuli.
Random systems of curves exhibiting fluctuating features on arbitrarily small scales ($δ$) are often encountered in critical models. For such systems it is shown that scale-invariant bounds on the probabilities of crossing events imply that typically all the realized curves admit Hölder continuous parametrizations with a common exponent and a common random prefactor, which in the scaling limit ($δ\to 0$) remains stochastically bounded. The regularity is used for the construction of scaling limits, formulated in terms of probability measures on the space of closed sets of curves. Under the hypotheses presented here the limiting measures are supported on sets of curves which are Hölder continuous but not rectifiable, and have Hausdorff dimensions strictly greater than one. The hypotheses are known to be satisfied in certain two dimensional percolation models. Other potential applications are also mentioned.