Overlap Synchronisation in Multipartite Random Energy Models

In a multipartite random energy model, made of a number of coupled generalised random energy models (GREMs), we determine the joint law of the overlaps in terms of the ones of the single GREMs. This provides the simplest example of the so-called overlap synchronisation.     

Large n Limit of Gaussian Random Matrices with External Source, Part I

We consider the random matrix ensemble with an external sourcedefined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a>1, we establish the universal behavior of local eigenvalue correlations in the limit n, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3×3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.Dedicated to Freeman Dyson on his eightieth birthdayThe first author was supported in part by NSF Grants DMS-9970625 and DMS-0354962.The second author was supported in part by projects G.0176.02 and G.0455.04 of FWO-Flanders, by K.U.Leuven research grant OT/04/24, and by INTAS Research Network NeCCA 03-51-6637.     

Central Limit Theorem for the Free Energy of the Random Field Ising Model

A central limit theorem is proved for the free energy of the random field Ising model with all plus or all minus boundary condition, at any temperature (including zero temperature) and any dimension. This solves a problem posed by Wehr and Aizenman (J Stat Phys 60:287–306, 1990). The proof uses a variant of Stein’s method.

     

On Epistemics in Expected Free Energy for Linear Gaussian State Space Models

Active Inference (AIF) is a framework that can be used both to describe information processing in naturally intelligent systems, such as the human brain, and to design synthetic intelligent systems (agents). In this paper we show that Expected Free Energy (EFE) minimisation, a core feature of the framework, does not lead to purposeful explorative behaviour in linear Gaussian dynamical systems. We provide a simple proof that, due to the specific construction used for the EFE, the terms responsible for the exploratory (epistemic) drive become constant in the case of linear Gaussian systems. This renders AIF equivalent to KL control. From a theoretical point of view this is an interesting result since it is generally assumed that EFE minimisation will always introduce an exploratory drive in AIF agents. While the full EFE objective does not lead to exploration in linear Gaussian dynamical systems, the principles of its construction can still be used to design objectives that include an epistemic drive. We provide an in-depth analysis of the mechanics behind the epistemic drive of AIF agents and show how to design objectives for linear Gaussian dynamical systems that do include an epistemic drive. Concretely, we show that focusing solely on epistemics and dispensing with goal-directed terms leads to a form of maximum entropy exploration that is heavily dependent on the type of control signals driving the system. Additive controls do not permit such exploration. From a practical point of view this is an important result since linear Gaussian dynamical systems with additive controls are an extensively used model class, encompassing for instance Linear Quadratic Gaussian controllers. On the other hand, linear Gaussian dynamical systems driven by multiplicative controls such as switching transition matrices do permit an exploratory drive.     

Exact Asymptotics of the Freezing Transition of a Logarithmically Correlated Random Energy Model

We consider a logarithmically correlated random energy model, namely a model for directed polymers on a Cayley tree, which was introduced by Derrida and Spohn. We prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the KPP-equation—thus translating Bramson’s work on the KPP-equation into a discrete time case. We also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point.     

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.     

Interfaces for Random Cluster Models

A random cluster measure on that is not translationally invariant is constructed for d3, the critical density p _c , and sufficiently large q. The resulting measure is proven to be a Gibbs state satisfying cluster model DLR- equations.     

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

We consider unitary random matrix ensembles on the space of Hermitian n × n matrices M, where the confining potential V _s,t is such that the limiting mean density of eigenvalues (as n→∞ and s,t→ 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P _I ² equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P _I ² equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights on . The special solution of the P _I ² equation pops up in the n ^−2/7-term of the asymptotics.     

Gaussian Fields and Random Packing

Consider sequential packing of unit volume balls in a large cube, in any dimension and with Poisson input. We show after suitable rescaling that the spatial distribution of packed balls tends to that of a Gaussian field in the thermodynamic limit. The results cover related applied models, including ballistic deposition and spatial birth-growth models.     

Thermodynamical Aspects of Modified Holographic Dark Energy Model

We investigate the unified first law and the generalized second law in a modified holographic dark energy model. The thermodynamical analysis on the apparent horizon can work and the corresponding entropy formula is extracted from the systematic algorithm. The entropy correction term depends on the extra-dimension number of the brane as expected, but the interplay between the correction term and the extra dimensions is more complicated. With the unified first law of thermodynamics well-founded, the generalized second law of thermodynamics is discussed and it is found that the second law can be violated in certain circumstances. Particularly, if the number of the extra dimensions is larger than one, the generalized law of thermodynamics is always satisfied; otherwise, the validity of the second law can only be guaranteed with the Hubble radius greatly smaller than the crossover scale r_c of the 5-dimensional DGP model.     

Thermodynamical Aspects of Modified Holographic Dark Energy Model

We investigate the unified first law and the generalized second law in a modified holographic dark energy model. The thermodynamical analysis on the apparent horizon can work and the corresponding entropy formula is extracted from the systematic algorithm. The entropy correction term depends on the extra-dimension number of the brane as expected, but the interplay between the correction term and the extra dimensions is more complicated. With the unified first law of thermodynamics well-founded, the generalized second law of thermodynamics is discussed and it is found that the second law can be violated in certain circumstances. Particularly, if the number of the extra dimensions is larger than one, the generalized law of thermodynamics is always satisfied; otherwise, the validity of the second law can only be guaranteed with the Hubble radius greatly smaller than the crossover scale rcof the 5-dimensional DGP model.     

Quenched Limit Theorems for Nearest Neighbour Random Walks in 1D Random Environment

It is well known that random walks in a one dimensional random environment can exhibit subdiffusive behavior due to the presence of traps. In this paper we show that the passage times of different traps are asymptotically independent exponential random variables with parameters forming, asymptotically, a Poisson process. This allows us to prove weak quenched limit theorems in the subdiffusive regime where the contribution of traps plays the dominating role.     

Transmission of Correlated Gaussian Packets Through a Delta-Potential

We study the evolution of the most general initial Gaussian packet with nonzero correlation coefficient between the coordinate and momentum operators in the presence of a repulsive delta-potential barrier, using the known exact propagator of the time-dependent Schrödinger equation. For the initial packet localized far enough from the barrier, we define the transmission coefficient as the probability of discovering the particle in the whole semi-axis on the other side of the barrier. It appears that the asymptotical transmission coefficient (calculated in the large time limit) depends on two dimensionless parameters: the normalized ratio of the potential strength to the initial mean value of momentum and the ratio of the initial momentum dispersion to the initial mean value of momentum. For small values of the second parameter, the result is reduced to the well-known formula for the transparency of the delta barrier, obtained in the plane-wave approximation by solving the stationary Schrödinger equation. For large values of the second parameter, the transmission coefficient can be much larger than that calculated in the plane-wave approximation. For a fixed initial spread of the packet in the coordinate space, the initial correlation coefficient influences the transparency of the barrier only indirectly, through the increase in the initial momentum dispersion.     

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

Universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace or the Hilbert-Schmidt ensemble. These universal limits have been proved before for determinantal Hermitian matrix ensembles and for some special classes of the Wigner random matrices. Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik”. Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik,” and grants RFBR-05-01-00911, DFG-RFBR-04-01-04000, and NS-638.2008.1.     

Gaussian Limits for Multidimensional Random Sequential Packing at Saturation

Consider the random sequential packing model with infinite input and in any dimension. When the input consists of non-zero volume convex solids we show that the total number of solids accepted over cubes of volume λ is asymptotically normal as λ → ∞. We provide a rate of approximation to the normal and show that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized Gaussian field. The method of proof involves showing that the collection of accepted solids satisfies the weak spatial dependence condition known as stabilization. Research partially supported by the Polish Minister of Scientific Research and Information Technology grant 1 P03A 018 28 (2005-2007) Research supported in part by NSF grant DMS-0203720     

Lyapunov Exponents for Products of Complex Gaussian Random Matrices

The exact value of the Lyapunov exponents for the random matrix product P _N =A _N A _N?1?A ₁ with each $A_{i} = \varSigma^{1/2} G_{i}^{\mathrm{c}}$ , where Σ is a fixed d×d positive definite matrix and $G_{i}^{\mathrm{c}}$ a d×d complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.