Large Deviations, Dynamics and Phase Transitions in Large Stochastic and Disordered Neural Networks

Neuronal networks are characterized by highly heterogeneous connectivity, and this disorder was recently related experimentally to qualitative properties of the network. The motivation of this paper is to mathematically analyze the role of these disordered connectivities on the large-scale properties of neuronal networks. To this end, we analyze here large-scale limit behaviors of neural networks including, for biological relevance, multiple populations, random connectivities and interaction delays. Due to the randomness of the connectivity, usual mean-field methods (e.g. coupling) cannot be applied, but, similarly to studies developed for spin glasses, we will show that the sequences of empirical measures satisfy a large deviation principle, and converge towards a self-consistent non-Markovian process. From a mathematical viewpoint, the proof differs from previous works in that we are working in infinite-dimensional spaces (interaction delays) and consider multiple cell types. The limit obtained formally characterizes the macroscopic behavior of the network. We propose a dynamical systems approach in order to address the qualitative nature of the solutions of these very complex equations, and apply this methodology to three instances in order to show how non-centered coefficients, interaction delays and multiple populations networks are affected by disorder levels. We identify a number of phase transitions in such systems upon changes in delays, connectivity patterns and dispersion, and particularly focus on the emergence of non-equilibrium states involving synchronized oscillations.     

Diffraction of Stochastic Point Sets: Explicitly Computable Examples

Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. The latter is based on the classical theory of point processes and the Palm distribution. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the Poisson summation formula for lattice Dirac combs.     

Derivation of Maximum Entropy Principles in Two-Dimensional Turbulence via Large Deviations

The continuum limit of lattice models arising in two-dimensional turbulence is analyzed by means of the theory of large deviations. In particular, the Miller–Robert continuum model of equilibrium states in an ideal fluid and a modification of that model due to Turkington are examined in a unified framework, and the maximum entropy principles that govern these models are rigorously derived by a new method. In this method, a doubly indexed, measure-valued random process is introduced to represent the coarse-grained vorticity field. The natural large deviation principle for this process is established and is then used to derive the equilibrium conditions satisfied by the most probable macrostates in the continuum models. The physical implications of these results are discussed, and some modeling issues of importance to the theory of long-lived, large-scale coherent vortices in turbulent flows are clarified.     

Large Deviations for Random Trees

We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.     

Large Deviations and Scaling Limit

We explore the behavior under scaling limits of large systems using methods from the theory large deviations. This is carried out through the examination of a few examples.     

Adaptive Sampling of Large Deviations

We introduce and test an algorithm that adaptively estimates large deviation functions characterizing the fluctuations of additive functionals of Markov processes in the long-time limit. These functions play an important role for predicting the probability and pathways of rare events in stochastic processes, as well as for understanding the physics of nonequilibrium systems driven in steady states by external forces and reservoirs. The algorithm uses methods from risk-sensitive and feedback control to estimate from a single trajectory a new process, called the driven process, known to be efficient for importance sampling. Its advantages compared to other simulation techniques, such as splitting or cloning, are discussed and illustrated with simple equilibrium and nonequilibrium diffusion models.     

Correction to: Adaptive Sampling of Large Deviations

Journal of Statistical Physics - The original version of this article unfortunately contained an error. The authors would like to correct the error with this erratum.     

Spherical Model on a Cayley Tree: Large Deviations

We study the spherical model of a ferromagnet on a Cayley tree and show that in the case of empty boundary conditions a ferromagnetic phase transition takes place at the critical temperature \(T_\mathrm{c} =\frac{6\sqrt{2}}{5}J\), where J is the interaction strength. For any temperature the equilibrium magnetization, \(m_n\), tends to zero in the thermodynamic limit, and the true order parameter is the renormalized magnetization \(r_n=n^{3/2}m_n\), where n is the number of generations in the Cayley tree. Below \(T_\mathrm{c}\), the equilibrium values of the order parameter are given by \(\pm \rho ^*\), where

$$\begin{aligned} \rho ^*=\frac{2\pi }{(\sqrt{2}-1)^2}\sqrt{1-\frac{T}{T_\mathrm{c}}}. \end{aligned}$$

One more notable temperature in the model is the penetration temperature

$$\begin{aligned} T_\mathrm{p}=\frac{J}{W_\mathrm{Cayley}(3/2)}\left( 1-\frac{1}{\sqrt{2}}\left( \frac{h}{2J}\right) ^2\right) . \end{aligned}$$

Below \(T_\mathrm{p}\) the influence of homogeneous boundary field of magnitude h penetrates throughout the tree. The main new technical result of the paper is a complete set of orthonormal eigenvectors for the discrete Laplace operator on a Cayley tree.

     

Large Deviations in Quantum Lattice Systems: One-Phase Region

We give large deviation upper bounds, and discuss lower bounds, for the Gibbs-KMS state of a system of quantum spins or an interacting Fermi gas on the lattice. We cover general interactions and general observables, both in the high temperature regime and in dimension one.     

Large Deviations for Quantum Spin Systems

We consider high temperature KMS states for quantum spin systems on a lattice. We prove a large deviation principle for the distribution of empirical averages , where the X _i 's are copies of a self-adjoint element X (level one large deviations). From the analyticity of the generating function, we obtain the central limit theorem. We generalize to a level two large deviation principle for the distribution of      

Large Deviations for Non-Uniformly Expanding Maps

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. 2000 Mathematics Subject Classification: 37D25, 37A50, 37B40, 37C40     

Large Deviations for Probabilistic Cellular Automata

We consider a generalized model of a probabilistic cellular automata described by a Markov chain on an infinite dimensional space and derive certain large deviations bounds for corresponding occupational measures.     

Large Deviations in Rarefied Quantum Gases

The probability of observing a large deviation (LD) in the number of particles in a region in a dilute quantum gas contained in a much larger region V is shown to decay as exp[–||F], where || is the volume of and F is the change in the appropriate free energy density, the same as in classical systems. However, in contrast with the classical case, where this formula holds at all temperatures and chemical potentials our proof is restricted to rarefied gases, both for the typical and observed density, at least for Bose or Fermi systems. The case of Boltzmann statistics with a bounded repulsive potential can be treated at all temperatures and densities. Fermions on a lattice in any dimension, or in the continuum in one dimension, can be treated at all densities and temperatures if the interaction is small enough (depending on density and temperature), provided one assumes periodic boundary conditions.     

Large Deviations in Quantum Spin Chains

We show the full large deviation principle for KMS-states and C*-finitely correlated states on a quantum spin chain. We cover general local observables. Our main tool is Ruelle’s transfer operator method.     

The Yang-Yang thermodynamic formalism and Large Deviations

The partition function for a one-dimensional system of Bosons with repulsive delta-function interaction is investigated. We prove that if the Bethe Ansatz eigenfunctions form a complete set then the grand canonical pressure is given by the Yang-Yang formula. The proof uses a probabilistic formalism to express the partition function as an expectation with respect to a probability measure on a Banach space of measures; the asymptotic behaviour of the expectation in the thermodynamic limit is determined by the Large Deviation Principle. This method is applicable in situations in which the Hamiltonian can be diagonalised using the Bethe Ansatz.     

Large Deviations for Probabilistic Cellular Automata II

We obtain an upper large deviations bound which shows that for some models of probabilistic cellular automata (which are far away from the product case) the lower large deviation bound derived in Eizenberg and Kifer J. Stat. Phys. 108: 1255–1280 (2002) is sharp, and so the corresponding large deviations phenomena cannot be described via the traditional Donsker–Varadhan form of the action functional. For models which are close to the product case we derive approximate large deviations bounds using the Donsker–Varadhan functional for the product case.     

Large Deviations,Fluctuations and Shrinking Intervals

This paper concerns the statistical properties of hyperbolic diffeomorphisms. We obtain a large deviation result with respect to slowly shrinking intervals for a large class of Hölder continuous functions. In case of time reversal symmetry, we obtain a corresponding version of the Fluctuation Theorem.     

Large Deviations for Gaussian Diffusions with Delay

Dynamical systems driven by nonlinear delay SDEs with small noise can exhibit important rare events on long timescales. When there is no delay, classical large deviations theory quantifies rare events such as escapes from metastable fixed points. Near such fixed points, one can approximate nonlinear delay SDEs by linear delay SDEs. Here, we develop a fully explicit large deviations framework for (necessarily Gaussian) processes \(X_t\) driven by linear delay SDEs with small diffusion coefficients. Our approach enables fast numerical computation of the action functional controlling rare events for \(X_t\) and of the most likely paths transiting from \(X_0 = p\) to \(X_T=q\). Via linear noise local approximations, we can then compute most likely routes of escape from metastable states for nonlinear delay SDEs. We apply our methodology to the detailed dynamics of a genetic regulatory circuit, namely the co-repressive toggle switch, which may be described by a nonlinear chemical Langevin SDE with delay.