Large Deviation Properties Research Articles

The stochastic behavior of a molecular switching circuit with feedback

BackgroundUsing a statistical physics approach, we study the stochastic switching behavior of a model circuit of multisite phosphorylation and dephosphorylation with feedback. The circuit consists of a kinase and phosphatase acting on multiple sites of a substrate that, contingent on its modification state, catalyzes its own phosphorylation and, in a symmetric scenario, dephosphorylation. The symmetric case is viewed as a cartoon of conflicting feedback that could result from antagonistic pathways impinging on the state of a shared component.ResultsMultisite phosphorylation is sufficient for bistable behavior under feedback even when catalysis is linear in substrate concentration, which is the case we consider. We compute the phase diagram, fluctuation spectrum and large-deviation properties related to switch memory within a statistical mechanics framework. Bistability occurs as either a first-order or second-order non-equilibrium phase transition, depending on the network symmetries and the ratio of phosphatase to kinase numbers. In the second-order case, the circuit never leaves the bistable regime upon increasing the number of substrate molecules at constant kinase to phosphatase ratio.ConclusionThe number of substrate molecules is a key parameter controlling both the onset of the bistable regime, fluctuation intensity, and the residence time in a switched state. The relevance of the concept of memory depends on the degree of switch symmetry, as memory presupposes information to be remembered, which is highest for equal residence times in the switched states.ReviewersThis article was reviewed by Artem Novozhilov (nominated by Eugene Koonin), Sergei Maslov, and Ned Wingreen.

The Lempel-Ziv Complexity of Non-Stationary Chaos in Infinite Ergodic Cases

The large deviation properties of the Lempel-Ziv complexity are studied using a onedimensional non-hyperbolic chaos map called the “modified Bernoulli map”, where the transition between stationary and non-stationary chaos is clearly observed. The upper limit of the Lempel-Ziv complexity in the non-stationary regime is theoretically evaluated, and the relationship between the algorithmic complexity and the Lempel-Ziv complexity is discussed. Non-stationary processes are universal phenomena in non-hyperbolic systems, and they are usually characterized by an infinite ergodic measure and intrinsic long time tails, such as 1/f ν spectral fluctuations. It is shown that the Lempel-Ziv complexity obeys universal scaling laws and that the Lempel-Ziv complexity has the L 1 -function property, which guarantees the Darling-Kac-Aaronson theorem for an infinite ergodic system. The most striking result is that the maximum diversity appears at the transition point from stationary chaos to non-stationary chaos where the exact 1/f spectral process is generated.

A Simple Discrete Model of Brownian Motors: Time-periodic Markov Chains

In this paper, we consider periodically inhomogeneous Markov chains, which can be regarded as a simple version of physical model—Brownian motors. We introduce for them the concepts of periodical reversibility, detailed balance, entropy production rate and circulation distribution. We prove the equivalence of the following statements: The time-periodic Markov chain is periodically reversible; It is in detailed balance; Kolmogorov's cycle condition is satisfied; Its entropy production rate vanishes; Every circuit and its reversed circuit have the same circulation weight. Hence, in our model of Markov chains, the directed transport phenomenon of Brownian motors, i.e. the existence of net circulation, can occur only in nonequilibrium and irreversible systems. Moreover, we verify the large deviation property and the Gallavotti-Cohen fluctuation theorem of sample entropy production rates of the Markov chain.

Large deviation and the tangent cone at infinity of a crystal lattice

We discuss a large deviation property of a periodic random walk on a crystal lattice in view of geometry, and relate it to a rational convex polyhedron in the first homology group of a finite graph, which, as we shall observe, has remarkable combinatorial features, and shows up also in the Gromov-Hausdorff limit of a crystal lattice.

The Lempel-Ziv Complexity of 1/f Spectral Chaos and the Infinite Ergodic Theory

A new large deviation property for the Lempel-Ziv complexity is numerically studied by using a one-dimesional non-hyperbolic ‘‘modified Bernoulli map’’, where the transition between stationary and non-stationary chaos is clearly observed. We will show that the Lempel-Ziv complexity and its fluctuations obey the universal scaling laws, and that the Lempel-Ziv complexity has the L1-function property of the infinite ergodic theory. One of the most striking results is that the 1/f spectral process reveals the maximum diversity at the transition point from the stationary chaos to the non-stationary one.

Large deviation approach to non equilibrium processes in stochastic lattice gases

We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.

Directed Network as a Chaotic Piece-Wise Linear One-Dimensional Map and Its Large-Deviation Properties

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius-Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large-deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure. Recurrence time statistics and their relationships to double time correlation functions and to power spectra are also considered.

On the Asymptotic Properties of a Nonparametric<tex>$L_1$</tex>-Test Statistic of Homogeneity

We present two simple and explicit procedures for testing homogeneity of two independent multivariate samples of size n. The nonparametric tests are based on the statistic T/sub n/, which is the L/sub 1/ distance between the two empirical distributions restricted to a finite partition. Both tests reject the null hypothesis of homogeneity if T/sub n/ becomes large, i.e., if T/sub n/ exceeds a threshold. We first discuss Chernoff-type large deviation properties of T/sub n/. This results in a distribution-free strong consistent test of homogeneity. Then the asymptotic null distribution of the test statistic is obtained, leading to an asymptotically /spl alpha/-level test procedure.

Large Deviations for Countable to One Markov Systems

In this paper, we study large deviation properties for countable to one Markov systems associated to weak Gibbs measures for non-Holder potentials. Furthermore, we establish multifractal large deviation laws for countable to one piecewise conformal Markov systems, which are derived systems constructed over hyperbolic regions for certain nonhyperbolic systems exhibiting intermittency. We apply our results to higher-dimensional number theoretical transformations.

Uniqueness for a weak nonlinear evolution equation and large deviations for diffusing particles with electrostatic repulsion

We use hydrodynamics techniques to study the large deviations properties of the McKean–Vlasov model with singular interactions introduced by Cépa and Lépingle (Probab. Theory Related Fields 107 (1997) 429). In a general framework, we prove upper bounds and exponential tightness, and study the action functional. The study of lower bounds is much harder and requires a uniqueness result for a class of nonlinear evolution equations. In the case of interacting Ornstein–Uhlenbeck particles, we prove a general uniqueness statement by extending techniques of Cabannal-Duvillard and Guionnet (Ann. Probab. 29 (2001) 1205). Using this result we deduce some lower bounds for interacting particles with constant diffusion coefficient and general drift terms.

On Large Deviation Properties of Erd�s?R�nyi Random Graphs

We show that large deviation properties of Erd\"os-R\'enyi random graphs can be derived from the free energy of the $q$-state Potts model of statistical mechanics. More precisely the Legendre transform of the Potts free energy with respect to $\ln q$ is related to the component generating function of the graph ensemble. This generalizes the well-known mapping between typical properties of random graphs and the $q\to 1$ limit of the Potts free energy. For exponentially rare graphs we explicitly calculate the number of components, the size of the giant component, the degree distributions inside and outside the giant component, and the distribution of small component sizes. We also perform numerical simulations which are in very good agreement with our analytical work. Finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges, without recourse to the thermodynamics of the Potts model.

Large Deviations for the Emprirical Measures of Reflecting Brownian Motion and Related Constrained Processes in $R_+$

We consider the large deviations properties of the empirical measure for one dimensional constrained processes, such as reflecting Brownian motion, the M/M/1 queue, and discrete time analogues. Because these processes do not satisfy the strong stability assumptions that are usually assumed when studying the empirical measure, there is significant probability (from the perspective of large deviations) that the empirical measure charges the point at infinity. We prove the large deviation principle and identify the rate function for the empirical measure for these processes. No assumption of any kind is made with regard to the stability of the underlying process.

Singularities in the fluctuation of on-off intermittency

For a two-dimensional piecewise linear map exhibiting on-off intermittency, the scaling property of fluctuation, i.e., the large deviation property is investigated. It is shown that there are three phases of fluctuation and the q-weighted average of an observed quantity has singularities such as jumps or a plateau due to transitions between the phases. At the onset of on-off intermittency, the width of the plateau vanishes due to the disappearance of one of the three phases and the singularity becomes weaker but more probable. The singularity at the onset of on-off intermittency is also examined on the coupled logistic map.

Integration of large-deviation kernels and applications to large deviations for evolutionary games

We investigate large-deviation properties for systems of probability kernels and these kernels when integrated with respect to a system of measures that also satisfy the large-deviation principle. The results of our analysis are then used to demonstrate the large-deviation property and derive rate functions for systems of measures derived from the distributions of evolutionary game processes.

ON THE LARGE DEVIATION PROPERTY OF RANDOM MEASURES ON THE d-DIMENSIONAL EUCLIDEAN SPACE

We give a formulation of the large deviation property for rescalings of random measures on the d-dimensional Euclidean space R . The approach is global in the sense that the objects are Radon measures on R and the dual objects are the continuous functions with compact support. This is applied to the cluster random measures with Poisson centers, a large class of random measures that includes the Poisson processes.

Large Deviations Upper Bounds for the Laws of Matrix-Valued Processes and Non-Communicative Entropies

Using Itô’s calculus, we study the large deviations properties of the law of the spectral measure of the Hermitian Brownian motion.We extend this strategy to the symmetric, unitary and Wishart processes. This dynamical approach is generalized to the study of the large deviations of the non-commutative laws of several independent Hermitian Brownian motions. As a consequence, we can bound from above entropies defined in the spirit of the microstates entropy introduced by Voiculescu.

On Large Deviations in the Gaussian Autoregressive Process: Stable, Unstable and Explosive Cases

For the Gaussian autoregressive process, the asymptotic behaviour of the Yule‐Walker estimator is totally different in the stable, unstable and explosive cases. We show that, irrespective of this trichotomy, this estimator shares quite similar large deviation properties in the three situations. However, in the explosive case, we obtain an unusual rate function with a discontinuity point at its minimum.

LARGE DEVIATIONS AND THE RANDOM ENERGY MODEL

We present a simple proof of the formula for the free energy of the random energy model using a large deviation property which holds almost surely with respect to the randomness. This proof is extended to the case with external magnetic field leading to the solution of a model with higher-order ferromagnetic term. It is shown that this model is useful for Sourlas' application to error-correcting codes as was already pointed out in a recent letter by the authors.

Complete Equivalence of the Gibbs Ensembles for One-Dimensional Markov Systems

We show the equivalence of the Gibbs ensembles at the level of measures for one-dimensional Markov-Systems with arbitrary boundary conditions. That is, the limit of the microcanonical Gibbs ensemble is a Gibbs measure with an interaction depending on the microcanonical constraint. In fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins (including the boundary spins) are fixed. When conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble. We show that the limit is a Gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures. For this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions. We establish analogous results for finite range potentials.

Tsallis maximum entropy principle and the law of large numbers

Tsallis has suggested a nonextensive generalization of the Boltzmann-Gibbs entropy, the maximization of which gives a generalized canonical distribution under special constraints. In this Brief Report, we show that the generalized canonical distribution so obtained may differ from that predicted by the law of large numbers when empirical samples are held to the same constraint. This conclusion is based on a result regarding the large deviation property of conditional measures and is confirmed by numerical evidence.