Narrow Results

For a general class of gas models — which includes discrete and continuous Gibbsian models as well as contour or polymer ensembles — we determine a diluteness condition that implies: (1) uniqueness of the infinite-volume equilibrium measure; (2) stability of this measure under perturbations of parameters and discretization schemes, and (3) existence of a coupled perfect-simulation scheme for the infinite-volume measure together with its perturbations and discretizations. Some of these results have previously been obtained through methods based on cluster expansions. In contrast, our treatment is purely probabilistic and its diluteness condition is weaker than existing convergence conditions for cluster expansions.

Extracting dynamics from point processes produced by different models describing spiking phenomena depends on several factors affecting the quality of reconstruction of nonuniformly sampled dynamical systems. Although its ability is verified by embedding theorems analogous to the Takens theorem for uniformly sampled time series, a limited amount of samples, a low firing rate and the presence of noise can provide significant computational errors and incorrect characterization of the analyzed oscillatory regimes. Here, we discuss how to improve the accuracy of the quantitative evaluation of complex oscillations from point processes using data resampling. This approach provides a more stable estimation of Lyapunov exponents for noisy datasets. The advantages of resampling-based reconstruction are confirmed by the analysis of various spiking mechanisms, including the generation of single firing events and chaotic bursts.

Detection of chaos in experimental data is a crucial issue in nonlinear science. Historically, one of the first evidences of a chaotic behavior in experimental recordings came from laser physics. In a recent work, a Minimal Universal Model of chaos was developed by revisiting the model of laser with feedback, and a first electronic implementation was discussed. Here, we propose an upgraded electronic implementation of the Minimal Universal Model, which allows for a precise and reproducible analysis of the model’s parameters space. As a marker of a possible chaotic behavior the variability of the spiking activity that characterizes one of the system’s coordinates was used. Relying on a numerical characterization of the relationship between spiking activity and maximum Lyapunov exponent at different parameter combinations, several potentially chaotic settings were selected. The analysis via divergence exponent method of experimental time series acquired by using those settings confirmed a robust chaotic behavior and provided values of the maximum Lyapunov exponent that are in very good agreement with the theoretical predictions. The results of this work further uphold the reliability of the Minimal Universal Model. In addition, the upgraded electronic implementation provides an easily controllable setup that allows for further developments aiming at coupling multiple chaotic systems and investigating synchronization processes.

In this paper we study the complex interactions involved in the incoming stimulus, from a gamma (γ) and/or an alpha (α) motoneuron, and the outgoing response from the muscle spindle transmitted by the Ia sensory afferent neuron to the spinal cord. The most interesting case is the γ and α coactivation to the function of the muscle spindle, while the effect from a single (γ or α) motoneuron is also presented as a comparison. The mathematical background of this analysis is based on the theory of stationary point processes. A kernel type method of estimating second- and third-order conditional densities is used. Under certain conditions the asymptotic distributions of these estimates are Normal and the construction of 95% approximate confidence intervals is feasible. The application of these asymptotic results to the system of the muscle spindle enables us to detect and interpret its excitatory and/or inhibitory behavior.

We model continuous-time information flows generated by a number of information sources that switch on and off at random times. By modulating a multi-dimensional Lévy random bridge over a random point field, our framework relates the discovery of relevant new information sources to jumps in conditional expectation martingales. In the canonical Brownian random bridge case, we show that the underlying measure-valued process follows jump-diffusion dynamics, where the jumps are governed by information switches. The dynamic representation gives rise to a set of stochastically-linked Brownian motions on random time intervals that capture evolving information states, as well as to a state-dependent stochastic volatility evolution with jumps. The nature of information flows usually exhibits complex behavior, however, we maintain analytic tractability by introducing what we term the effective and complementary information processes, which dynamically incorporate active and inactive information, respectively. As an application, we price a financial vanilla option, which we prove is expressed by a weighted sum of option values based on the possible state configurations at expiry. This result may be viewed as an information-based analogue of Merton’s option price, but where jump-diffusion arises endogenously. The proposed information flows also lend themselves to the quantification of asymmetric informational advantage among competitive agents, a feature we analyze by notions of information geometry.

Rényi’s parking problem (or 1D sequential interval packing problem) dates back to 1958, when Rényi studied the following random process: Consider an interval $I$ of length $x$, and sequentially and randomly pack disjoint unit intervals in $I$ until the remaining space prevents placing any new segment. The expected value of the measure of the covered part of $I$ is $M(x)$, so that the ratio $M(x)/x$ is the expected filling density of the random process. Following recent work by Gargano et al. [4], we studied the discretized version of the above process by considering the packing of the 1D discrete lattice interval $\{1,2,\dots ,n+2k-1\}$ with disjoint blocks of $(k+1)$ integers but, as opposed to the mentioned [4] result, our exclusion process is symmetric, hence more natural. Furthermore, we were able to obtain useful recursion formulas for the expected number of $r$-gaps ($0\le r\le k$) between neighboring blocks. We also provided very fast converging series and extensive computer simulations for these expected numbers, so that the limiting filling density of the long line segment (as $n\to \infty$) is Rényi’s famous parking constant, $0.7475979203\dots .$

Let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of all non-singleton balls $B$ of $X$, we consider the hierarchical Laplacian $L=L_C$. The operator $L$ acts in ${\mathcal{ℒ}}^2(X,m),$ is essentially self-adjoint and has a purely point spectrum. Choosing a sequence $\{𝜀(B)\}$ of i.i.d. random variables, we consider the perturbed function $C(B,\omega )$ and the perturbed hierarchical Laplacian $L^{\omega }=L_{C(\omega )}.$ Under certain conditions, the density of states $𝔪$ exists and it is a continuous function. We choose a point $\lambda$ such that $𝔪(\lambda )>0$ and build a sequence of point processes defined by the eigenvalues of $L^{\omega }$ located in the vicinity of $\lambda$. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity $𝔪(\lambda )$.

We express the gap probabilities of the tacnode process as the ratio of two Fredholm determinants; the denominator is the standard Tracy–Widom distribution, while the numerator is the Fredholm determinant of a very explicit kernel constructed with Airy functions and exponentials. The formula allows us to apply the theory of numerical evaluation of Fredholm determinants and thus produce numerical results for the gap probabilities. In particular we investigate numerically how, in different regimes, the Pearcey process degenerates to the Airy one, and the tacnode degenerates to the Pearcey and Airy ones.

We model continuous-time information flows generated by a number of information sources that switch on and off at random times. By modulating a multi-dimensional Lévy random bridge over a random point field, our framework relates the discovery of relevant new information sources to jumps in conditional expectation martingales. In the canonical Brownian random bridge case, we show that the underlying measure-valued process follows jump-diffusion dynamics, where the jumps are governed by information switches. The dynamic representation gives rise to a set of stochastically-linked Brownian motions on random time intervals that capture evolving information states, as well as to a state-dependent stochastic volatility evolution with jumps. The nature of information flows usually exhibits complex behavior, however, we maintain analytic tractability by introducing what we term the effective and complementary information processes, which dynamically incorporate active and inactive information, respectively. As an application, we price a financial vanilla option, which we prove is expressed by a weighted sum of option values based on the possible state configurations at expiry. This result may be viewed as an information-based analogue of Merton’s option price, but where jump-diffusion arises endogenously. The proposed information flows also lend themselves to the quantification of asymmetric informational advantage among competitive agents, a feature we analyze by notions of information geometry.

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and give rise to a variety of questions pertaining to calculus of variations, Partial Differential Equations and probability. We will review these as well as “the mean-field limit” results that allow to derive effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order beyond the mean-field limit, giving information on the system at the microscopic level. In the setting of statistical mechanics, this allows for instance to observe the effect of the temperature and to connect with crystallization questions.