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The variational principle for a marked Gibbs point process with infinite-range multibody interactions
Get PDFWe prove the Gibbs variational principle for the Asakura?Oosawa model in which particles of random size obey a hardcore constraint of non-overlap and are additionally subject to a temperature-dependent area interaction. The particle size is unbounded, leading to infinite-range interactions, and the potential cannot be written as a k-body interaction for fixed k. As a byproduct, we also prove the existence of infinite-volume Gibbs point processes satisfying the DLR equations. The essential control over the influence of boundary conditions can be established using the geometry of the model and the hard-core constraint
Survival and extinction for a contact process with a density-dependent birth rate
Get PDFTo study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a density-dependent birth rate. The players are located on the d-dimensional integer lattice, with natural birth rate λ and natural death rate one. The process also depends on a payoff a11 = a modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by $λ times the exponential of a times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when a > > 0, in which case the payoff a models mutual cooperation, whereas the birth rate decreases with the local density when a < 0, in which case the payoff a models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case a = 0), we prove that, for all payoffs a , there is a phase transition from extinction to survival in the direction of λ. Using various block constructions, we also prove that, for all birth rates λ, there is a phase transition in the direction of a. This last result is in sharp contrast with the behavior of the nonspatial deterministic mean-field model in which the stability of the extinction state only depends on λ . This underlines the importance of space (local interactions) and stochasticity in our model
Large and moderate deviations in Poisson navigations
Get PDFWe derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizonal displacement as well as renewal-process arguments
An ergodic and isotropic zero-conductance model with arbitrarily strong local connectivity
Get PDFWe exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all non-trivial choices of the connectivity parameter. The model is based on the so-called randomly stretched lattice where we additionally elongate layers containing few open edges
Modeling, Analysis, and Simulation of All-semiconductor Photonic-crystal Surface-emitting Lasers
Get PDFOptimality Conditions in Control Problems with Random State Constraints in Probabilistic or almost Sure Form
No full textIn this paper, we discuss optimality conditions for optimization problems subject to random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution
Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli
No full textWe discuss computing with hierarchies of families of (potentially weighted) semiclassical Jacobi polynomials which arise in the construction of multivariate orthogonal polynomials. In particular, we outline how to build connection and differentiation matrices with optimal complexity and compute analysis and synthesis operations in quasi-optimal complexity. We investigate a particular application of these results to constructing orthogonal polynomials in annuli, called the generalized Zernike annular polynomials, which lead to sparse discretizations of partial differential equations (PDEs). We compare against a scaled-and-shifted Chebyshev–Fourier series showing that in general the annular polynomials converge faster when approximating smooth functions and have better conditioning. We also construct a sparse spectral element method by combining disk and annulus cells, which is highly effective for solving PDEs with radially discontinuous variable coefficients and data
Additive splitting methods for the generalized nonlinear Schrödinger equation
Get PDFSplitting methods provide an efficient approach to solving evolutionary wave equations, especially in situations where dispersive and nonlinear effects on wave propagation can be separated, as in the generalized nonlinear Schrödinger equation (GNLSE). However, such methods are explicit and can lead to numerical instabilities. We study these instabilities in the context of the GNLSE. Results previously obtained for multiplicative splitting methods are extended to additive splittings. An easy-to-use estimate of the largest possible integration step is derived and confirmed by numerical experiments
Polarized frequency combs in a mode-locked VECSEL
No full textIn this paper, we present a detailed and rigorous derivation of the delay differential equations of the spin-flip model for vertical external cavity lasers with a semiconductor saturable absorption mirror. This model describe mode-locked semiconductor lasers in the ring-resonator geometry with unidirectional lasing. This contribution completes a previous communication [Vladimirov et al. Opt. Lett., 45, 252 (2020)], and we further complete the analytical derivation by taking into account phase and amplitude anisotropies and the resulting different delay times for orthogonal linear polarizations. We show evidence of the coexistence of two linearly polarized frequency combs generation with slightly different repetition rates due to the birefringence-induced time-of- flight difference
Singularities in L1-supercritical Fokker–Planck equations: A qualitative analysis
No full textA class of nonlinear Fokker--Planck equations with superlinear drift is investigated in the L1-supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis with main focus on the isotropic case, where solutions will be shown to converge to the unique minimiser of the free energy as time tends to infinity. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the three-dimensional Kaniadakis--Quarati model for Bose--Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model