1,720,999 research outputs found
Free-energy fluctuations and chaos in the Sherrington-Kirkpatrick model
No full textThe sample-to-sample fluctuations Delta F(N) of the free-energy in the Sherrington-Kirkpatrick model are shown rigorously to be related to bond chaos. Via this connection, the fluctuations become analytically accessible by replica methods. The replica calculation for bond chaos shows that the exponent mu governing the growth of the fluctuations with system size N, Delta F(N) similar to N(mu), is bounded by mu <= 1/4
Sample-to-sample fluctuations and bond chaos in the m-component spin glass
No full textWe calculate the finite-size scaling of the sample-to-sample fluctuations of the free energy Delta F of the m component vector spin glass in the large-m limit. This is accomplished using a variant of the interpolating Hamiltonian technique which is used to establish a connection between the free energy fluctuations and bond chaos. The calculation of bond chaos then shows that the scaling of the free-energy fluctuations with system size N is Delta F similar to N(mu) with 1/5 <= mu <= 3/10, and very likely mu = 1/5 exactly.German Science Foundation (DFG) [AS 136/2-1
The m-component spin glass on a Bethe lattice
No full textWe study the m-component vector spin glass in the limit m ->infinity on a Bethe lattice. The cavity method allows for a solution of the model in a self-consistent field approximation and for a perturbative solution of the full problem near the phase transition. The low-temperature phase of the model is analyzed numerically and a generalized Bose-Einstein condensation is found, as in the fully connected model. Scaling relations between four distinct zero-temperature exponents are found
Local Linear Convergence of the ADMM/Douglas - Rachford Algorithms without Strong Convexity and Application to Statistical Imaging
No full textWe consider the problem of minimizing the sum of a convex function and a convex function composed with an injective linear mapping. For such problems, subject to a coercivity condition at fixed points of the corresponding Picard iteration, iterates of the alternating directions method of multipliers converge locally linearly to points from which the solution to the original problem can be computed. Our proof strategy uses duality and strong metric subregularity of the Douglas--Rachford fixed point mapping. Our analysis does not require strong convexity and yields error bounds to the set of model solutions. We show in particular that convex piecewise linear-quadratic functions naturally satisfy the requirements of the theory, guaranteeing eventual linear convergence of both the Douglas--Rachford algorithm and the alternating directions method of multipliers for this class of objectives under mild assumptions on the set of fixed points. We demonstrate this result on quantitative image deconvolution and denoising with multiresolution statistical constraints
Modern Statistical Challenges in High-Resolution Fluorescence Microscopy
No full textConventional light microscopes have been used for centuries for the study of small length scales down to approximately 250 nm. Images from such a microscope are typically blurred and noisy, and the measurement error in such images can often be well approximated by Gaussian or Poisson noise. In the past, this approximation has been the focus of a multitude of deconvolution techniques in imaging. However, conventional microscopes have an intrinsic physical limit of resolution. Although this limit remained unchallenged for a century, it was broken for the first time in the 1990s with the advent of modern superresolution fluorescence microscopy techniques. Since then, superresolution fluorescence microscopy has become an indispensable tool for studying the structure and dynamics of living organisms. Current experimental advances go to the physical limits of imaging, where discrete quantum effects are predominant. Consequently, this technique is inherently of a non-Gaussian statistical nature, and we argue that recent technological progress also challenges the long-standing Poisson assumption. Thus, analysis and exploitation of the discrete physical mechanisms of fluorescent molecules and light, as well as their distributions in time and space, have become necessary to achieve the highest resolution possible. This article presents an overview of some physical principles underlying modern fluorescence microscopy techniques from a statistical modeling and analysis perspective. To this end, we develop a prototypical model for fluorophore dynamics and use it to discuss statistical methods for image deconvolution and more complicated image reconstruction and enhancement techniques. Several examples are discussed in more detail, including variational multiscale methods for confocal and stimulated emission depletion (STED) microscopy, drift correction for single marker switching (SMS) microscopy, and sparse estimation and background removal for superresolution by polarization angle demodulation (SPoD). We illustrate that such methods benefit from advances in large-scale computing, for example, from recent tools from convex optimization. We argue that in the future, even higher resolutions will require more detailed models that delve into sub-Poissonian statistics
Dynamics of a one-dimensional granular gas with a stochastic coefficient of restitution
No full textRecently, we have modelled inelastic collisions of one-dimensional rods [1,2] by the absorption of translational energy E-tr through internal degrees of freedom, in particular elastic vibrations. We arrived at a stochastic description of collision processes, characterised by a stochastic coefficient of restitution a. In this paper, we construct an analytic approximation for the transition probability E-tr --> E-tr' =(1 - epsilon(2))E-tr. This allows us to perform much longer simulations of large, strongly inelastic granular systems and study relaxation to the true equilibrium state. If the internal vibrations are undamped, equilibrium is characterised by propagating sound waves. In the case of damping, the system develops towards a final state which consists of one big cluster, containing all particles at rest. (C) 2000 Elsevier Science B.V. All rights reserved
Universal aspects of vacancy-mediated disordering dynamics: The effect of external fields
No full textWe investigate the disordering of an initially phase-segregated binary alloy, due to a highly mobile defect which couples to an electric or gravitational field. Using both mean-field and Monte Carlo methods, we show that the late stages of this process exhibit dynamic scaling, characterized by a set of exponents and scaling functions. A new scaling variable emerges, associated with the field. While the scaling functions carry information about the field and the boundary conditions, the exponents are universal. They can be computed analytically, in excellent agreement with simulation results
Nature of perturbation theory in spin glasses
No full textThe high-order behaviour of the perturbation expansion in the cubic replica field theory of spin glasses in the paramagnetic phase has been investigated. The study starts with the zero-dimensional version of the replica field theory and this is shown to be equivalent to the problem of finding finite-size corrections in a modified spherical spin glass near the critical temperature. We find that the high-order behaviour of the perturbation series is described, to leading order, by coefficients of alternating signs (suggesting that the cubic field theory is well defined) but that there are also subdominant terms with a complicated dependence of their sign on the order. Our results are then extended to the d-dimensional field theory and in particular used to determine the high-order behaviour of the terms in the expansion of the critical exponents in a power series in is an element of = 6 - d. We have also corrected errors in the existing 6 expansions at third order
The Integrated Density of States of the Random Graph Laplacian
Full text linkWe analyse the density of states of the random graph Laplacian in the percolating regime. A symmetry argument and knowledge of the density of states in the nonpercolating regime allows us to isolate the density of states of the percolating cluster (DSPC) alone, thereby eliminating trivially localised states due to finite subgraphs. We derive a nonlinear integral equation for the integrated DSPC and solve it with a population dynamics algorithm. We discuss the possible existence of a mobility edge and give strong evidence for the existence of discrete eigenvalues in the whole range of the spectrum
Stress relaxation of near-critical gels
No full textThe time-dependent stress relaxation for a Rouse model of a cross-linked polymer melt is completely determined by the spectrum of eigenvalues of the connectivity matrix. The latter has been computed analytically for a mean-field distribution of cross-links. It shows a Lifshitz tail for small eigenvalues and all concentrations below the percolation threshold, giving rise to a stretched exponential decay of the stress relaxation function in the sol phase. At the critical point the density of states is finite for small eigenvalues, resulting in a logarithmic divergence of the viscosity and an algebraic decay of the stress relaxation function. Numerical diagonalization of the connectivity matrix supports the analytical findings and has furthermore been applied to cluster statistics corresponding to random bond percolation in two and three dimensions
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