Finite dimensional corrections to mean field in a short-range p-spin glassy model

In this work we discuss a short range version of the $p$-spin model. The model is provided with a parameter that allows to control the crossover with the mean field behaviour. We detect a discrepancy between the perturbative approach and numerical simulation. We attribute it to non-perturbative effects due to the finite probability that each particular realization of the disorder allows for the formation of regions where the system is less frustrated and locally freezes at a higher temperature.Comment: 18 pages, 5 figures, submitted to Phys Rev

On Equilibrium Dynamics of Spin-Glass Systems

We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical $2+p$ spin glass model we test the asymptotic static limit of the Sompolinsky solution showing that it fails to yield a thermodynamically stable solution. We then present an alternative formulation, based on the Crisanti, H\"orner and Sommers [Z. f\"ur Physik {\bf 92}, 257 (1993)] dynamical solution of the spherical $p$-spin spin glass model, reproducing a stable static limit that coincides, in the case of a one step Replica Symmetry Breaking Ansatz, with the solution at the dynamic free energy threshold at which the relaxing system gets stuck off-equilibrium. We formally extend our analysis to any number of Replica Symmetry Breakings $R$. In the limit $R\to\infty$ both formulations lead to the Parisi anti-parabolic differential equation. This is the special case, though, where no dynamic blocking threshold occurs. The new formulation does not contain the additional order parameter $\Delta$ of the Sompolinsky theory.Comment: 24 pages, 6 figure

Local excitations in mean field spin glasses

We address the question of geometrical as well as energetic properties of local excitations in mean field Ising spin glasses. We study analytically the Random Energy Model and numerically a dilute mean field model, first on tree-like graphs, equivalent to a replica symmetric computation, and then directly on finite connectivity random lattices. In the first model, characterized by a discontinuous replica symmetry breaking, we found that the energy of finite volume excitation is infinite whereas in the dilute mean field model, described by a continuous replica symmetry breaking, it slowly decreases with sizes and saturates at a finite value, in contrast with what would be naively expected. The geometrical properties of these excitations are similar to those of lattice animals or branched polymers. We discuss the meaning of these results in terms of replica symmetry breaking and also possible relevance in finite dimensional systems.Comment: 7 pages, 4 figures, accepted for publicatio

Is the droplet theory for the Ising spin glass inconsistent with replica field theory?

Symmetry arguments are used to derive a set of exact identities between irreducible vertex functions for the replica symmetric field theory of the Ising spin glass in zero magnetic field. Their range of applicability spans from mean field to short ranged systems in physical dimensions. The replica symmetric theory is unstable for d>8, just like in mean field theory. For 6<d<8 and d<6 the resummation of an infinite number of terms is necessary to settle the problem. When d<8, these Ward-like identities must be used to distinguish an Almeida-Thouless line from the replica symmetric droplet phase.Comment: 4 pages. Accepted for publication in J.Phys.A. This is the accepted version with the following minor changes: one extra sentence in the abstract; footnote 2 slightly extended; last paragraph somewhat reformulate

Bayesian joint models with INLA exploring marine mobile predator-prey and competitor species habitat overlap

EPSRC grant Ecowatt 2050 EP/K012851/1 ACKNOWLEDGMENTS We would like to thank the associate editor and the anonymous reviewers for their useful and constructive suggestions which led to a considerable improvement of the manuscript. The authors would also like to thank the following people/organizations for making large datasets available for use in this paper: Mark Lewis (Joint Nature Conservation Committee), Philip Hammond (Scottish Oceans Institute, University of St. Andrews), Susan Lusseau (Marine Scotland Science), Darren Stevens (The Sir Alister Hardy Foundation for Ocean Science, PML), and Yuri Artioli (Plymouth Marine Laboratory). This work was supported by the Engineering and Physical Sciences Research Council (EcoWatt250; EPSRC EP/K012851/1).Peer reviewedPublisher PD

Statistical mechanics of the random K-SAT model

The Random K-Satisfiability Problem, consisting in verifying the existence of an assignment of N Boolean variables that satisfy a set of M=alpha N random logical clauses containing K variables each, is studied using the replica symmetric framework of diluted disordered systems. We present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest K=2, K>= 3 and K>>1. The calculation of the number of solutions, which allowed us [Phys. Rev. Lett. 76, 3881 (1996)] to predict a first order jump at the threshold where the Boolean expressions become unsatisfiable with probability one, is thoroughly displayed. In the case K=2, the (rigorously known) critical value (alpha=1) of the number of clauses per Boolean variable is recovered while for K>=3 we show that the system exhibits a replica symmetry breaking transition. The annealed approximation is proven to be exact for large K.Comment: 34 pages + 1 table + 8 fig., submitted to Phys. Rev. E, new section added and references update

Computing minimal finite free resolutions

AbstractIn this paper we address the basic problem of computing minimal finite free resolutions of homogeneous submodules of graded free modules over polynomial rings. We develop a strategy, which keeps the resolution minimal at every step. Among the relevant benefits is a marked saving of time, as the first reported experiments in CoCoA show. The algorithm has been optimized using a variety of techniques, such as minimizing the number of critical pairs and employing an “ad hoc” Hilbert-driven strategy. The algorithm can also take advantage of various a priori pieces of information, such as the knowledge of the Castelnuovo regularity

Quenched Random Graphs

Spin models on quenched random graphs are related to many important optimization problems. We give a new derivation of their mean-field equations that elucidates the role of the natural order parameter in these models.Comment: 9 pages, report CPTH-A264.109

The Complexity of Ising Spin Glasses

We compute the complexity (logarithm of the number of TAP states) associated with minima and index-one saddle points of the TAP free energy. Higher-index saddles have smaller complexities. The two leading complexities are equal, consistent with the Morse theorem on the total number of turning points, and have the value given in [A. J. Bray and M. A. Moore, J. Phys. C 13, L469 (1980)]. In the thermodynamic limit, TAP states of all free energies become marginally stable.Comment: Typos correcte