Dynamics versus replicas in the random field Ising model

In a previous article we have shown, within the replica formalism, that the conventional picture of the random field Ising model breaks down, by the effect of singularities in the interactions between fields involving several replicas, below dimension eight. In the zero-replica limit several coupling constants have thus to be considered, instead of just one. As a result we had found that there is no stable fixed point in the vicinity of dimension six. It is natural to reconsider the problem in a dynamical framework, which does not require replicas, although the equilibrium properties should be recovered in the large time limit. Singularities in the zero-replica limit are a priori not visible in a dynamical picture. In this note we show that in fact new interactions are also generated in the stochastic approach. Similarly these interactions are found to be singular below dimension eight. These critical singularities require the introduction of a time origin $t_0$ at which initial data are given. The dynamical properties are thus dependent upon the waiting time. It is shown here that one can indeed find a complete correspondence between the equilibrium singularities in the $n=0$ limit, and the singularities in the dynamics when the initial time $t_0$ goes to minus infinity, with $n$ replaced by $-\frac{1}{t_0}$. There is thus complete coherence between the two approaches.Comment: 8 pages, latex, no figur

Twist Free Energy

One may impose to a system with spontaneous broken symmetry, boundary conditions which correspond to different pure states at two ends of a sample. For a discrete Ising-like broken symmetry, boundary conditions with opposite spins in two parallel limiting planes, generate an interface and a cost in free energy per unit area of the interface. For continuum symmetries the order parameter interpolates smoothly between the end planes carrying two different directions of the order parameter. The cost in free energy is then proportional to $L^{d-2}$ for a system of characteristic size L. The power of $L$ is related to the lower critical dimension, and the coefficient of this additional free energy vanishes at the critical temperature. In this note it is shown within a loop expansion that one does find the expected behavior of this twist free energy. This is a preamble to the study of situations where the broken continuum symmetry is believed to be more complex, as in Parisi's ansatz for the Edwards-Anderson spin glass.Comment: 15 pages, latex, no figur

Twist Free Energy in a Spin Glass

The field theory of a short range spin glass with Gaussian random interactions, is considered near the upper critical dimension six. In the glassy phase, replica symmetry breaking is accompanied with massless Goldstone modes, generated by the breaking of reparametrization invariance of a Parisi type solution. Twisted boundary conditions are thus imposed at two opposite ends of the system in order to study the size dependence of the twist free energy. A loop-expansion is performed to first order around a twisted background. It is found, as expected but it is non trivial, that the theory does renormalize around such backgrounds, as well as for the bulk. However two main differences appear, in comparison with simple ferromagnetic transitions : (i) the loop expansion yields a (negative) anomaly in the size dependence of the free energy, thereby lifting the lower critical dimension to a value greater than two given by $d_c = 2-\eta(d_c)$ (ii) the free energy is lowered by twisting the boundary conditions. This sign may reflect a spontaneous spatial non-uniformity of the order parameter.Comment: 15 pages, latex, no figur

On infrared divergences in spin glasses

By studying the structure of infrared divergences in a toy propagator in the replica approach to the Ising spin glass below $T_c$, we suggest a possible cancellation mechanism which could decrease the degree of singularity in the loop expansion.Comment: 13 pages, Latex , revised versio

La religione nei platonici di Cambridge

I platonici di Cambridge sono autori legati da stima e amicizia, ma anche da impegni di studio e di insegnamento a Cambridge: all\u2019Emmanuel College, B. Whichcote (1609-1683), comunemente ritenuto il caposcuola, J. Smith (1616-1652), N. Culverwell (1618-1651), R. Cudworth (1617-1688); al Christ\u2019s College, H. More (1614-1687). Di ispirazione platonica, hanno un atteggiamento molto critico verso l\u2019empirismo baconiano (contro il sapere volto principalmente all\u2019azione rivendicano il valore della contemplazione), il puritanesino (al legalismo religioso e all\u2019arbitrarismo teologico oppongono la religione come interiorit\ue0 e oggettiva moralit\ue0), il meccanicismo cartesiano e il materialismo hobbesiano (li giudicano inadeguati a spiegare il mondo degli esseri viventi e pericolosi in quanto possono aprire la via all\u2019ateismo). Per sensibilit\ue0 e formazione, essi ritengono centrale il tema della religione, sia in se stessa, che nei suoi rapporti con la ragione, con la prassi, con l\u2019antropologia. 1. Religione e ragione. Per i platonici di Cambridge, la religione \ue8 intima comunione con Dio, via del ritorno dell\u2019uomo a Dio. La ragione \ue8 the candle of the Lord, lumen Domini (Whichcote), lume derivato: lumen de lumine, participata similitudo rationis aeternae (Smith). Seguire la ragione \ue8 seguire Dio (Smith), e rifiutare la ragione \ue8 disonorare Dio (Culverwell). La ragione \ue8 vera amica della religione (Smith): \ue8 in grado di confutare l\u2019ateismo e di combattere la superstizione; ha la stessa fonte della religione; \ue8 condizione imprescindibile per la comprensione della rivelazione. Ma la ragione, se \ue8 lumen Domini, non finisce per svalutare la religione? Effettivamente, i platonici di Cambridge, nell\u2019opporsi al fideismo dei puritani, accentuano il valore della ragione, ma lo fanno unicamente in funzione apologetica: la religione \ue8 l\u2019esito pi\uf9 vero della ragione (Whichcote); chi \ue8 guidato dalla religione vive in comunione con la propria ragione (Smith); si pu\uf2 riconoscere Cristo nella natura, nella ragione, come nei doni della grazia (Whichcote). 2. Religione prassi. Per i platonici di Cambridge, la moralit\ue0, gi\ue0 importante per la ragione in ordine alla conoscenza in s\ue9 (l\u2019immoralit\ue0 \ue8 sviante), e soprattutto per la conoscenza di Dio (l\u2019occhio non pu\uf2 contemplare il sole se non \ue8 simile al sole (Smith)), \ue8 essenziale per la religione, che ha come fine la deificazione (th\ue9osis) dell\u2019uomo, la sua assimilazione a Dio attraverso l\u2019esercizio delle virt\uf9, l\u2019adesione attiva al volere divino, l\u2019imitazione di Cristo, modello di santit\ue0. 3. Religione e antropologia. Per i platonici di Cambridge, la ragione \ue8 in certo modo \uabreligiosa\ubb, in quanto vinculum Dei et hominis (Smith). Ma se \ue8 cos\uec, la religione \ue8 naturale, strutturale nell\u2019uomo in quanto essere razionale, per cui la definizione di animal rationale pu\uf2, deve, essere sostituita da animal capax religionis. Niente infatti \ue8 specifico dell\u2019uomo quanto la capacit\ue0 di religione (Whichcote): immagine di Dio, in relazione ontologica e spirituale con Dio, l\u2019uomo \ue8 costitutivamente religioso. Di qui: l\u2019ateismo \ue8 contro ragione e contro natura, \ue8 non liberazione ma alienazione dell\u2019uomo; se Dio non esistesse, niente di peggio della religione (More). I platonici di Cambridge, con le loro posizioni forse hanno favorito dei processi che hanno portato, ad esempio, al razionalismo teologico di Toland o alla dissoluzione della teologia nell\u2019antropologia in Feuerbach: esiti, per la verit\ue0, impropri e non voluti. Ben altri i loro meriti nella storia della filosofia: l\u2019aver conservato una tradizione \uabclassica\ubb di pensiero (platonismo, neoplatonismo, patristica), l\u2019aver dato l\u2019avvio per qualche verso alla \uabfilosofia della religione\ubb, l\u2019aver trattato problemi di sicuro interesse speculativo ed esistenziale

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

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

Dynamic critical behaviour in Ising spin glasses

The critical dynamics of Ising spin glasses with Bimodal, Gaussian, and Laplacian interaction distributions are studied numerically in dimensions 3 and 4. The data demonstrate that in both dimensions the critical dynamic exponent $z_{\rm c}$, the non-equilibrium autocorrelation decay exponent $\lambda_c/z_{\rm c}$, and the critical fluctuation-dissipation ratio $X_{\infty}$ all vary strongly and systematically with the form of the interaction distribution.Comment: 8 pages, 4 figures, version to appear in Phys. Rev.

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

Universality classes in anisotropic non-equilibrium growth models

We study the effect of generic spatial anisotropies on the scaling behavior in the Kardar-Parisi-Zhang equation. In contrast to its "conserved" variants, anisotropic perturbations are found to be relevant in d > 2 dimensions, leading to rich phenomena that include novel universality classes and the possibility of first-order phase transitions and multicritical behavior. These results question the presumed scaling universality in the strong-coupling rough phase, and shed further light on the connection with generalized driven diffusive systems.Comment: 4 pages, revtex, 2 figures (eps files enclosed