arXiv subjects

Jack Raymond

Publications and source records attributed to Jack Raymond.

42 records · Page 3Linked to original sources

arXiv2009

Equilibrium properties of disordered spin models with two scale interactions

Methods for understanding classical disordered spin systems with interactions conforming to some idealized graphical structure are well developed. The equilibrium properties of the Sherrington-Kirkpatrick model, which has a densely connected structure, have become well understood. Many features generalize to sparse Erdos-Renyi graph structures above the percolation threshold, and to Bethe lattices when appropriate boundary conditions apply. In this paper we consider spin states subject to a combination of sparse strong interactions with weak dense interactions, which we term a composite model. The equilibrium properties are examined through the replica method, with exact analysis of the high temperature paramagnetic, spin glass and ferromagnetic phases by perturbative schemes. We present results of a replica symmetric variational approximations where perturbative approaches fail at lower temperature. Results demonstrate novel reentrant behaviors from spin glass to ferromagnetic phases as temperature is lowered, including transitions from replica symmetry broken to replica symmetric phases. The nature of high temperature transitions is found to be sensitive to the connectivity profile in the sparse sub-graph, with regular connectivity a discontinuous transition from the paramagnetic to ferromagnetic phases is apparent.

cond-mat.dis-nn↗

arXiv2008

Randomness and metastability in CDMA paradigms

Code Division Multiple Access (CDMA) in which the signature code assignment to users contains a random element has recently become a cornerstone of CDMA research. The random element in the construction is particularly attractive in that it provides robustness and flexibility in application, whilst not making significant sacrifices in terms of multiuser efficiency. We present results for sparse random codes of two types, with and without modulation. Simple microscopic consideration on system samples would suggest differences in the phase space of the two models, but we demonstrate that the thermodynamic results and metastable states are equivalent in the minimum bit error rate detector. We analyse marginal properties of interactions and also make analogies to constraint satisfiability problems in order to understand qualitative features of detection and metastable states. This may have consequences for developing algorithmic methods to escape metastable states, thus improving decoding performance.

cs.IT↗

arXiv2008

Sparsely-spread CDMA - a statistical mechanics based analysis

Sparse Code Division Multiple Access (CDMA), a variation on the standard CDMA method in which the spreading (signature) matrix contains only a relatively small number of non-zero elements, is presented and analysed using methods of statistical physics. The analysis provides results on the performance of maximum likelihood decoding for sparse spreading codes in the large system limit. We present results for both cases of regular and irregular spreading matrices for the binary additive white Gaussian noise channel (BIAWGN) with a comparison to the canonical (dense) random spreading code.

cs.IT↗

arXiv2008

On composite systems of dilute and dense couplings

Composite systems, where couplings are of two types, a combination of strong dilute and weak dense couplings of Ising spins, are examined through the replica method. The dilute and dense parts are considered to have independent canonical disordered or uniform bond distributions; mixing the models by variation of a parameter $γ$ alongside inverse temperature $β$ we analyse the respective thermodynamic solutions. We describe the variation in high temperature transitions as mixing occurs; in the vicinity of these transitions we exactly analyse the competing effects of the dense and sparse models. By using the replica symmetric ansatz and population dynamics we described the low temperature behaviour of mixed systems.

cond-mat.dis-nn↗

arXiv2007

The Phase Diagram of 1-in-3 Satisfiability Problem

We study the typical case properties of the 1-in-3 satisfiability problem, the boolean satisfaction problem where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized by average connectivity and probability of negation of a variable in a clause. Random 1-in-3 Satisfiability and Exact 3-Cover are special cases of this ensemble. We interpolate between these cases from a region where satisfiability can be typically decided for all connectivities in polynomial time to a region where deciding satisfiability is hard, in some interval of connectivities. We derive several rigorous results in the first region, and develop the one-step--replica-symmetry-breaking cavity analysis in the second one. We discuss the prediction for the transition between the almost surely satisfiable and the almost surely unsatisfiable phase, and other structural properties of the phase diagram, in light of cavity method results.

cond-mat.stat-mech↗

arXiv2007

A Hike in the Phases of the 1-in-3 Satisfiability

We summarise our results for the random $ε$--1-in-3 satisfiability problem, where $ε$ is a probability of negation of the variable. We employ both rigorous and heuristic methods to describe the SAT/UNSAT and Hard/Easy transitions.

cond-mat.stat-mech↗