Nicolas Macris

Coupled graphical models and their thresholds

The excellent performance of convolutional low-density parity-check codes is the result of the spatial coupling of individual underlying codes across a window of growing size, but much smaller than the length of the individual codes. Remarkably, the belief-propagation threshold of the coupled ensemble is boosted to the maximum-a-posteriori one of the individual system. We investigate the generality of this phenomenon beyond coding theory: we couple general graphical models into a one-dimensional chain of large individual systems. For the later we take the Curie-Weiss, random field Curie-Weiss, If-satisfiability, and Q-coloring models. We always find, based on analytical as well as numerical calculations, that the message passing thresholds of the coupled systems come very close to the static ones of the individual models. The remarkable properties of convolutional low-density parity-check codes are a manifestation of this very general phenomenon.

Tight Bounds on the Capacity of Binary Input Random CDMA Systems

In this paper, we consider code-division multiple-access (CDMA) communication over a binary input additive white Gaussian noise (AWGN) channel using random spreading. For a general class of symmetric distributions for spreading sequences, in the limit of a large number of users, we prove an upper bound to the capacity. The bound matches the formula obtained by Tanaka using the replica method. We also show concentration of various relevant quantities including mutual information and free energy. The mathematical methods are quite general and allow us to discuss extensions to other multiuser scenarios.

Griffith–Kelly–Sherman Correlation Inequalities: A Useful Tool in the Theory of Error Correcting Codes

It is shown that a correlation inequality of statistical mechanics can be applied to linear low-density parity-check codes. Thanks to this tool we prove that, under a natural assumption, the exponential growth rate of regular low-density parity-check (LDPC) codes, can be computed exactly by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. Then, considering communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code we prove that, under a natural assumption, part of the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation algorithm. The correlation inequality yields a sharp lower bound on the GEXIT curve. We also make an extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem

Chains of mean-field models

We consider a collection of Curie-Weiss (CW) spin systems, possibly with a random field, each of which is placed along the positions of a one-dimensional chain. The CW systems are coupled together by a Kac-type interaction in the longitudinal direction of the chain and by an infinite-range interaction in the direction transverse to the chain. Our motivations for studying this model come from recent findings in the theory of error-correcting codes based on spatially coupled graphs. We find that, although much simpler than the codes, the model studied here already displays similar behavior. We are interested in the van der Waals curve in a regime where the size of each Curie-Weiss model tends to infinity, and the length of the chain and range of the Kac interaction are large but finite. Below the critical temperature, and with appropriate boundary conditions, there appears a series of equilibrium states representing kink-like interfaces between the two equilibrium states of the individual system. The van der Waals curve oscillates periodically around the Maxwell plateau. These oscillations have a period inversely proportional to the chain length and an amplitude exponentially small in the range of the interaction; in other words, the spinodal points of the chain model lie exponentially close to the phase transition threshold. The amplitude of the oscillations is closely related to a Peierls-Nabarro free energy barrier for the motion of the kink along the chain. Analogies to similar phenomena and their possible algorithmic significance for graphical models of interest in coding theory and theoretical computer science are pointed out.

Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

Abstract: We consider a random Schrödinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the spectrum in the lowest N Landau bands of this random Hamiltonian when the magnetic field is sufficiently strong, depending on N. We show that the spectrum in these bands is entirely pure point, that the energies coinciding with the Landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length. By relating the Hamiltonian to a lattice operator we are able to use the Aizenman–Molchanov method to prove localization.

Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems

We consider chains of random constraint satisfaction models that are spatially coupled across a finite window along the chain direction. We investigate their phase diagram at zero temperature using the survey propagation formalism and the interpolation method. We prove that the SAT-UNSAT phase transition threshold of an infinite chain is identical to the one of the individual standard model, and is therefore not affected by spatial coupling. We compute the survey propagation complexity using population dynamics as well as large degree approximations, and determine the survey propagation threshold. We find that a clustering phase survives coupling. However, as one increases the range of the coupling window, the survey propagation threshold increases and saturates towards the phase transition threshold. We also briefly discuss other aspects of the problem. Namely, the condensation threshold is not affected by coupling, but the dynamic threshold displays saturation towards the condensation one. All these features may provide a new avenue for obtaining better provable algorithmic lower bounds on phase transition thresholds of the individual standard model.

Ground States and Flux Configurations of the Two-Dimensional Falicov-Kimball Model

The Falicov-Kimball model is a lattice model of itinerant spinless fermions (“electrons”) interacting by an on-site potential with classical particles (“ions”). We continue the investigations of the crystalline ground states that appear for various filling of electrons and ions for large coupling. We investigate the model for square as well as triangular lattices. New ground states are found and the effects of a magnetic flux on the structure of the phase diagram are studied. The flux phase problem where one has to find the optimal flux configurations and the nuclei configurations is also solved in some cases. Finally we consider a model where the fermions are replaced by hard-core bosons. This model also has crystalline ground states. Therefore their existence does not require the Pauli principle, but only the on-site hard-core constraint for the itinerant particles.

Sharp Bounds for Optimal Decoding of Low-Density Parity-Check Codes

Consider communication over a binary-input memoryless output-symmetric channel with low-density parity-check (LDPC) codes and maximum a posteriori (MAP) decoding. The replica method of spin glass theory allows to conjecture an analytic formula for the average input-output conditional entropy per bit in the infinite block length limit. Montanari proved a lower bound for this entropy, in the case of LDPC ensembles with convex check degree polynomial, which matches the replica formula. Here we extend this lower bound to any irregular LDPC ensemble. The new feature of our work is an analysis of the second derivative of the conditional input-output entropy with respect to noise. A close relation arises between this second derivative and correlation or mutual information of codebits. This allows us to extend the realm of the ldquointerpolation method,rdquo in particular, we show how channel symmetry allows to control the fluctuations of the ldquooverlap parametersrdquo.

On edge states in semi-infinite quantum Hall systems

We consider an electron in two dimensions submitted to a magnetic field and to the potential of impurities. We show that when the electron is confined to a half-space by a planar wall described by a smooth increasing potential, the total Hamiltonian necessarily has a continuous spectrum in some intervals in between the Landau levels provided that both the amplitude and spatial variation of the impurity potential are sufficiently weak. The spatial decay of the impurity potential is not needed. In particular, this proves the occurrence of edge states in semi-infinite quantum Hall systems.

On the flux phase conjecture at half-filling: An improved proof

We present a simplification of Liebs proof of the flux phase conjecture for interacting fermion systems—such as the Hubbard model—at half-filling on a general class of graphs. The main ingredient is a procedure which transforms a class of fermionic Hamiltonians into reflection-positive form. The method can also be applied to other problems, which we briefly illustrate with two examples concerning thet−V model and an extended Falicov-Kimball model.