Publications by Nicolas Macris

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We study the diamagnetic surface currents of particles in thermal equilibrium submitted to a constant magnetic field. The current density of independent electrons with Boltzmann (respectively Fermi) statistics has a gaussian (respectively exponential) bound for its fall off into the bulk. For a system of interacting particles at low activity with Boltzmann statistics, the current density is localized near to the boundary and integrable when the two-body potential decays as |x|–agr, agr >4, agr>4, in three d

We consider generalized linear models in regimes where the number of nonzero components of the signal and accessible data points are sublinear with respect to the size of the signal. We prove a variational formula for the asymptotic mutual information per ...

We consider an electron-phonon system in two and three dimensions on square, hexagonal, and cubic lattices. The model is a modification of the standard Holstein model where the optical branch is appropriately curved in order to have a reflection positive Hamiltonian. Using infrared bounds together with a recent result on the coexistence of long-range order for electron and phonon fields, we prove that, at sufficiently low temperatures and sufficiently strong electron-phonon coupling, there is a Peierls inst

Recently the decay of correlations between bits of low density generator matrix (LDGM) codes have been investigated by using high temperature expansions from statistical physics \cite{KuMa07}. In this work we apply these ideas to a special class of low den ...

We consider increasingly complex models of matrix denoising and dictionary learning in the Bayes-optimal setting, in the challenging regime where the matrices to infer have a rank growing linearly with the system size. This is in contrast with most existin ...

We consider communication over a binary input memoryless output symmetric channel with low density generator matrix codes and optimal maximum a posteriori decoding. It is known that the problem of computing the average conditional entropy, over such code e ...

We consider spatially coupled systems governed by a set of scalar density evolution equations. Such equations track the behavior of message-passing algorithms used, for example, in coding, sparse sensing, or constraint-satisfaction problems. Assuming that ...

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 aver ...

We consider the density of states of Schr6dmger operators with a uniform magnetic field and a random potential with a Gaussian distribution. We show that the restriction to the states of the first Landau level is eqmvalent to a scaling hmit where one looks at the density of states near to the energy of the frst Landau level and simultaneously lets the strength of the coupling to the random potential go to zero. We also consider a different limit where we look at the suitably normalised density of states nea

Efficient deterministic algorithms are proposed with logarithmic step complexities for the generation of entangled GHZ(N) and W-N states useful for quantum networks, and an implementation on the IBM quantum computer up to N = 16 is demonstrated. Improved q ...

We present a class of diffusion-based algorithms to draw samples from high-dimensional probability distributions given their unnormalized densities. Ideally, our methods can transport samples from a Gaussian distribution to a specified target distribution in finite time. Our approach relies on the stochastic interpolants framework to define a time-indexed collection of probability densities that bridge a Gaussian distribution to the target distribution. Subsequently, we derive a diffusion process that obeys

Convolutional Low-Density-Parity-Check (LDPC) ensembles have excellent performance. Their iterative threshold increases with their average degree, or with the size of the coupling window in randomized constructions. In the latter case, as the window size grows, the Belief Propagation (BP) threshold attains the maximum-a-posteriori (MAP) threshold of the underlying ensemble. In this contribution we show that a similar phenomenon happens for the growth rate of coupled ensembles. Loosely speaking, we observe t

The inference of a large symmetric signal-matrix S ϵ RN × N corrupted by additive Gaussian noise, is considered for two regimes of growth of the rank M as a function of N. For sub-linear ranks M=Θ (Nα) with α in (0,1) the mutual information and minimum mea ...

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 on

In recent years important progress has been achieved towards proving the validity of the replica predictions for the (asymptotic) mutual information (or free energy) in Bayesian inference problems. The proof techniques that have emerged appear to be quite ...

The loop series provides a formal way to write down corrections to the Bethe entropy (and/or free energy) of graphical models. We provide methods to rigorously control such expansions for low-density parity-check codes used over a highly noisy binary symme ...

We investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled low-density generator-matrix codes. The degree of the check nodes is regular and the one of code-bits is Poisson distributed with an aver ...

There has been definite progress recently in proving the variational single-letter formula given by the heuristic replica method for various estimation problems. In particular, the replica formula for the mutual information in the case of noisy linear esti ...

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 eige

We consider a statistical model for finite-rank symmetric tensor factorization and prove a single-letter variational expression for its mutual information when the tensor is of even order. The proof uses the adaptive interpolation method, for which rank-on ...

Nicolas Macris

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Diamagnetic currents

Information theoretic limits of learning a sparse rule

Charge density wave and quantum fluctuations in a molecular crystal

Decay of Correlations: An Application to Low-Density Parity Check codes

Statistical limits of dictionary learning: Random matrix theory and the spectral replica method

Proof of replica formulas in the high noise regime for communication using LDGM codes

The Velocity of the Propagating Wave for General Coupled Scalar Systems

Sharp Bounds for Optimal Decoding of Low Density Parity Check Codes

Density of states of random Schroedinger operators with a uniform magnetic field

Efficient Quantum Algorithms for GHZ and W States, and Implementation on the IBM Quantum Computer

Sampling in High-dimensions Using Stochastic Interpolants and Forward -backward Stochastic Differential Equations

Near concavity of the growth rate for coupled LDPC chains

Matrix Inference in Growing Rank Regimes

Ground states and flux configurations of the two dimensional Falicov-Kimball model

The adaptive interpolation method: a simple scheme to prove replica formulas in Bayesian inference

Beyond the Bethe Free Energy of LDPC Codes via Polymer Expansions

Approaching the Rate-Distortion Limit With Spatial Coupling, Belief Propagation, and Decimation

The Mutual Information in Random Linear Estimation Beyond i.i.d. Matrices

Characterisation of the spectrum of the Landau Hamiltonian with delta impurities

Mutual Information for Low-Rank Even-Order Symmetric Tensor Factorization