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Marius A. Schmidt

Publications and source records attributed to Marius A. Schmidt.

11 recordsLinked to original sources
arXiv2023

AMP algorithms and Stein's method: Understanding TAP equations with a new method

We propose a new iterative construction of solutions of the classical TAP equations for the Sherrington-Kirkpatrick model, i.e. with finite-size Onsager correction. The algorithm can be started in an arbitrary point, and converges up to the AT line. The analysis relies on a novel treatment of mean field algorithms through Stein's method. As such, the approach also yields weak convergence of the effective fields at all temperatures towards Gaussians, and can be applied, upon proper alterations, to all models where TAP-like equations and a Stein-operator are available.

math.PR↗
arXiv2023

On concavity of TAP free energy in the SK model

We analyse the Hessian of the Thouless-Anderson-Palmer (TAP) free energy for the Sherrington-Kirkpatrick model, below the de Almeida-Thouless line, evaluated in Bolthausen's approximate solutions of the TAP equations. We show that the empirical spectral distribution weakly converges to a measure with negative support below the AT line, an that the support includes zero on the AT line. In this ``macroscopic'' sense, TAP free energy is concave in the order parameter of the theory, i.e. the random spin-magnetisations. This proves a spectral interpretation of the AT line. However, for specific magnetizations, the Hessian of the TAP free energy can have positive outlier eigenvalues. The question whether such outliers may also occur close to the TAP solutions is left open. In a simplified setting where the magnetizations are independent of the disorder, we prove that Plefka's second condition is equivalent to all eigenvalues being negative.

math.PR↗
arXiv2022

Complexity of local maxima of given radial derivative for mixed $p$-spin Hamiltonians

We study the number of local maxima with given radial derivative of spherical mixed $p$-spin models and prove that the second moment matches the square of the first moment on exponential scale for arbitrary mixtures and any radial derivative. This is surprising, since for the number of local maxima with given radial derivative and given energy the corresponding result is only true for specific mixtures [Sub17; BSZ20]. We use standard Kac-Rice computations to derive formulas for the first and second moment at exponential scale, and then find a remarkable analytic argument that shows that the second moment formula is bounded by twice the first moment formula in this general setting. This also leads to a new proof of a central inequality used to prove concentration of the number critical points of pure $p$-spin models of given energy in [Sub17] and removes the need for the computer assisted argument used in that paper for $3 \leq p \leq 10$.

math.PR↗
arXiv2022

Phase diagram for the tap energy of the $p$-spin spherical mean field spin glass model

We solve the Thouless-Anderson-Palmer (TAP) variational principle associated to the spherical pure $p$-spin mean field spin glass Hamiltonian and present a detailed phase diagram. In the high temperature phase the maximum of variational principle is the annealed free energy of the model. In the low temperature phase the maximum, for which we give a formula, is strictly smaller. The high temperature phase consists of three subphases. (1) In the first phase $m=0$ is the unique relevant TAP maximizer. (2) In the second phase there are exponentially many TAP maximizers, but $m=0$ remains dominant. (3) In the third phase, after the so called dynamic phase transition, $m=0$ is no longer a relevant TAP maximizer, and exponentially many non-zero relevant TAP solutions add up to give the annealed free energy. Finally in the low temperature phase a subexponential number of TAP maximizers of near-maximal TAP energy dominate.

cond-mat.dis-nn↗
arXiv2019

From Parisi to Boltzmann

We sketch a new framework for the analysis of disordered systems, in particular mean field spin glasses, which is variational in nature and within the formalism of classical thermodynamics. For concreteness, only the Sherrington-Kirkpatrick model is considered here. For this we show how the Parisi solution (replica symmetric, or when replica symmetry is broken) emerges, in large but finite volumes, from a high temperature expansion to second order of the Gibbs potential with respect to order parameters encoding the law of the effective fields. In contrast with classical systems where convexity in the order parameters is the default situation, the functionals employed here are, at infinite temperature, concave: this feature is eventually due to the Gaussian nature of the interaction and implies, in particular, that the canonical Boltzmann-Gibbs variational principles must be reversed. The considerations suggest that thermodynamical phase transitions are intimately related to the divergence of the infinite expansions.

math.PR↗
arXiv2018

Oriented first passage percolation in the mean field limit, 2. The extremal process

This is the second, and last paper in which we address the behavior of oriented first passage percolation on the hypercube in the limit of large dimensions. We prove here that the extremal process converges to a Cox process with exponential intensity. This entails, in particular, that the first passage time converges weakly to a random shift of the Gumbel distribution. The random shift, which has an explicit, universal distribution related to modified Bessel functions of the second kind, is the sole manifestation of correlations ensuing from the geometry of Euclidean space in infinite dimensions. The proof combines the multiscale refinement of the second moment method with a conditional version of the Chen-Stein bounds, and a contraction principle.

math.PR↗
arXiv2018

First passage percolation in the mean field limit

The Poisson clumping heuristic has lead Aldous to conjecture the value of the first passage percolation on the hypercube in the limit of large dimensions. Aldous' conjecture has been rigorously confirmed by Fill and Pemantle [Annals of Applied Prob- ability 3 (1993)] by means of a variance reduction trick. We present here a streamlined and, we believe, more natural proof based on ideas emerged in the study of Derrida's random energy models.

math.PR↗
arXiv2017

On McKean's martingale in the Bovier-Hartung extremal process

It has been proved by Bovier & Hartung [Elect. J. Probab. 19 (2014)] that the maximum of a variable-speed branching Brownian motion (BBM) in the weak correlation regime converges to a randomly shifted Gumbel distribution. The random shift is given by the almost sure limit of McKean's martingale, and captures the early evolution of the system. In the Bovier-Hartung extremal process, McKean's martingale thus plays a role which parallels that of the derivative martingale in the classical BBM. In this note, we provide an alternative interpretation of McKean's martingale in terms of a law of large numbers for high-points of BBM, i.e. particles which lie at a macroscopic distance from the edge. At such scales, 'McKean-like martingales' are naturally expected to arise in all models belonging to the BBM-universality class.

math.PR↗
arXiv2015

On the replica symmetry phase of the independent set problem

The independent set problem, ISP for short, asks for the maximal number of vertices in a (large) graph which can be occupied such that none of them are neighbors. We address the question from a statistical mechanics perspective, in the case of Erdoes-Renyi random graphs. We thereby introduce a Hamiltonian penalizing configurations which do not satisfy the non-neighboring constraint: the ground state of the ensuing disordered system corresponds to the solution of the ISP. Identifying the ground state amounts, in turns, to control the phase where replica symmetry is broken, which is way beyond our current understanding. By means of Talagrand's cavity method, we rigorously establish the existence of a replica symmetry phase, computing, in particular, the free energy in the limit of large graphs. A conjectural formula for the ground state, hence for the solution of the ISP, is also derived. Being based on the Parisi theory, the emerging picture is that of a staggering complexity.

math.CO↗
arXiv2015

From Derrida's random energy model to branching random walks: from 1 to 3

We study the extremes of a class of Gaussian fields with in-built hierarchical structure. The number of scales in the underlying trees depends on a parameter alpha in [0,1]: choosing alpha=0 yields the random energy model by Derrida (REM), whereas alpha=1 corresponds to the branching random walk (BRW). When the parameter alpha increases, the level of the maximum of the field decreases smoothly from the REM- to the BRW-value. However, as long as alpha<1 strictly, the limiting extremal process is always Poissonian.

math.PR↗