Large Deviations of Extreme Eigenvalues of Random Matrices

Dean, David S.; Majumdar, Satya N.

doi:10.1103/physrevlett.97.160201

Cited by 218 publications

(425 citation statements)

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“…largest and smallest) eigenvalues of a random matrix. For a large class of random matrix ensembles, including the W W W ensemble [469], one finds [470][471][472] 158) where the function Ψ(ζ), which depends on the particular ensemble and is computable in simple cases, is N -independent at leading order in large N . In summary, eigenvalue repulsion dictates that the probability that a Hessian matrix in the ensemble defined by (3.149) has only positive eigenvalues is given by…”

Section: Random Supergravitymentioning

confidence: 99%

Inflation and String Theory

2015

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Section: Random Supergravitymentioning

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Inflation and String Theory

2015

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“…Note that in the usual Gaussian ensembles the second term of S 2 is absent, thus if the average value of the eigenvalues is shifted from zero (and thus we shall see later the value of α is shifted from 1/2) then the constraint increases the action S 2 by a finite value and thus the probability of finding a Gaussian matrix with α = 1/2 is of order exp(−N 2 θ(α)) making the probability of minima for instance extremely small [13,14,15]. This is not the case for the Hessian of a Gaussian random field at a critical point due to the presence of this zero mode.…”

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“…We now adopt the constrained Coulomb gas approach recently introduced in [13] and using standard methods of functional integration [10] we may write…”

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Statistics of Critical Points of Gaussian Fields on Large-Dimensional Spaces

2007

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We calculate the average number of critical points of a Gaussian field on a high-dimensional space as a function of their energy and their index. Our results give a complete picture of the organization of critical points and are of relevance to glassy and disordered systems and landscape scenarios coming from the anthropic approach to string theory. 12.40.Ee, 78.55.Q The statistics of random fields have been studied in detail in the mathematics and physics literature [1]. Gaussian random fields are of particular importance, as they can be argued to occur spontaneously in systems with a large number of degrees of freedom by appealing to the central limit theorem. Recently string theorists have postulated that the effective theory arising from string theory is highly complex leading to a string landscape with a huge number of possible minima, each of which could describe a potential universe [2], thus a statistical study of this complex landscape may well be necessary. Gaussian fields arise generically in the study of classical complex systems such as spin glasses, optimization problems and protein folding [3]. In many of these systems, jamming or the onset of a glass transition occurs due to the presence of a large number of metastable states or local minima in the energy landscape. High-dimensional complex potentials also occur as the potential energy of interacting particle systems which exhibit a structural glass transition (no quenched disorder is present). Many numerical results exist on these potential energy landscapes, notably for Lennard-Jones and soft-sphere glass forming mixtures [4]. Scaling arguments relating the number of critical points (points where the gradient of the potential energy vanishes) of various types also exist [5] and the onset of the glass transition is often identified with the point where the critical points in the free energy landscape become dominated by minima [6]. In this Letter we compute the average number of critical points of a Gaussian potential, on a high dimensional space, at fixed value of the potential and at fixed number of negative eigenvalues at the critical point.We shall consider a Gaussian field φ defined over volume V of an N -dimensional Euclidean space. The field has zero mean and correlation function:This scaling in N ensures the existence of the thermodynamic limit (as N → ∞) of what is known as the toy model, whose statics [7] and dynamics [8] have been studied in the literature. The version of the toy model corresponding to our study here is a single particle in the potential V (x) = φ(x). In the harmonically confined version of the toy model the potential energy of the particle is V (x) = φ(x) + 1 2 µx 2 , and the presence of the confining potential means that the volume V of the system can be taken to infinity. The total number of critical points for the toy model with confining potential was studied by Fyodorov [9] as a function of µ. Here we study the case without confining potential in a finite volume. We give explicit expressions for the average...

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“…Energy landscapes provide a simple conceptual framework for modeling these systems but in fact their use goes much beyond that. For instance the vacua of string theories are expected to proliferate enormously, and the problem of estimating the number of local minima [2,3] [8,9] and the associated problems in random matrix theory [3,10,11]. Typically these systems consider a particle, a configuration of particles, or even a manifold, subject to a random potential.…”

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“…For instance the vacua of string theories are expected to proliferate enormously, and the problem of estimating the number of local minima [2,3] of string energy landscapes is still an open problem. Other examples include rugged (random) energy landscapes in evolutionary biology [4], quantum cosmology [5], manifolds in random media [6], glassy systems [1,7] random potentials [8,9] and the associated problems in random matrix theory [3,10,11]. Typically these systems consider a particle, a configuration of particles, or even a manifold, subject to a random potential.…”

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Statistics of the number of minima in a random energy landscape

Majumdar¹,

Martin²

2006

Phys. Rev. E

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We consider random energy landscapes constructed from d-dimensional lattices or trees. The distribution of the number of local minima in such landscapes follows a large deviation principle and we derive the associated law exactly for dimension 1. Also of interest is the probability of the maximum possible number of minima; this probability scales exponentially with the number of sites. We calculate analytically the corresponding exponent for the Cayley tree and the two-leg ladder; for 2 to 5 dimensional hypercubic lattices, we compute the exponent numerically and compare to the Cayley tree case.PACS numbers: 02.50.Cw (Probability theory), 05.50.+q For many materials that are glassy [1] local minima of the energy (or of the free energy) trap the system for long times, leading to subtle equilibrium and out of equilibrium properties. Energy landscapes provide a simple conceptual framework for modeling these systems but in fact their use goes much beyond that. For instance the vacua of string theories are expected to proliferate enormously, and the problem of estimating the number of local minima [2,3] [8,9] and the associated problems in random matrix theory [3,10,11]. Typically these systems consider a particle, a configuration of particles, or even a manifold, subject to a random potential. One extreme case for its visual simplicity is that of a point particle in a random Gaussian potential; on the opposite extreme are energy landscapes associated with the configuration of a many body system. Examples in this last category include: (1) the p-spin glass model [12], which in the limit of large p reduces to the far simpler random energy model of Derrida [13]; (2) atomic clusters [14] and other glassy systems with no quenched disorder [15]; (3) random manifolds in random media [6].For any landscape, it is desirable to know the statistical properties of the minima (or more generally of the saddle points). A quantity frequently considered is the expected number of minima as a function of their energy [9]. It may also be of interest to consider how the set of minima are organized topologically, e.g., whether the barrier tree [16] is ultrametric. Our focus here is to better understand the statistics of the total number of local minima in random energy landscapes; our underlying space is a regular (Euclidean) lattice on each site of which resides a random energy. Let M denote the total number of local minima for given values of the energies on each site. Evidently M is a random variable as it varies from one realization of the landscape to another. We are interested in the statistics of M as a function of the number N of lattice sites. Similar questions were studied recently in the context of random permutations [17,18], ballistic deposition [19], and in simple models of glasses [20]. In this paper, we provide a number of analytical results on the distribution of the total number of local minima for random energy landscapes on several lattices; the moments of M are easily derived, so our focus concerns mainly the probabil...

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Large Deviations of Extreme Eigenvalues of Random Matrices

Cited by 218 publications

References 20 publications

Inflation and String Theory

Inflation and String Theory

Statistics of Critical Points of Gaussian Fields on Large-Dimensional Spaces

Statistics of the number of minima in a random energy landscape

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