On the Distribution of Surface Extrema in Several One- and Two-dimensional Random Landscapes

Hivert, Florent; Nechaev, Sergei; Oshanin, Gleb; Vasilyev, Oleg A.

doi:10.1007/s10955-006-9231-7

Cited by 14 publications

(23 citation statements)

References 24 publications

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“…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 probability of large deviations of M from its mean value, i.e., the probabilities of atypical configurations.…”

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confidence: 61%

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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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confidence: 61%

“…Near the mean M and within a region |M − M | = O( √ N ), the distribution P (M, N ) is expected to be a Gaussian [19] with mean and variance given in Eq. (11)…”

Section: General Properties Of the Distribution Of Mmentioning

confidence: 99%

Statistics of the number of minima in a random energy landscape

Majumdar¹,

Martin²

2006

Phys. Rev. E

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“…The understanding of the topology of such random surfaces is of much importance [40,41]. For example, the number of peaks determines the number of possible nonsatisfied bonds in a spin glass [42,43] or the "roof" of the surface in ballistic growth models [44].…”

Section: Random Surfacesmentioning

confidence: 99%

Partition of networks into basins of attraction

2008

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We study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree. Each node is associated with, or "attracted" to its neighbor of maximal degree, as long as the degree is increasing. A node that has no neighbors of higher degree is a peak, attracting all the nodes in its basin. Maximally random scale-free networks exhibit different behavior based on their degree distribution exponent gamma : For small gamma (broad distribution) networks are dominated by a giant basin, whereas for large gamma (narrow distribution) there are numerous basins, with peaks attracting mainly their nearest neighbors. We derive expressions for the first two moments of the number of basins. We also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale-free networks that resemble random nets. Finally, we generalize the problem to regular networks and lattices where all degrees are equal, and thus the attractiveness of a node must be determined by an assigned weight, rather than the degree. We derive the complete distribution of basins of attraction resulting from randomly assigned weights in one-dimensional chains.

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“…Generalization to two-dimensional square lattice with L = m × m, (m is an integer) sites is straightforward (see Fig.2): the only difference here is that we call as local surface peaks such sites j the numbers at which are greater than numbers appearing at four adjacent sites. For these models, our goal is to evaluate the probability P (M, L) that the surface created in such a way has exactly M peaks on a lattice containing L sites [10]. In one dimension this can be done exactly and provides also a distribution function of the number of right U-turns of the PGRW trajectories.…”

Section: Random Surfaces Generated By Random Permutationsmentioning

confidence: 99%

“…In one dimension this can be done exactly [10]. We are also able to calculate the "correlation function" p(l) defining the conditional probability that two peaks are separated by the interval l under the condition that the interval l does not contain other peaks.…”

Section: Introductionmentioning

confidence: 99%

Random patterns generated by random permutations of natural numbers

Oshanin

Voituriez

Nechaev

et al. 2007

Eur. Phys. J. Spec. Top.

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We survey recent results on some one-and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time n, whose moves to the right or to the left are induced by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site X at time n, obtain the probability measure of different excursions and define the asymptotic distribution of the number of "U-turns" of the trajectories -permutation "peaks" and "through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers 1, 2, 3, . . . , L on sites of a 1d or 2d square lattices containing L sites. We calculate the distribution function of the number of local "peaks" -sites the number at which is larger than the numbers appearing at nearest-neighboring sites -and discuss some surprising collective behavior emerging in this model.

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On the Distribution of Surface Extrema in Several One- and Two-dimensional Random Landscapes

Cited by 14 publications

References 24 publications

Statistics of the number of minima in a random energy landscape

Statistics of the number of minima in a random energy landscape

Partition of networks into basins of attraction

Random patterns generated by random permutations of natural numbers

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