1,107 research outputs found
Random matrix model for QCD_3 staggered fermions
We show that the lowest part of the eigenvalue density of the staggered
fermion operator in lattice QCD_3 at small lattice coupling constant beta has
exactly the same shape as in QCD_4. This observation is quite surprising, since
universal properties of the QCD_3 Dirac operator are expected to be described
by a non-chiral matrix model. We show that this effect is related to the
specific nature of the staggered fermion discretization and that the eigenvalue
density evolves towards the non-chiral random matrix prediction when beta is
increased and the continuum limit is approached. We propose a two-matrix model
with one free parameter which interpolates between the two limits and very well
mimics the pattern of evolution with beta of the eigenvalue density of the
staggered fermion operator in QCD_3.Comment: 8 pages 4 figure
Asymmetric correlation matrices: an analysis of financial data
We analyze the spectral properties of correlation matrices between distinct
statistical systems. Such matrices are intrinsically non symmetric, and lend
themselves to extend the spectral analyses usually performed on standard
Pearson correlation matrices to the realm of complex eigenvalues. We employ
some recent random matrix theory results on the average eigenvalue density of
this type of matrices to distinguish between noise and non trivial correlation
structures, and we focus on financial data as a case study. Namely, we employ
daily prices of stocks belonging to the American and British stock exchanges,
and look for the emergence of correlations between two such markets in the
eigenvalue spectrum of their non symmetric correlation matrix. We find several
non trivial results, also when considering time-lagged correlations over short
lags, and we corroborate our findings by additionally studying the asymmetric
correlation matrix of the principal components of our datasets.Comment: Revised version; 11 pages, 13 figure
A Random Matrix Approach to VARMA Processes
We apply random matrix theory to derive spectral density of large sample
covariance matrices generated by multivariate VMA(q), VAR(q) and VARMA(q1,q2)
processes. In particular, we consider a limit where the number of random
variables N and the number of consecutive time measurements T are large but the
ratio N/T is fixed. In this regime the underlying random matrices are
asymptotically equivalent to Free Random Variables (FRV). We apply the FRV
calculus to calculate the eigenvalue density of the sample covariance for
several VARMA-type processes. We explicitly solve the VARMA(1,1) case and
demonstrate a perfect agreement between the analytical result and the spectra
obtained by Monte Carlo simulations. The proposed method is purely algebraic
and can be easily generalized to q1>1 and q2>1.Comment: 16 pages, 6 figures, submitted to New Journal of Physic
Emergence of a 4D World from Causal Quantum Gravity
Causal Dynamical Triangulations in four dimensions provide a
background-independent definition of the sum over geometries in nonperturbative
quantum gravity, with a positive cosmological constant. We present evidence
that a macroscopic four-dimensional world emerges from this theory dynamically.Comment: 11 pages, 3 figures; some short clarifying comments added; final
version to appear in Phys. Rev. Let
Number statistics for -ensembles of random matrices: applications to trapped fermions at zero temperature
Let be the probability that a
-ensemble of random matrices with confining potential
has eigenvalues inside an interval of the real
line. We introduce a general formalism, based on the Coulomb gas technique and
the resolvent method, to compute analytically for large . We show that this probability scales for large
as , where is the Dyson index of the
ensemble. The rate function , independent of ,
is computed in terms of single integrals that can be easily evaluated
numerically. The general formalism is then applied to the classical
-Gaussian (), -Wishart () and
-Cauchy () ensembles. Expanding the rate function
around its minimum, we find that generically the number variance exhibits a non-monotonic behavior as a function of the size
of the interval, with a maximum that can be precisely characterized. These
analytical results, corroborated by numerical simulations, provide the full
counting statistics of many systems where random matrix models apply. In
particular, we present results for the full counting statistics of zero
temperature one-dimensional spinless fermions in a harmonic trap.Comment: 34 pages, 19 figure
Signal from noise retrieval from one and two-point Green's function - comparison
We compare two methods of eigen-inference from large sets of data, based on
the analysis of one-point and two-point Green's functions, respectively. Our
analysis points at the superiority of eigen-inference based on one-point
Green's function. First, the applied by us method based on Pad?e approximants
is orders of magnitude faster comparing to the eigen-inference based on
uctuations (two-point Green's functions). Second, we have identified the source
of potential instability of the two-point Green's function method, as arising
from the spurious zero and negative modes of the estimator for a variance
operator of the certain multidimensional Gaussian distribution, inherent for
the two-point Green's function eigen-inference method. Third, we have presented
the cases of eigen-inference based on negative spectral moments, for strictly
positive spectra. Finally, we have compared the cases of eigen-inference of
real-valued and complex-valued correlated Wishart distributions, reinforcing
our conclusions on an advantage of the one-point Green's function method.Comment: 14 pages, 8 figures, 3 table
Maximal entropy random walk in community finding
The aim of this paper is to check feasibility of using the maximal-entropy
random walk in algorithms finding communities in complex networks. A number of
such algorithms exploit an ordinary or a biased random walk for this purpose.
Their key part is a (dis)similarity matrix, according to which nodes are
grouped. This study encompasses the use of the stochastic matrix of a random
walk, its mean first-passage time matrix, and a matrix of weighted paths count.
We briefly indicate the connection between those quantities and propose
substituting the maximal-entropy random walk for the previously chosen models.
This unique random walk maximises the entropy of ensembles of paths of given
length and endpoints, which results in equiprobability of those paths. We
compare performance of the selected algorithms on LFR benchmark graphs. The
results show that the change in performance depends very strongly on the
particular algorithm, and can lead to slight improvements as well as
significant deterioration.Comment: 7 pages, 4 figures, submitted to European Physical Journal Special
Topics following the 4-th Conference on Statistical Physics: Modern Trends
and Applications, July 3-6, 2012 Lviv, Ukrain
Portfolio Optimization and the Random Magnet Problem
Diversification of an investment into independently fluctuating assets
reduces its risk. In reality, movement of assets are are mutually correlated
and therefore knowledge of cross--correlations among asset price movements are
of great importance. Our results support the possibility that the problem of
finding an investment in stocks which exposes invested funds to a minimum level
of risk is analogous to the problem of finding the magnetization of a random
magnet. The interactions for this ``random magnet problem'' are given by the
cross-correlation matrix {\bf \sf C} of stock returns. We find that random
matrix theory allows us to make an estimate for {\bf \sf C} which outperforms
the standard estimate in terms of constructing an investment which carries a
minimum level of risk.Comment: 12 pages, 4 figures, revte
Universal microscopic correlation functions for products of independent Ginibre matrices
We consider the product of n complex non-Hermitian, independent random
matrices, each of size NxN with independent identically distributed Gaussian
entries (Ginibre matrices). The joint probability distribution of the complex
eigenvalues of the product matrix is found to be given by a determinantal point
process as in the case of a single Ginibre matrix, but with a more complicated
weight given by a Meijer G-function depending on n. Using the method of
orthogonal polynomials we compute all eigenvalue density correlation functions
exactly for finite N and fixed n. They are given by the determinant of the
corresponding kernel which we construct explicitly. In the large-N limit at
fixed n we first determine the microscopic correlation functions in the bulk
and at the edge of the spectrum. After unfolding they are identical to that of
the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic
correlations we find at the origin differ for each n>1 and generalise the known
Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.Comment: 20 pages, v2 published version: typos corrected and references adde
Multiplication law and S transform for non-hermitian random matrices
We derive a multiplication law for free non-hermitian random matrices
allowing for an easy reconstruction of the two-dimensional eigenvalue
distribution of the product ensemble from the characteristics of the individual
ensembles. We define the corresponding non-hermitian S transform being a
natural generalization of the Voiculescu S transform. In addition we extend the
classical hermitian S transform approach to deal with the situation when the
random matrix ensemble factors have vanishing mean including the case when both
of them are centered. We use planar diagrammatic techniques to derive these
results.Comment: 25 pages + 11 figure
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