2,984 research outputs found
A Numerical Analyst Looks at the "Cutoff Phenomenon" in Card Shuffling and Other Markov Chains
Diaconis and others have shown that certain Markov chains exhibit a "cutoff phenomenon" in which, after an initial period of seemingly little progress, convergence to the steady state occurs suddenly. Since Markov chains are just powers of matrices, how can such effects be explained in the language of applied linear algebra? We attempt to do this, focusing on two examples: random walk on a hypercube, which is essentially the same as the problem of Ehrenfest urns, and the celebrated case of riffle shuffling of a deck of cards. As is typical with transient phenomena in matrix processes, the reason for the cutoff is not readily apparent from an examination of eigenvalues or eigenvectors, but it is reflected strongly in pseudosprectra - provided they are measured in the 1-norm, not the 2-norm. We illustrate and explain the cutoff phenomenon with Matlab computations based in part on a new explicit formula for the entries of the "riffle shuffle matrix", and note that while the normwise cutoff may occur at one point, such as for the riffle shuffle, weak convergence may occur at an equally precise earlier point such as
Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks
Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations
Quantification of reachable attractors in asynchronous discrete dynamics
Motivation: Models of discrete concurrent systems often lead to huge and
complex state transition graphs that represent their dynamics. This makes
difficult to analyse dynamical properties. In particular, for logical models of
biological regulatory networks, it is of real interest to study attractors and
their reachability from specific initial conditions, i.e. to assess the
potential asymptotical behaviours of the system. Beyond the identification of
the reachable attractors, we propose to quantify this reachability.
Results: Relying on the structure of the state transition graph, we estimate
the probability of each attractor reachable from a given initial condition or
from a portion of the state space. First, we present a quasi-exact solution
with an original algorithm called Firefront, based on the exhaustive
exploration of the reachable state space. Then, we introduce an adapted version
of Monte Carlo simulation algorithm, termed Avatar, better suited to larger
models. Firefront and Avatar methods are validated and compared to other
related approaches, using as test cases logical models of synthetic and
biological networks.
Availability: Both algorithms are implemented as Perl scripts that can be
freely downloaded from http://compbio.igc.gulbenkian.pt/nmd/node/59 along with
Supplementary Material.Comment: 19 pages, 2 figures, 2 algorithms and 2 table
Information theoretic aspects of the two-dimensional Ising model
We present numerical results for various information theoretic properties of
the square lattice Ising model. First, using a bond propagation algorithm, we
find the difference between entropies on cylinders of
finite lengths and 2L with open end cap boundaries, in the limit
. This essentially quantifies how the finite length correction for
the entropy scales with the cylinder circumference . Secondly, using the
transfer matrix, we obtain precise estimates for the information needed to
specify the spin state on a ring encircling an infinite long cylinder.
Combining both results we obtain the mutual information between the two halves
of a cylinder (the "excess entropy" for the cylinder), where we confirm with
higher precision but for smaller systems results recently obtained by Wilms et
al. -- and we show that the mutual information between the two halves of the
ring diverges at the critical point logarithmically with . Finally we use
the second result together with Monte Carlo simulations to show that also the
excess entropy of a straight line of spins in an infinite lattice diverges
at criticality logarithmically with . We conjecture that such logarithmic
divergence happens generically for any one-dimensional subset of sites at any
2-dimensional second order phase transition. Comparing straight lines on square
and triangular lattices with square loops and with lines of thickness 2, we
discuss questions of universality.Comment: 12 pages, including 17 figure
Numerically optimized Markovian coupling and mixing in one-dimensional maps
Algorithms are introduced that produce optimal Markovian couplings for large finite-state-space discrete-time Markov chains with sparse transition matrices; these algorithms are applied to some toy models motivated by fluid-dynamical mixing problems at high Peclét number. An alternative definition of the time-scale of a mixing process is suggested. Finally, these algorithms are applied to the problem of coupling diffusion processes in an acute-angled triangle, and some of the simplifications that occur in continuum coupling problems are discussed
Follow the fugitive: an application of the method of images to open dynamical systems
Borrowing and extending the method of images we introduce a theoretical
framework that greatly simplifies analytical and numerical investigations of
the escape rate in open dynamical systems. As an example, we explicitly derive
the exact size- and position-dependent escape rate in a Markov case for holes
of finite size. Moreover, a general relation between the transfer operators of
closed and corresponding open systems, together with the generating function of
the probability of return to the hole is derived. This relation is then used to
compute the small hole asymptotic behavior, in terms of readily calculable
quantities. As an example we derive logarithmic corrections in the second order
term. Being valid for Markov systems, our framework can find application in
information theory, network theory, quantum Weyl law and via Ulam's method can
be used as an approximation method in more general dynamical systems.Comment: 9 pages, 1 figur
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