45 research outputs found
Growth and Decay in Life-Like Cellular Automata
We propose a four-way classification of two-dimensional semi-totalistic
cellular automata that is different than Wolfram's, based on two questions with
yes-or-no answers: do there exist patterns that eventually escape any finite
bounding box placed around them? And do there exist patterns that die out
completely? If both of these conditions are true, then a cellular automaton
rule is likely to support spaceships, small patterns that move and that form
the building blocks of many of the more complex patterns that are known for
Life. If one or both of these conditions is not true, then there may still be
phenomena of interest supported by the given cellular automaton rule, but we
will have to look harder for them. Although our classification is very crude,
we argue that it is more objective than Wolfram's (due to the greater ease of
determining a rigorous answer to these questions), more predictive (as we can
classify large groups of rules without observing them individually), and more
accurate in focusing attention on rules likely to support patterns with complex
behavior. We support these assertions by surveying a number of known cellular
automaton rules.Comment: 30 pages, 23 figure
Hydrodynamic limit for a zero-range process in the Sierpinski gasket
We prove that the hydrodynamic limit of a zero-range process evolving in
graphs approximating the Sierpinski gasket is given by a nonlinear heat
equation. We also prove existence and uniqueness of the hydrodynamic equation
by considering a finite-difference scheme.Comment: 24 pages, 1 figur
Expected length of the longest common subsequence for large alphabets
We consider the length L of the longest common subsequence of two randomly
uniformly and independently chosen n character words over a k-ary alphabet.
Subadditivity arguments yield that the expected value of L, when normalized by
n, converges to a constant C_k. We prove a conjecture of Sankoff and Mainville
from the early 80's claiming that C_k\sqrt{k} goes to 2 as k goes to infinity.Comment: 14 pages, 1 figure, LaTe
Symmetrized models of last passage percolation and non-intersecting lattice paths
It has been shown that the last passage time in certain symmetrized models of
directed percolation can be written in terms of averages over random matrices
from the classical groups , and . We present a theory of
such results based on non-intersecting lattice paths, and integration
techniques familiar from the theory of random matrices. Detailed derivations of
probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure
Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition
The one-dimensional totally asymmetric simple exclusion process (TASEP) is
considered. We study the time evolution property of a tagged particle in TASEP
with the step-type initial condition. Calculated is the multi-time joint
distribution function of its position. Using the relation of the dynamics of
TASEP to the Schur process, we show that the function is represented as the
Fredholm determinant. We also study the scaling limit. The universality of the
largest eigenvalue in the random matrix theory is realized in the limit. When
the hopping rates of all particles are the same, it is found that the joint
distribution function converges to that of the Airy process after the time at
which the particle begins to move. On the other hand, when there are several
particles with small hopping rate in front of a tagged particle, the limiting
process changes at a certain time from the Airy process to the process of the
largest eigenvalue in the Hermitian multi-matrix model with external sources.Comment: 48 pages, 8 figure
An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution
We compute exactly the asymptotic distribution of scaled height in a
(1+1)--dimensional anisotropic ballistic deposition model by mapping it to the
Ulam problem of finding the longest nondecreasing subsequence in a random
sequence of integers. Using the known results for the Ulam problem, we show
that the scaled height in our model has the Tracy-Widom distribution appearing
in the theory of random matrices near the edges of the spectrum. Our result
supports the hypothesis that various growth models in dimensions that
belong to the Kardar-Parisi-Zhang universality class perhaps all share the same
universal Tracy-Widom distribution for the suitably scaled height variables.Comment: 5 pages Revtex, 3 .eps figures included, new references adde
Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections
In this note we analyze an anisotropic, two-dimensional bootstrap percolation
model introduced by Gravner and Griffeath. We present upper and lower bounds on
the finite-size effects. We discuss the similarities with the semi-oriented
model introduced by Duarte.Comment: Key words: Bootstrap percolation, anisotropy, finite-size effect
Convergence of nonlocal threshold dynamics approximations to front propagation
In this note we prove that appropriately scaled threshold dynamics-type
algorithms corresponding to the fractional Laplacian of order converge to moving fronts. When the resulting interface
moves by weighted mean curvature, while for the normal velocity is
nonlocal of ``fractional-type.'' The results easily extend to general nonlocal
anisotropic threshold dynamics schemes.Comment: 19 page
Scaling Limits for Internal Aggregation Models with Multiple Sources
We study the scaling limits of three different aggregation models on Z^d:
internal DLA, in which particles perform random walks until reaching an
unoccupied site; the rotor-router model, in which particles perform
deterministic analogues of random walks; and the divisible sandpile, in which
each site distributes its excess mass equally among its neighbors. As the
lattice spacing tends to zero, all three models are found to have the same
scaling limit, which we describe as the solution to a certain PDE free boundary
problem in R^d. In particular, internal DLA has a deterministic scaling limit.
We find that the scaling limits are quadrature domains, which have arisen
independently in many fields such as potential theory and fluid dynamics. Our
results apply both to the case of multiple point sources and to the
Diaconis-Fulton smash sum of domains.Comment: 74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in
v2: added "least action principle" (Lemma 3.2); small corrections in section
4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version);
expanded section 6.