653 research outputs found
Rooted Spiral Trees on Hyper-cubical lattices
We study rooted spiral trees in 2,3 and 4 dimensions on a hyper cubical
lattice using exact enumeration and Monte-Carlo techniques. On the square
lattice, we also obtain exact lower bound of 1.93565 on the growth constant
. Series expansions give and on a square lattice. With Monte-Carlo simulations we get the
estimates as , and . These results
are numerical evidence against earlier proposed dimensional reduction by four
in this problem. In dimensions higher than two, the spiral constraint can be
implemented in two ways. In either case, our series expansion results do not
support the proposed dimensional reduction.Comment: replaced with published versio
Walks confined in a quadrant are not always D-finite
We consider planar lattice walks that start from a prescribed position, take
their steps in a given finite subset of Z^2, and always stay in the quadrant x
>= 0, y >= 0. We first give a criterion which guarantees that the length
generating function of these walks is D-finite, that is, satisfies a linear
differential equation with polynomial coefficients. This criterion applies,
among others, to the ordinary square lattice walks. Then, we prove that walks
that start from (1,1), take their steps in {(2,-1), (-1,2)} and stay in the
first quadrant have a non-D-finite generating function. Our proof relies on a
functional equation satisfied by this generating function, and on elementary
complex analysis.Comment: To appear in Theoret. Comput. Sci. (special issue devoted to random
generation of combinatorial objects and bijective combinatorics
Winding of simple walks on the square lattice
A method is described to count simple diagonal walks on with a
fixed starting point and endpoint on one of the axes and a fixed winding angle
around the origin. The method involves the decomposition of such walks into
smaller pieces, the generating functions of which are encoded in a commuting
set of Hilbert space operators. The general enumeration problem is then solved
by obtaining an explicit eigenvalue decomposition of these operators involving
elliptic functions. By further restricting the intermediate winding angles of
the walks to some open interval, the method can be used to count various
classes of walks restricted to cones in of opening angles that
are integer multiples of .
We present three applications of this main result. First we find an explicit
generating function for the walks in such cones that start and end at the
origin. In the particular case of a cone of angle these walks are
directly related to Gessel's walks in the quadrant, and we provide a new proof
of their enumeration. Next we study the distribution of the winding angle of a
simple random walk on around a point in the close vicinity of
its starting point, for which we identify discrete analogues of the known
hyperbolic secant laws and a probabilistic interpretation of the Jacobi
elliptic functions. Finally we relate the spectrum of one of the Hilbert space
operators to the enumeration of closed loops in with fixed
winding number around the origin.Comment: 46 pages, 16 figures. Version accepted for publicatio
Trees of self-avoiding walks
We consider the biased random walk on a tree constructed from the set of
finite self-avoiding walks on a lattice, and use it to construct probability
measures on infinite self-avoiding walks. The limit measure (if it exists)
obtained when the bias converges to its critical value is conjectured to
coincide with the weak limit of the uniform SAW. Along the way, we obtain a
criterion for the continuity of the escape probability of a biased random walk
on a tree as a function of the bias, and show that the collection of escape
probability functions for spherically symmetric trees of bounded degree is
stable under uniform convergence
Asymptotic Behavior of Inflated Lattice Polygons
We study the inflated phase of two dimensional lattice polygons with fixed
perimeter and variable area, associating a weight to a
polygon with area and bends. For convex and column-convex polygons, we
show that , where , and . The
constant is found to be the same for both types of polygons. We argue
that self-avoiding polygons should exhibit the same asymptotic behavior. For
self-avoiding polygons, our predictions are in good agreement with exact
enumeration data for J=0 and Monte Carlo simulations for . We also
study polygons where self-intersections are allowed, verifying numerically that
the asymptotic behavior described above continues to hold.Comment: 7 page
Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk - Monte Carlo Tests
Simulations of the self-avoiding walk (SAW) are performed in a half-plane and
a cut-plane (the complex plane with the positive real axis removed) using the
pivot algorithm. We test the conjecture of Lawler, Schramm and Werner that the
scaling limit of the two-dimensional SAW is given by Schramm's Stochastic
Loewner Evolution (SLE). The agreement is found to be excellent. The
simulations also test the conformal invariance of the SAW since conformal
invariance would imply that if we map the walks in the cut-plane into the half
plane using the conformal map z -> sqrt(z), then the resulting walks will have
the same distribution as the SAW in the half plane. The simulations show
excellent agreement between the distributions.Comment: Second version added more simulations and a proof of irreducibility.
25 pages, 16 figure
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