5 research outputs found
Exact solution of two classes of prudent polygons
Prudent walks are self-avoiding walks on the square lattice which never step
into the direction of an already occupied vertex. We study the closed version
of these walks, called prudent polygons, where the last vertex is adjacent to
the first one. More precisely, we give the half-perimeter generating functions
of two subclasses of prudent polygons, which turn out to be algebraic and
non-D-finite, respectively.Comment: 11 pages, 3 figures; 14 pages, 4 figures, improved exposition,
additional figure; 23 pages, 12 figures, additional section and figure
Scaling Limit of the Prudent Walk
We describe the scaling limit of the nearest neighbour prudent walk on the
square lattice, which performs steps uniformly in directions in which it does
not see sites already visited. We show that the scaling limit is given by the
process Z(u) = s_1 theta^+(3u/7) e_1 + s_2 theta^-(3u/7) e_2, where e_1, e_2 is
the canonical basis, theta^+(t), resp. theta^-(t), is the time spent by a
one-dimensional Brownian motion above, resp. below, 0 up to time t, and s_1,
s_2 are two random signs. In particular, the asymptotic speed of the walk is
well-defined in the L^1-norm and equals 3/7.Comment: Better exposition, stronger claim, simpler description of the
limiting process; final version, to appear in Electr. Commun. Probab
The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics
Prudent walks are special self-avoiding walks that never take a step towards
an already occupied site, and \emph{-sided prudent walks} (with )
are, in essence, only allowed to grow along directions. Prudent polygons
are prudent walks that return to a point adjacent to their starting point.
Prudent walks and polygons have been previously enumerated by length and
perimeter (Bousquet-M\'elou, Schwerdtfeger; 2010). We consider the enumeration
of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find
that the generating function is expressed in terms of a -hypergeometric
function, with an accumulation of poles towards the dominant singularity. This
expression reveals an unusual asymptotic structure of the number of polygons of
area , where the critical exponent is the transcendental number
and and the amplitude involves tiny oscillations. Based on numerical data, we
also expect similar phenomena to occur for 4-sided polygons. The asymptotic
methodology involves an original combination of Mellin transform techniques and
singularity analysis, which is of potential interest in a number of other
asymptotic enumeration problems.Comment: 27 pages, 6 figure