




Rigorous Studies in the Statistical Mechanics of Lattice Models /
Études rigoureuses dans la mécanique statistique des
modèles de réseaux (Chris Soteros and Stu Whittington, Organizers)
 MIREILLE BOUSQUETMÉLOU, LaBRI, Universite Bordeaux 1. 351 cours de la Liberation.
33405 Talence Cedex. France
The siteperimeter of bargraphs

The site perimeter of a lattice animal is the number of vertices of the
underlying grid that are adjacent to the animal (but do not belong to
it). This parameter plays an important role in percolation models.
As selfavoiding polygons can also be seen as polyominoes (hence
animals), one can also study their site perimeter. The case of polygons
that are both column and rowconvex is easy to deal with, and we shall
explain why it always yields an algebraic generating function. But the
main result of the talk will be the exact solution of a model of
polygons that are ONLY columnconvex: bargraphs. This model can also be
described as a selfinteracting partially directed walk attached to a
surface.
We obtain the site perimeter generating function for bargraphs as a
qseries in which an algebraic series is plugged: to our knowledge,
this type of series has not appeared before in the zoo of animals
generating functions.
(joint work with Andrew Rechnitzer)
 RICHARD BRAK, Melbourne
Combinatorial aspects of the simple asymmetric exclusion process

The simple asymmetric exclusion process (ASEP) is a finite one
dimensional system of particles hopping with excluded volume. It is a
dynamical model for problems such as driven diffusive systems, traffics
flows and the kinetics of biopolymerization. Much work has been done
on exact calculations for the stationary distribution for the many
variants of the model. All these methods are all algebraic. In this
talk we show that there is a very rich combinatorial dimension to the
ASEP. The infinite dimensional matrices that arise as representations
of the algebras in the problem can be interpreted as lattice path
problems. This interpretation leads to combinatorial solutions to many
of the problems. These solutions use new involutions and bijections.
Interestingly, one of the lattice path representations is related to
compact percolation with a damp wall.
 FRANK DEN HOLLANDER, Eurandom, Eindhoven, The Netherlands
Heteropolymers near interfaces

We consider a heteropolymer, consisting of an i.i.d. concatenation of
hydrophilic and hydrophobic monomers, in the presence of water and oil
arranged in alternating layers. The heteropolymer is modelled by a
directed path
where the vertical component lives on Z, and the layers are
horizontal with equal width. The path measure for the vertical
component is given by that of simple random walk multiplied by an
exponential weight factor that favors the combinations
hydrophilic/water and hydrophobic/oil and disfavors the combinations
hydrophilic/oil and hydrophobic/water. We study the vertical motion of
the heteropolymer as a function of its total length n when the width
of the layers is d_{n} and the parameters in the exponential weight
factor are such that the heteropolymer tends to stay close to an
interface (the socalled ``localized regime''). In the limit as n ®¥ and under the condition that

lim
n ® ¥


d_{n} loglogn

= ¥, 
lim
n ® ¥


d_{n} logn

= 0, 

we show that the vertical motion is a diffusive hopping between
neighboring interfaces on a time scale
where c is computed explicitly in terms of a variational problem.
An analysis of this variational problem sheds light on the optimal
hopping strategy.
(Joint work with M. Wüthrich, Winterthur, Switzerland.)
 YUANAN DIAO, University of North Carolina at Charlotte, Charlotte, North
Carolina 28269, USA
Upper bounds on linking numbers of thick links

The maximum of the linking number between two lattice polygons of
lengths n_{1}, n_{2} (with n_{1} £ n_{2}) is proven to be of the order
of n_{1}(n_{2})^{[1/3]}. This result is generalized to smooth
links of unit thickness. The result also implies that the writhe of a
lattice knot K of length n is at most 26n^{4/3} /p. In the
second half of the paper examples are given to show that linking
numbers of order n_{1}(n_{2})^{[1/3]} can be obtained when
n_{1}^{3} ³ n_{2}. When n_{1}^{3} < n_{2}, it is further shown that the maximum
of the linking number between these two polygons is bounded by cn_{1}^{2} for some constant c > 0. Finally the maximal total linking
number of lattice links with more than 2 components is generalized to
k components.
 BERTRAND DUPLANTIER, Institut Henri
Poincaré 11, rue Pierre et Marie Curie, F75231 Paris Cedex 05
and Service de Physique Théorique, CEA/ Saclay,
F91191 GifSurYvette Cedex, France
Distribution of potential near conformally invariant
boundaries

The distribution of the electrostatic potential near any conformally
invariant fractal boundary is exactly solved in two dimensions. This
class of boundaries appears in a natural way in the statistical
mechanics of any random cluster at a critical point, like a percolation
or an Ising cluster, a Brownian path, or a selfavoiding walk.
Consider a single charged random cluster, generically
called C. Let H (z) be the potential at exterior point
z, with Dirichlet boundary conditions
H(w Î ¶C)=0 on the outer boundary
¶C of C.
The multifractal formalism
characterizes subsets ¶C_{a} of boundary sites
w by a local exponent a, such that the potential scales at
distance r=zw from w as
H (r w Î ¶C_{a}) » r ^{a}, r ® 0. 
 (1) 
The subset ¶C_{a} has a varying Hausdorff
``multifractal'' dimension f(a)=dim(¶C_{a}). An electrostatic exponent a
corresponds geometrically to a local equivalent opening angle
q = p/a along the fractal boundary, and the Hausdorff
dimension [^(f)](q)=f(a = p/q) of the boundary
subset with such angle q is found to be

^ f

(q)= 
p q

 
25c 12


(pq)^{2} q(2pq)

, 
 (2) 
with c is a parameter describing the critical model. This naturally
solves the potential theory near a random fractal in a statistical,
i.e., probabilistic sense. Surprisingly, this classical
problem is solved in the simplest way by a method of ``quantum
gravity'', borrowed from string theory.
Values of c are, for instance, c=1/2 for an Ising cluster; c=0
for the frontier of a Brownian motion [2], for a selfavoiding
walk [3], as well as for a critical percolation cluster
[4]. One thus finds for c=0 that these three boundaries all
have the statistics of a selfavoiding walk, with a unique external
perimeter dimension D_{EP}=sup_{q}[^(f)](q)=4/3, which
establishes and generalizes a wellknown conjecture by Mandelbrot. This
dimension is identical to the external perimeter dimension for
percolation, directly derived in [5].
For any value of c, the Hausdorff dimension of the frontier D_{EP} = sup_{q}[^(f)](q)=[^(f)]([^(q)]), and the typical
harmonic angle [^(q)] satisfy [^(q)]=p(32D_{EP}).
For a critical Potts cluster, the dimensions D_{EP} of the external
perimeter (which is a simple curve) and D_{H} of the cluster's
hull (which possesses double points) obey the duality equation
[1]
(D_{EP}1)(D_{H}1)= 
1 4

, 
 (3) 
independently of the model.
A related covariant MF spectrum is obtained for any critical system
near the cluster boundary.
References
{1.} B. Duplantier, Conformally Invariant Fractals and Potential Theory.
Phy. Rev. Lett. 84(2000), 13631367.
{2.}
, Random Walks and Quantum Gravity in Two Dimensions.
Phy. Rev. Lett. 81(1998), 54895492.
{3.}
, TwoDimensional Copolymers and Exact Conformal
Multifractality. Phy. Rev. Lett. 82(1999), 880883.
{4.}
, Harmonic Measure Exponents for 2D Percolation.
Phy. Rev. Lett. 82(1999), 39403943.
{5.}
A. Aizenman, B. Duplantier and A. Aharony,
Path Crossing Exponents and the External Perimeter in 2D
Percolation. Phys. Rev. Lett. 83(1999), 13591362.
 TONY GUTTMANN, The University of Melbourne, Victoria 3010, Australia
The scaling function of selfavoiding polygons

We derive the scaling function for rooted selfavoiding polygons, on
the square and triangular lattices, enumerated by both perimeter and
area, defined up to a single constant. The scaling function satisfies
a Ricatti equation, and its solution is given by the logarithmic
derivative of an Airy function. The scaling function for unrooted
selfavoiding polygons is given by the logarithm of an Airy function.
While believed to be exact, the results are not rigorous. The result
suggests that the generating function may be the solution of a
qalgebraic difference equation of unknown degree. For
twodimensional percolation theory, by contrast, the generating
function cannot be expressed as a qalgebraic difference equation.
 EDNA JAMES, University of Saskatchewan
Critical exponents for trails in Z^{2} with a
fixed number of vertices of degree 4

For N large, the possible number of Nedge embeddings in
Z^{d} of an arbitrary connected graph t, is believed to grow
asymptotically as A_{t} e^{kN} N^{gt1}, where A_{t} is
a constant, depending on the type of graph t (a walk, polygon,
figureeight, etc.), and k and g_{t} are called the
connective constant and critical exponent, respectively.
Assuming the above asymptotic form, we prove for d = 2, that the
critical exponent for trailgons (closed trails) with k vertices of
degree 4 is given by g_{o} + k, where g_{o} is the critical
exponent for selfavoiding polygons. A similar relation holds between
the critical exponents for all trails with k vertices of degree 4
and for selfavoiding walks.
 BUKS JANSE VAN RENSBURG, York University, Toronto, Ontario M3J 1P3
Knots in adsorbing polygons

A model for a ring polymer adsorbing onto a solid wall is a polygon
confined in the half space z ³ 0 of the cubic lattice, and
adsorbing onto the z=0 plane. Such a polygon is also an embedding of
the circle in real space, and its knot type is well defined. In this
talk I shall present rigorous and numerical results (including the
energy and mean size) of adsorbing polygons, with particular reference
to the entanglement complexity of these polygons, as measured by
considering their knot types. Numerical data were obtained by a
multiple Markov Chain implementation of the pivot algorithm for
polygons confined in halfspace z ³ 0 interacting with the z=0
plane and in the canonical ensemble.
 ANTAL JARAI, PIMSUBC, 1933 West Mall, Vancouver, British Columbia V6T 1Z2
Incipient infinite clusters in 2D percolation

Percolation models at the critical point produce huge but finite
connected clusters that have sometimes been referred to as ``incipient
infinite clusters'' or ``infinite clusters at criticality''. In 1986
H. Kesten proposed the definition of an object, which we call the IIC,
that makes sense of the above loose terminology. In his definition the
origin is conditioned to be connected to the boundary of a large box
(at p = p_{c}), and the size of the box goes to infinity. In 2D the
conditional measures can be shown to have a weak limit, and under the
limiting measure the cluster of the origin is an infinite fractallike
set resembling the large critical clusters.
As noted by M. Aizenman, the IIC provides the microscopic (lattice
scale) description of large percolation clusters at p_{c}. In this
talk we give a proof of this observation in several settings. For
example, consider the largest cluster in a finite box and pick a site
from it uniformly at random. Then we show that the law of the cluster,
when viewed from the random site, converges weakly to the IIC as the
box gets large. Similar results hold, if the largest cluster is
replaced by spanning clusters, the ChayesChayesDurrett cluster or the
invaded region in invasion percolation. All of these objects have been
proposed as alternative definitions of the ``incipient infinite
cluster''; therefore our results show the equivalence of several
natural definitions of the IIC.
 PIERRE LEROUX, Université du Québec, Montréal, Québec
Enumeration of symmetry classes of convex and parallelogram
polyominoes

Several families of polynominoes, defined by convexity or directedness
properties, have been enumerated according to area, perimeter and more
refined parameters, on the square lattice. In this context,
polyominoes are taken up to translations. However, it is also natural,
from a geometric point of view, to consider polyominoes up to rotations
and reflections, that is as ``tiles'' which can move freely in space.
These congruencetype polyominoes can be seen as orbits of the dihedral
group D_{4} acting on (translationtype) polyominoes. Using Burnside's
Lemma, we are led into the enumeration of the various symmetry classes
of polyominoes associated with each element of D_{4}. Together with my
students E. Rassart and A. Robitaille, we have solved this problem in
the cases of convex and of parallelogram polyominoes. Using Moebius
inversion, we have also enumerated the asymmetric polyominoes of these
two sorts. Many of the symmetry classes which occur are closely
related to existing families of discrete models in statistical
mechanics, for example compact source directed convex polyominoes, and
their study raises challenging questions. In this talk, I will survey
this work and mention some open problems.
 NEAL MADRAS, Department of Mathematics and Statistics,
York University, Toronto, Ontario M3J 1P3
Selfavoiding walks on hyperbolic graphs

This talk considers graphs that correspond to regular tilings of the
hyperbolic plane. One example is the infinite planar graph in which
every face is a triangle and eight triangles meet at every vertex. To
a physicist, such graphs display ``infinite dimensional''
characteristics. In particular, selfavoiding walks should behave like
ordinary random walks on these graphs. This predicts, for example,
that there is exponentially small probability for a random Nstep
selfavoiding walk to end at a neighbour of the initial point. I shall
discuss work in this direction.
 ALEKS OWCZAREK, The University of Melbourne, Victoria, Australia 3010
An infinite hierarchy of exact scaling functions

The partition functions for various problems concerning n directed
nonintersecting walks on a square lattice are known to be given by an
n×n determinant. The analysis of the asymptotic behaviour is
often an outstanding problem because of the difficulty with analysing
such an expression. In this talk we focus on the particular problem of
such walks interacting via contact potentials with a wall parallel to
the direction of the walks, and derive the asymptotics of the partition
function of a watermelon network of n such walks for all interaction
parameters. The importance of underlying combinatorial results are
highlighted in this calculation. As a result we give results for the
associated network exponents in the three regimes: desorbed, adsorbed,
and at the adsorption transition. Furthermore, we derive the full
scaling function around the adsorption transition for all n. At the
adsorption transition we also derive a simple ``product form'' for the
partition function.
 NICHOLAS PIPPENGER, Computer Science Department, University of British
Columbia, Vancouver, British Columbia V6T 1Z4
Random Boolean functions

We consider random Boolean functions of n arguments, f :B^{n} ® B (where B = {0, 1}), for which
each value f(x_{1}, ¼, x_{n}) is independently 1 with probability
p (and thus 0 with probability 1p). We study the expected
length [`(L)](n) of the shortest representation of such a
function in disjunctive normal form (that is, as the disjunction of
zero or more terms, each of which is the conjunction of zero or more
literals, each of which is either an argument or its complement). The
bounds we obtain for this problem involve correlation inequalities,
cluster expansions, and other tools of statistical physics.
 ANDREW RECHNITZER, Department of Mathematics and Statistics, York University, York,
Ontario
Polymer adsorption and Dyckpaths

One of the many places that lattice animals and their specialisations
appear is in the modeling of the polymers in solution. The
selfavoiding walk, for example, is used to model long chain polymers
in solution; by altering the properties of the walk, different physical
situations can be modeled, such as polymer collapse and adsorption.
Though the selfavoiding walk model is a very good model of such
behaviour, it has one major drawbackit is (as yet) unsolved. It is
possible to obtain a solvable model by considering directedwalk
models, such as Dyckpaths.
We review the methods for solving the Dyckpath model, and show that it
undergoes an adsorption transition. We then extend this model to the
problem of nonhomogeneous polymers and explain the difficulty of
finding general solutions. For an infinite family of cases we are able
to find solutions and locate the adsorption transition.
 YVAN SAINTAUBIN, Université de Montréal, CP 6128 succ centreville,
Montréal, Quebéc H3C 3J7
Nonunitarity of some observables of the critical 2d Ising model

The distribution of spins at the end of a halfinfinite cylinder is
obtained rigorously for the critical 2d Ising model on a square
lattice. As the number of sites along the circumference goes to
infinity, the ratio of number of sign flips to number of sites at the
boundary tends to a constant. Using conformal field theory, one can ask
the finer question, which pair of sign flips at the boundary of the
cylinder are the endpoints of the same cluster inside the cylinder.
This leads to a critical exponent related to a nonunitary
representation of the Virasoro algebra.
 GORDON SLADE, Department of Mathematics, University of British Columbia
Vancouver, British Columbia V6T 1Z2
Highdimensional networks of selfavoiding walks

We discuss recent work with Remco van der Hofstad that proves that a
sufficiently spreadout network of mutuallyavoiding selfavoiding
walks having the structure of a tree behaves like a system of
independent Brownian motions above four dimensions. Extensions to
networks having the shape of a general graph, rather than a tree, will
also be discussed. The proof uses a generalisation of the lace
expansion, in which the time variable is indexed by a tree or a graph
rather than by an interval.
 ALAN SOKAL, New York University, New York, New York, USA
Potts models, chromatic polynomials, and all that

The qstate Potts model is a statisticalmechanical model that
generalizes the wellknown Ising model. It can be defined on an
arbitrary finite graph, and its partition function encodes much
important information about that graph (including its chromatic
polynomial and its reliability polynomial). The complex zeros of the
Pottsmodel partition function are of interest both to statistical
mechanicians (in connection with the LeeYang picture of phase
transitions) and to combinatorists. I begin by giving an introduction
to all these problems. I then sketch two recent results: (a) Proof
of a universal upper bound on the qplane zeros of the chromatic
polynomial (or antiferromagnetic Pottsmodel partition function) in
terms of the graph's maximum degree (maximum number of nearest
neighbors to any site). (b) Construction of a countable family of
planar graphs whose chromatic zeros are dense in the whole complex
qplane except possibly for the disc q1 < 1. This talk is
intended to be understandable to both mathematicians and physicists;
no prior knowledge of either graph theory or statistical mechanics is
required.
 CHRIS SOTEROS, Department of Mathematics and Statistics, University of
Saskatchewan, Saskatoon, Saskatchewan S7N 5E6
Random knotting and linking in lattice models of polymers

Selfavoiding polygons in Z^{3} can be used to represent the possible
conformations of a ring polymer. Studies of random knotting in such a
lattice model have been motivated by the conjecture (due to Frisch and
Wasserman (1968) and Delbruck (1962)) that almost all sufficiently long
ring polymers are knotted, and more recently by evidence that the
presence of knots in closed circular DNA can provide information about
enzyme action on DNA. More generally, embeddings of graphs (not
necessarily connected) in Z^{3} can be used to represent the possible
conformations of a polymer network, however, it remains a challenge to
appropriately characterize the entanglement complexity of such
networks. This has motivated studies of random linking in systems of
selfavoiding polygons and random knotting in embeddings of graphs.
Recent rigorous results about random knotting and linking in lattice
models of polymers will be reviewed. The results are focussed on
understanding i) the effects of geometrical constraints on the
probability of knotting in selfavoiding polygons, ii) the probability
of linking in systems of k selfavoiding polygons, and iii) the
extent to which these results can be extended to the study of polymer
networks.
 DE WITT SUMNERS, Florida State University, Tallahassee, Florida, USA
Using knots to analyze DNA packing in viral capsids

Bacteriophages are viruses that infect bacteria; the linear phage DNA
is tightly packed in a protein icosahedron. When the viral DNA is
released from the confining capsid, it circularizes, and these DNA
circles are knotted with very high (8095%) probability. We use the
spectrum of DNA knots produced, together with simulation of knots in
confined volumes to infer packing geometry. In particular, we conclude
that DNA packing in the viral capsid is not random.
 STU WHITTINGTON, Department of Chemistry, University of Toronto, Toronto, Ontario
Localization transition in a randomly coloured selfavoiding walk

Consider a selfavoiding walk on the simple cubic lattice with vertices
coloured A and B uniformly and independently. The walk is assigned
a weight which is determined by the number of A vertices having
positive zcoordinate and the number of B vertices having negative
zcoordinate. This can be regarded as a model of a random copolymer
with two types of monomers, each of which prefers one of two immiscible
solvents. We shall discuss the existence of the limiting quenched
average free energy for this model, and thermodynamic selfaveraging.
The system can be shown to undergo a phase transition corresponding to
localization at the interface (i.e. to a phase in which a
nonzero fraction of the vertices of the walk have zero
zcoordinate). Recent results about the nature of the phase diagram
will be presented.

