The main area of my research interests is the scale-invariance that
appears in various problems of statistical physics. Many of these
problems occur in condensed matter or solid state physics but in general
they need not be confined to that area and indeed the concept of
scale invariance may be familiar to many in biology, mathematics,
geosciences and other disciplines. I mean by scale invariance
the characteristics of some object (either in real space or in some
abstract space) that it contains its own approximate replicas within itself,
at least *on average*. Thus when the length-scale of observation
is changed by a factor then an identical object is recovered,
at least *statistically*.

The best known example of this in condensed matter area occurs at thermodynamic
critical points such as the liquid-gas critical point. Scale invariance
occurs in abundance there. For example, the liquid droplets that form
out of the vapor phase have a power-law size distribution while the
density-density correlation functions decay by power-laws as functions of
distance, and in this sense they exhibit statistical scale invariance.
This idea was expanded and supplemented to become the celebrated
renormalization group theory of Kenneth Wilson. If we examine a statistical
model of the liquid-gas critical point known as the lattice-gas model
(or the Ising model), we find that the clusters formed by *connecting*
the particles of the material close to each other are very fragmented and
possess rugged perimeters. In fact in two dimensions these clusters
are themselves scale-invariant directly in real space at the critical point.
Mandelbrot coined the term *fractal* to describe these objects which are
scale-invariant in real space, and their study is an important part of
the research of scale invariance in statistical physics today.

Much of my own recent and current research concerns scale invariance in one way or another, and most can be grouped in one of the following areas:

(1) Percolation and other clustering phenomena

(2) Effect of surfaces, interfaces and finite sizes on statistical properties

(3) Kinetics of disorderly growth processes

(4) Statistics of self-avoiding walks and polymer conformation

(5) Statistics of random and self-avoiding walks in random media

In addition, I am interested in the application of massively parallel computing to these areas. I have an on-going collaboration with some computer scientists in this pursuit, which already won us an international award called the Gordon Bell Prize in 1992.

In the following detailed description of my work, numerical
notation corresponds to the items on the publication list, and whenever
*we* appears, it should be interpreted to indicate my work in collaboration
with other individuals appearing as coauthors of corresponding articles.

**I. Percolation and other clustering phenomena**

The first area is that of my Ph.D. thesis work, and has been of continuing
interest to me ever since. The term *percolation* refers in general to
the problem of connectivity for diverse phenomena featuring
geometrical disorder. It occurs in its purest form
in a random network of open and closed (or occupied and empty) bonds
on a regular lattice. The problem concerns how the global connectivity
of the network depends on the local connectivity.
It turns out that the global properties have a sharp singularity as a
function of the local property just as in usual critical phenomena. In
fact, this analogy is more than superficial, and one can formulate detailed
correspondences between percolation and thermal critical phenomena.
A good example of a system for which percolation plays an important role is
a ferromagnet which is diluted randomly with non-magnetic impurities.
These general developments are contained in a broad review by Prof. Stanley
and his group.

Our main contributions in this area have been to demonstrate the close
parallels to critical phenomena by numerical and analytical studies of
*scaling* in percolation. These studies confirmed the validity
of scaling hypotheses for percolation and showed how the analogs of
thermodynamic functions depend on spatial dimensionality of the lattice.
Our studies of the effects of varying the way connectivity propagates
showed, e.g., that site randomness and bond randomness lead to the same
*universality class*, i.e., essentially the same critical
behavior. Corrections to asymptotic scaling has also been
studied by a simulation on a very large scale, resulting
in an accurate determination of the so-called correction-to-scaling
exponent.

A variant of percolation where only unlike species connect was
also studied, which should be relevant for certain gels.
This work resolved a conflict that existed among previous studies by
others and showed that such problem has the same set of critical exponents
as usual percolation but its scaling functions do *not* reduce to the
usual ones and thus it is not in the same *universality class* as
ordinary percolation in the strict sense. In another work, a description
of percolation clusters as a form of branched polymer was given in order
to understand the apparent breakdown of finite-size scaling (see below)
for percolation in dimensions greater than six.

Another more recent area of our contribution is the precise characterization of the so-called backbone part of the critical percolation cluster. This is the part of the cluster which is connected to the opposite extreme points of the cluster by mutually exclusive paths and would be the current carrying part if the cluster were made of resistive material with a potential difference applied to the two ends. By precision Monte Carlo simulations our work was one of the first to resolve then-existing discrepancies in the estimates of the backbone fractal dimension in two dimensions. Another work in three dimensions proposes a substantial correction to the previously accepted value of the backbone fractal dimension as well as calculates the dynamic critical exponents.

**II. Effect of surfaces, interfaces and finite sizes
on statistical properties**

There are two parts to the work in this area which are intimately related
in some cases. The emphasis in the first part is on so-called surface wetting
transitions. These phase transitions are associated with the formation of
a macroscopically thick *wetting* layer on substrate which occurs, e.g.,
in binary fluid mixtures in contact with a bounding solid. Similar
phenomena also occur, and are now recognized to be analyzable in a similar
manner, for the adsorption of gasses on solid surfaces. While the experimental
realizations of the latter have been known for many years (e.g., ethylene on
exfoliated graphite), wetting transitions of binary fluids were observed
only relatively recently, motivated by the theoretical development.

We are generally credited, at least in part, for formulating scaling
hypotheses for wetting, the identification of the wetting tricritical point
and the elucidation of the relationships between wetting and more classic
surface *magnetic* transitions.
Some extensions for polymer solutions and blends near a wall were also
given , where metastability was studied and *extended wetting*
was introduced. More recently, Prof. Ascarelli of our department
discovered anomalies in the density of fluid argon confined between
capacitor plates and we collaborated on its interpretation from
the point of view of surface phase transitions. Prof. Ascarelli later found
substantial surface irregularities by electron microscopy and it is possible
that the stress induced by capillary condensation around the irregularities
might be responsible for the anomaly observed.

Separate but related problems in the critical behavior of semi-infinite systems include those of critical adsorption and surface ordering. I studied these problems as applied to self-avoiding walks and percolation using a real-space renormalization method.

The second part is centered around a concept that the effects of finite
size on a system near criticality can be *scaled* just like
thermodynamic variables. The original idea is due to Prof. Michael Fisher
and subsequently it has been confirmed in many model systems
of varying complexity, either analytically or numerically.
This idea is most useful in extrapolating various theoretical calculations
based on small systems to thermodynamic limit; however, recent advances in
thin film experimental techniques have made it possible to directly test
the theory against critical phenomena occurring in films. In thin films,
both the finite thickness and the surfaces are important and they cannot be
separated. We have introduced the extensions of the theory which
incorporate non-neutral boundary conditions necessary in order to perform
the comparison with the experimental results and additionally calculated
the details within a mean field framework. This work suggested the possibility
that some experimental results actually correspond to a cross-over region.

More recently, we pointed out some serious shortcomings of methods called real-space renormalization as applied to the problem of random walks, and as an alternative, proposed a finite-size scaling approach for the random walk. We have also proposed and numerically verified finite-size scaling for the largest non-trivial eigenvalue associated with the random walk in random media. This latter work is mainly concerned with the effect of environmental disorder and will be discussed again in the section on random walks in disordered media below.

**III. Kinetics of disorderly growth processes**

The third area deals with a question of whether the ideas of critical phenomena (such as renormalization group) is applicable to the various irreversible, kinetic growth processes. These include the so-called diffusion limited aggregation (DLA) (introduced as a model of smoke and dust particle aggregation) and invasion percolation (introduced as a model of water penetration into porous, oil-bearing rocks). Our approach of a modified cell-renormalization group to these problems are capable of obtaining certain critical exponents to a precision only achievable otherwise by a much larger scale Monte Carlo simulation. The success of our methods then indicate the correctness of the assumption of scale invariance in this problem. Since our work on a discrete lattice was completed, much has come to be known about the diffusion limited aggregates. For example, it is now known that a DLA grown in continuous space has different asymptotic properties than a very late stage DLA grown on a lattice. However, a lattice DLA of intermediate size is recognized to share most of the essential features with a much larger scale DLA in continuum.

In a separate work, I studied the applicability of the well-known cell renormalization procedure to the systems of dimensionality in the range where mean field behavior is expected. Despite its close relationship to the finite-size scaling, which does break down in a certain sense, I argued and showed numerically that the validity of the renormalization procedure is not affected in those high dimensionalities.

**IV. Statistics of self-avoiding walks and polymer conformation**

The fourth area considers the static and dynamic conformation of
macromolecular chains. This area turns out to be intimately related
to the ideas of scale invariance introduced in the study of critical
phenomena and proves to be a field with potentially important applications.
Microscopically, a linear polymer has a well defined potential energy
as a function of the dihedral angle, or the angle made by the planes
defined by successive carbon-carbon backbone bond pairs.
The minimum of this energy is assumed by the *trans*
configuration, which gives an essentially straight chain segment.
However, thermal fluctuations allow a finite probability that the chain
takes a *gauche* configuration, or a *turn*.
This introduces the persistence length for the chain which is the measure
of over how many monomers the chain retains its stiff, one-dimensional
conformation.

We have studied the crossover
from a flexible chain to a stiff one as the persistence length
increases, say, when the temperature is lowered. We have found that
this crossover is interrelated with the crossover between non-excluded volume
to excluded volume statistics and that simple scaling theory needs to be
modified. In particular, for extremely stiff limit, the excluded-volume
effect becomes irrelevant above *two* dimensions rather than the usual
*four*. Also studied is the effect of stiffness of the chains
on the concentration profile in so-called depletion region near a neutral
surface. This work showed numerically that the stiffness can be scaled
away in a rather simple way and the exponent that describes the concentration
profile near the surface remains the same as for flexible chains.

A closely related problem concerns a polymer whose persistence length is not well defined but has a broad distribution. If this distribution were of a power law type, then we may have a very peculiar and interesting morphology, which we may call an excluded-volume Lévy flight, an extension of the special random walk introduced by Mandelbrot. It would have scale-invariance characterized by a dimension-like parameter continuously depending on the power law index. We have introduced a mapping between this problem and a model ferromagnet with a long-range interaction and also gave Monte Carlo simulation results. The mapping immediately allows the application of the renormalization group techniques to this problem and in fact serves to check the validity of the so-called expansion, for the first time, for small where it is supposed to work best. We have also obtained the exact asymptotic behavior of the one-dimensional limit of a variant of this problem, and, upon comparison with the result of the magnetic mapping for the original model, discovered surprising complexities in their morphology. Extensions of this work were made to the correction to scaling at upper marginal dimensions and to the ensemble described by the contour length.

There is also a work which does not fit in to the two major categories discussed in this section. In this work, we propose a scaling picture for a single but very long chain which occupy a finite fraction of the space available.

**V. Random and self-avoiding walks in disordered media**

A few years ago, we started studying the random walk as a model of
diffusion in media with ordered or disordered impurities (or scatterers),
first by using mainly Monte Carlo simulation.
We paid particular attention to media which are *fractal*
(i.e., those with scale invariance, especially with
effective dimensionality lower than the embedding Euclidean dimensionality).
In an early simulation, we discovered unexpected
regular oscillations in certain autocorrelation functions and a power-law
behavior which appears to have a universal character.
We later elucidated this behavior in terms of the mathematical
subtleties involved in the differentiations of asymptotic functions.
In the same work we proposed scaling hypotheses for the
power-law behaviors of the autocorrelation functions and confirmed
some aspects of them by exact summation. This work was later further
extended to study the effects of geometrical anisotropy, and
the same numerical method was applied to the problem of equally
weighted Gaussian trajectories (called ideal chains), which established
its behavior to be distinct from the usual random walk.

More recently, we developed methods based on the approximate diagonalization
of a dynamical matrix associated with the random walk. This matrix defines
a so-called Markov chain, and its eigenspectrum contains essentially all
the information of the random walk in the particular medium. The main
contribution here was the development and application of a practical way
to approximately diagonalize very large matrices (representing large media),
which aids in the study of the asymptotic long time behavior of
the random walk. This method was used successfully to obtain
the dynamic critical exponents accurately for the critical percolation
cluster, the backbone part of a percolation cluster,
a loop-enhanced percolation cluster, a *tree* structure
with no loops, , and the DLA cluster. Additional work on
a loop-suppressed percolation cluster has been completed.
The eigenspectrum method has also been used in the study of the ideal chain
in its relation to the localization problem.

The information of the velocity-velocity and other autocorrelation functions and frequency dependent diffusivity is more detailed than that of the displacement (which is more commonly studied), and is expected to be relevant to understanding the AC conductivity measurements on disordered metals and semi-conductors. Through a mathematical mapping between the diffusion and elasticity problems, these studies have consequences for certain vibrational properties of the corresponding elastic systems as well.

A somewhat related study concerns self-avoiding random walks
(SAW), a model for linear polymers, confined to randomly diluted media.
Initially, the interest was focused on the relationship between the length
of the chain and the coherence length of the medium. Our early numerical
work suggested that the end-to-end distance exponent is unchanged
when the chain is constrained to a medium even at the critical dilution.
This was contrary to then-accepted notion that the SAW at the critical
dilution is in a different universality class than that at less dilution
and thus our work caused some controversy and a flurry of works by others.
We have gone back to this problem and performed a much larger scale,
complete enumeration calculation to avoid the possibility of biases in
sampling methods and found that the relevant exponents in
fact do change (considerably in 3D but only slightly in 2D) and that our
previous simulations were biased by using too few samples for longer walks.
To accomplish this level of accuracy, we have extensively used a parallel
computing toolkit called *EcLiPSe*,
working closely with its developers at Purdue.

Our work also confirmed the most recent renormalization group prediction by Machta and Le Doussal that the asymptotic behavior of the long SAW is different at and below critical dilution and again at no dilution. It also suggested some unexpectedly complex classes of statistical behavior which depend on the type of the ensemble used. This last point was not previously appreciated.

We have reviewed some aspects of the random walks in random media,
and there is a broader review including the SAW, which is to appear as a
chapter in the forthcoming book called *Annual Reviews of Computational
Physics* (edited by D. Stauffer, World Scientific). A different perspective,
from the point of view of the mean field Langevin equation, for diffusion
in fractal media was also offered in our work.