arXiv:cond-mat/0212526v2 [cond-mat.dis-nn] 19 Aug 2003
EPJ manuscript No.
(will be inserted by the editor)
Universal properties of shortest paths in isotropically correlated
Roland Schorr and Heiko Rieger
Theoretische Physik, Universit¨ at des Saarlandes, 66041 Saarbr¨ ucken, Germany
Received: date / Revised version: date
Abstract. We consider the optimal paths in a d-dimensional lattice, where the bonds have isotropically
correlated random weights. These paths can be interpreted as the ground state configuration of a simplified
polymer model in a random potential. We study how the universal scaling exponents, the roughness and
the energy fluctuation exponent, depend on the strength of the disorder correlations. Our numerical results
using Dijkstra’s algorithm to determine the optimal path in directed as well as undirected lattices indicate
that the correlations become relevant if they decay with distance slower than 1/r in d = 2 and 3. We show
that the exponent relation 2ν −ω = 1 holds at least in d = 2 even in case of correlations. Both in two and
three dimensions, overhangs turn out to be irrelevant even in the presence of strong disorder correlations.
PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 05.50.+q Lattice
theory and statistics (Ising, Potts, etc.) – 64.60 Ak Renormalization-group, fractal, and percolation studies
of phase transitions – 68.35.Rh Phase transitions and critical phenomena
Optimal paths have been a subject of intensive studies
during the recent years. Besides being one of the simplest
problems involving disorder, this interest can be traced
back to the relevance of this problem to various fields,
such as polymer models [1,2,3], surface growth , ran-
dom bond ferromagnets [5,6,7], spin glasses , and the
traveling salesman problem .
The model under considerationis easily sketched: given
an arbitrary weighted graph, each edge has a particular
cost. The optimal or shortest path connecting two sites is
the one of minimal weight, which is the sum of all costs
along that path. We do not restrict to a particular geome-
try yet as well as we do not specify the costs more precisely
so far. In the simplest case, we choose them from a set of
random numbers that are uniformly distributed. In this
context, the directed polymer model has drawn the most
significant attention during the past years [1,2,10,11,12],
where one assumes in d = 2 a simple square lattice being
cut along its diagonal and oriented as a triangle with the
diagonal as its base. One allows only paths in the direc-
tion to the base, i.e., the path cannot turn backwards. The
costs of the edges that belong to the shortest path are in-
terpreted as potential energies for a polymer configuration
that passes through these edges (or bonds).
We may now ask whether the properties (scaling ex-
ponents) of the shortest path are either influenced by the
distribution of the random numbers or the geometry of
the lattice. The former is still discussed [1,13,14,15,16].
As far as the latter is concerned, it seems to be clear that
the universal properties are not changed if the random-
ness is uncorrelated. For this case Schwartz et al.  in-
vestigated directed and undirected paths in d = 2,3 using
Dijkstra’s algorithm to find the shortest path and Marsili
and Zhang  used a transfer matrix method approach
considering directed and undirected paths up to d = 6.
Both state that overhangs exist but nevertheless they sug-
gest that both problems belong to the same universality
class, even in high dimensions where overhanging config-
urations are entropically favored. It is not a priori clear
that this observation remains true for correlated disorder.
In fact, as we will point out below, the average number of
overhangs increases for strongly correlated disorder indi-
cating that they might become relevant for strong enough
In the present study, we study the universal proper-
ties of shortest paths and focus on the effect of isotropi-
cally correlated random weights on the scaling exponents.
To this end, we consider directed and undirected lattice-
graphs in two and three dimensions with bond weights ηj,
where the d-dimensional index vector j = i1,...,id ∈ Zd
denotes the position of a particular bond in the lattice.
The total energy or cost of a path P from one end of the
lattice (e.g. from one special site or node in the top layer)
to the opposite end (e.g. to an arbitrary site or node in
the bottom layer) is simply the sum of these bond weights
The weights ηj are correlated positive random variables,
which we define below. We choose the index vector j of the
8Roland Schorr and Heiko Rieger: Universal properties of shortest paths in isotropically correlated random potentials
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