Lattice trails. I. Exact results

Guttman, A J

doi:10.1088/0305-4470/18/4/008

Cited by 39 publications

(5 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) with positive u. Such walks (most clearly on trees) are closely related to Pólya urn processes and similar problems with reinforcement that can be solved exactly (note that walks with bond reinforcement are often called 'trails' in the physics literature [15]). For the models with multiple site reinforcement discussed above (called vertex-reinforced random walks or VRRW), the related urn process is Friedman-like and less tractable [13,14], see endnote [28].…”

Section: Background: Mathematicsmentioning

confidence: 99%

Reinforced walks in two and three dimensions

2009

View full text Add to dashboard Cite

In probability theory, reinforced walks are random walks on a lattice (or more generally a graph) that preferentially revisit neighboring 'locations' (sites or bonds) that have been visited before. In this paper, we consider walks with one-step reinforcement, where one preferentially revisits locations irrespective of the number of visits. Previous numerical simulations [A. Ordemann et al., Phys. Rev. E 64, 046117 (2001)] suggested that the site model on the lattice shows a phase transition at finite reinforcement between a random-walk like and a collapsed phase, in both 2 and 3 dimensions. The very different mathematical structure of bond and site models might also suggest different phenomenology (critical properties, etc.). We use high statistics simulations and heuristic arguments to suggest that site and bond reinforcement are in the same universality class, and that the purported phase transition in 2 dimensions actually occurs at zero coupling constant. We also show that a quasi-static approximation predicts the large time scaling of the end-to-end distance in the collapsed phase of both site and bond reinforcement models, in excellent agreement with simulation results.PACS numbers:

show abstract

Section: Background: Mathematicsmentioning

confidence: 99%

Reinforced walks in two and three dimensions

2009

View full text Add to dashboard Cite

show abstract

“…For L < 15, we tested all initial shapes that (i) do not reuse lattice bonds and (ii) do not cause immediate jamming. Notice that the number of paths selected by (i) scales exponentially with L [Guttmann, 1985]. Disregarding differences in absolute position and orientation, we find no dependence on the initial condition.…”

Section: Resultsmentioning

confidence: 65%

Self-Avoiding Chiral Walks

Dutta

Steinbock

2010

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

We describe a simple, discrete model of deterministic chiral motion on a square lattice. The model is based on rotating walkers with trailing tails spanning L lattice bonds. These tail segments cannot overlap and their leading A segments cannot be crossed. As prescribed by their chirality, walkers must turn if possible, or go straight, or else correct earlier steps recursively. The resulting motion traces unbound trajectories and complex periodic orbits with various symmetries. Periods tend to decrease with increasing L and vary between L and L2. Interacting walkers can form intricate pair states. Some orbits match pinned spiral tip trajectories observed experimentally in excitable systems.

show abstract

“…Hence the number of n-taus increases exponentially with n, at a rate which is independent of τ , and at the same rate as that for n-SAWs. Guttmann [23] proved the existence of the following limit for lattice trails and Zhao and Lookman [8] proved the second inequality in…”

Section: Terminology and Statement Of Resultsmentioning

confidence: 99%

The statistics of collapsing square lattice trails with a fixed number of vertices of degree 4

James

Soteros

2007

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

A trail on the square lattice with a fixed number, k, of vertices of degree 4 is called a k-trail. We model polymer collapse using k-trails by incorporating an interaction energy which is proportional to the number of nearest-neighbour contact edges of the trail. It is known that the number of square lattice nedge closed (open) k-trails can be bounded above and below (to O(n k )) by the number of n-step self-avoiding circuits (walks). This along with pattern theorems for self-interacting self-avoiding circuits and walks are used herein to establish upper and lower bounds (to O(n k )) for the collapsing free energy of k-trails in terms of self-avoiding circuits or walks, as appropriate. We also use pattern theorems to obtain bounds on the limiting nearest-neighbour contact density for collapsing k-trails. Finally, we investigate k-trails with a fixed density of nearest-neighbour contacts and show that their limiting entropy per monomer is independent of k.

show abstract

Lattice trails. I. Exact results

Cited by 39 publications

References 16 publications

Reinforced walks in two and three dimensions

Reinforced walks in two and three dimensions

Self-Avoiding Chiral Walks

The statistics of collapsing square lattice trails with a fixed number of vertices of degree 4

Contact Info

Product

Resources

About