On directed interacting animals and directed percolation

Knežević, Milan; Vannimenus, J.

doi:10.1088/0305-4470/35/12/303

Cited by 6 publications

(14 citation statements)

References 49 publications

(161 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have found that the number n ( )  of required configurations grows roughly by a factor of 1.9 (for n20) when passing from n to n 1 ( ) + , and for the largest strip this number is 24 352 697 ( )  = . It is interesting to note that size of the transfer matrix T n (x) for directed animals on cylinders can be further significantly reduced [4], since its rank is smaller than dimension n ( )  . Let us note, however, that the size of the matrix elements of such a reduced matrix (polynomials in x) grows much more rapidly with n than in the case of T n (x); for simplicity, in this paper we used T n (x) matrices only.…”

Section: Directed Lattice Animalsmentioning

confidence: 99%

“…Due to a lack of exact results, a considerable portion of research activity on directed percolation has been devoted to the development of various numerical approaches. For two-dimensional systems quite accurate results were obtained using the transfer-matrix technique [2][3][4], Monte Carlo simulations [5] and, in particular, using the methods of series expansions [6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A transfer-matrix study of directed lattice animals and directed percolation on a square lattice

Knežević

2016

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We studied the large-scale properties of directed lattice animals and directed percolation on a square lattice. Using a transfer-matrix approach on strips of finite widths, we generated relatively long sequences of estimates for effective values of critical fugacity, percolation threshold and correlation length critical exponents. We applied two different extrapolation methods to obtain estimates for infinite systems. The precision of our final estimates is comparable to (or better than) the precision of the best currently available results.

show abstract

Section: Directed Lattice Animalsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A transfer-matrix study of directed lattice animals and directed percolation on a square lattice

Knežević

2016

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…The related problem of directed lattice animals (DA) has been studied extensively [17,18,19,20,21,22,14,15,23,24,25,26,27]. It is generally believed that the loop-free condition does not affect the critical behavior, so that both systems should have the same exponents.…”

Section: D+1mentioning

confidence: 99%

Dimensional reduction for directed branched polymers

Imbrie

2004

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Dimensional reduction occurs when the critical behavior of one system can be related to that of another system in a lower dimension. We show that this occurs for directed branched polymers (DBP) by giving an exact relationship between DBP models in D + 1 dimensions and repulsive gases at negative activity in D dimensions. This implies relations between exponents of the two models: The phenomenon of dimensional reduction has attracted considerable attention over the years. The first example was the controversial random field Ising model (RFIM), whose critical behavior was conjectured to be the same as the pure Ising model in two fewer dimensions [1]. A proof of long-range order for the RFIM in three dimensions [2,3] showed that dimensional reduction fails there, and recent work [4,5,6] has elucidated what goes wrong. A second example is the Parisi-Sourlas reduction of branched polymers (BP) in D + 2 dimensions to the Yang-Lee edge or iϕ 3 field theory in D dimensions [7]. This was recently confirmed with the discovery of an exact relationship between BP models and repulsive gases at negative activity in two fewer dimensions [8,9]. The failure of the heuristic arguments for dimensional reduction in the RFIM underscores the importance of having an exact result. In this letter we give a third example, in which directed branched polymers (DBP) reduce to the repulsive gas at negative activity in one fewer dimension.We consider directed branched polymers as self-avoiding tree graphs embedded in Z D+1or R D+1 so that every vertex can be reached from the root at 0 by a sequence of links which move forward with respect to a preferred direction. See 1

show abstract

“…The model of interacting directed animals on infinite strips of a square lattice has been studied in detail by using the transfer-matrix method coupled with the phenomenological renormalization group (PRG) approach [23]. Using this approach quite precise estimates of various quantities of interest have been obtained.…”

Section: Introductionmentioning

confidence: 99%

Transverse size of interacting directed lattice animals studied by Yang–Lee approach

Knežević¹,

Knežević²

2021

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Distributions of zeros of the grand canonical partition function in the complex fugacity plane are examined numerically for a model of self interacting directed lattice animals on infinite strips of the square lattice. It is shown that patterns of these zeros have a simple circular-like forms for large values of the interaction strength, with a small fraction of points scattered over the plane. As the interaction weakens, the zeros make up a more complicated structure which consists of a large number of branches. For weak interaction strength, the zeros lying closest to the real axis approach it by following the power-law behavior as the function of strip width. This allowed us to get quite good estimate of transverse size critical exponent for animals in the swollen state as well as at the point of their collapse transition. For stronger interactions, we found that these zeros move much faster toward the real axis—following an exponential law, which leads us to conjecture that animal transverse size diverges in a logarithmic way in the thermodynamic limit. An additional analysis, based on finite-size scaling analysis of the animal longitudinal correlation length, supports this conjecture.

show abstract

On directed interacting animals and directed percolation

Cited by 6 publications

References 49 publications

A transfer-matrix study of directed lattice animals and directed percolation on a square lattice

A transfer-matrix study of directed lattice animals and directed percolation on a square lattice

Dimensional reduction for directed branched polymers

Transverse size of interacting directed lattice animals studied by Yang–Lee approach

Contact Info

Product

Resources

About