arXiv subjects

Iwan Jensen

Publications and source records attributed to Iwan Jensen.

50 records · Page 3Linked to original sources

arXiv2000

A transfer matrix approach to the enumeration of plane meanders

A closed plane meander of order $n$ is a closed self-avoiding curve intersecting an infinite line $2n$ times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on transfer matrix methods, for the enumeration of plane meanders. While the algorithm has exponential complexity, its rate of growth is much smaller than that of previous algorithms. The algorithm is easily modified to enumerate various systems of closed meanders, semi-meanders, open meanders and many other geometries.

cond-mat.stat-mech↗

arXiv2000

Statistics of lattice animals (polyominoes) and polygons

We have developed an improved algorithm that allows us to enumerate the number of site animals (polyominoes) on the square lattice up to size 46. Analysis of the resulting series yields an improved estimate, $τ= 4.062570(8)$, for the growth constant of lattice animals and confirms to a very high degree of certainty that the generating function has a logarithmic divergence. We prove the bound $τ> 3.90318.$ We also calculate the radius of gyration of both lattice animals and polygons enumerated by area. The analysis of the radius of gyration series yields the estimate $ν= 0.64115(5)$, for both animals and polygons enumerated by area. The mean perimeter of polygons of area $n$ is also calculated. A number of new amplitude estimates are given.

cond-mat.stat-mech↗

arXiv2000

Critical exponents of plane meanders

Meanders form a set of combinatorial problems concerned with the enumeration of self-avoiding loops crossing a line through a given number of points, $n$. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We use a recently developed algorithm, based on transfer matrix methods, to enumerate plane meanders. This allows us to calculate the number of closed meanders up to $n=48$, the number of open meanders up to $n=43$, and the number of semi-meanders up to $n=45$. The analysis of the series yields accurate estimates of both the critical point and critical exponent, and shows that a recent conjecture for the exact value of the semi-meander critical exponent is unlikely to be correct, while the conjectured exponent value for closed and open meanders is not inconsistent with the results from the analysis.

cond-mat.stat-mech↗

arXiv2000

Size and area of square lattice polygons

We use the finite lattice method to calculate the radius of gyration, the first and second area-weighted moments of self-avoiding polygons on the square lattice. The series have been calculated for polygons up to perimeter 82. Analysis of the series yields high accuracy estimates confirming theoretical predictions for the value of the size exponent, $ν=3/4$, and certain universal amplitude combinations. Furthermore, a detailed analysis of the asymptotic form of the series coefficients provide the firmest evidence to date for the existence of a correction-to-scaling exponent, $Δ= 3/2$.

cond-mat.stat-mech↗

arXiv2000

Punctured polygons and polyominoes on the square lattice

We use the finite lattice method to count the number of punctured staircase and self-avoiding polygons with up to three holes on the square lattice. New or radically extended series have been derived for both the perimeter and area generating functions. We show that the critical point is unchanged by a finite number of punctures, and that the critical exponent increases by a fixed amount for each puncture. The increase is 1.5 per puncture when enumerating by perimeter and 1.0 when enumerating by area. A refined estimate of the connective constant for polygons by area is given. A similar set of results is obtained for finitely punctured polyominoes. The exponent increase is proved to be 1.0 per puncture for polyominoes.

cond-mat.stat-mech↗

arXiv1999

Enumerations of plane meanders

A closed plane meander of order n is a closed self-avoiding loop intersecting an infinite line 2n times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on transfer matrix methods, for the enumeration of plane meanders. This allows us to calculate the number of closed meanders up to n=24. The algorithm is easily modified to enumerate various systems of closed meanders, semi-meanders or open meanders.

cond-mat.stat-mech↗

arXiv1999

Low-density series expansions for directed percolation I: A new efficient algorithm with applications to the square lattice

A new algorithm for the derivation of low-density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems. Numerical evidence shows that the computational complexity grows exponentially, but with a growth factor $λ< \protect{\sqrt[8]{2}}$, which is much smaller than the growth factor $λ= \protect{\sqrt[4]{2}}$ of the previous best algorithm. For bond (site) percolation on the directed square lattice the series has been extended to order 171 (158). Analysis of the series yields sharper estimates of the critical points and exponents.

cond-mat.stat-mech↗

arXiv1999

Low-density series expansions for directed percolation II: The square lattice with a wall

A new algorithm for the derivation of low-density expansions has been used to greatly extend the series for moments of the pair-connectedness on the directed square lattice near an impenetrable wall. Analysis of the series yields very accurate estimates for the critical point and exponents. In particular, the estimate for the exponent characterizing the average cluster length near the wall, $τ_1=1.00014(2)$, appears to exclude the conjecture $τ_1=1$. The critical point and the exponents $ν_{\parallel}$ and $ν_{\perp}$ have the same values as for the bulk problem.

cond-mat.stat-mech↗

arXiv1999

Self-avoiding polygons on the square lattice

We have developed an improved algorithm that allows us to enumerate the number of self-avoiding polygons on the square lattice to perimeter length 90. Analysis of the resulting series yields very accurate estimates of the connective constant $μ=2.63815852927(1)$ (biased) and the critical exponent $α= 0.5000005(10)$ (unbiased). The critical point is indistinguishable from a root of the polynomial $581x^4 + 7x^2 - 13 =0.$ An asymptotic expansion for the coefficients is given for all $n.$ There is strong evidence for the absence of any non-analytic correction-to-scaling exponent.

cond-mat.stat-mech↗

arXiv1995

Series expansions of the percolation probability on the directed triangular lattice

We have derived long series expansions of the percolation probability for site, bond and site-bond percolation on the directed triangular lattice. For the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35. For the site-bond problem, which has not been studied before, we have derived the series to order 32. Our estimates of the critical exponent $β$ are in full agreement with results for similar problems on the square lattice, confirming expectations of universality. For the critical probability and exponent we find in the site case: $q_c = 0.4043528 \pm 0.0000010$ and $β= 0.27645 \pm 0.00010$; in the bond case: $q_c = 0.52198\pm 0.00001$ and $β= 0.2769\pm 0.0010$; and in the site-bond case: $q_c = 0.264173 \pm 0.000003$ and $β= 0.2766 \pm 0.0003$. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent $Δ= 1$.

cond-mat↗

arXiv1995

Series expansions of the percolation probability for directed square and honeycomb lattices

We have derived long series expansions of the percolation probability for site and bond percolation on directed square and honeycomb lattices. For the square bond problem we have extended the series from 41 terms to 54, for the square site problem from 16 terms to 37, and for the honeycomb bond problem from 13 terms to 36. Analysis of the series clearly shows that the critical exponent $β$ is the same for all the problems confirming expectations of universality. For the critical probability and exponent we find in the square bond case, $q_c = 0.3552994\pm 0.0000010$, $β= 0.27643\pm 0.00010$, in the square site case $q_c = 0.294515 \pm 0.000005$, $β= 0.2763 \pm 0.0003$, and in the honeycomb bond case $q_c = 0.177143 \pm 0.000002$, $β= 0.2763 \pm 0.0002$. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent $Δ= 1$.

cond-mat↗

arXiv1994

Critical behavior of nonequilibrium models with infinitely many absorbing states

I study the critical behavior of a two-dimensional dimer-trimer lattice model, introduced by Köhler and ben-Avraham [J. Phys. A {\bf 24}, L621 (1991)], for heterogeneous catalysis of the reaction $\frac{1}{2}A_{2}+\frac{1}{3}B_{3} \rightarrow AB$. The model possesses infinitely many absorbing states in which the lattice is saturated by adsorbed particles and reactions cease because only isolated vacancies are left. Results for various critical exponents show that the model exhibits the same critical behavior as directed percolation, contrary to earlier findings by Köhler and ben-Avraham. Together with several other studies, reviewed briefly in this article, this confirms that directed percolation is the generic universality class for models with infinitely many absorbing states.

cond-mat↗

arXiv1994

Critical Exponents for Branching Annihilating Random Walks with an Even Number of Offspring

Recently, Takayasu and Tretyakov [Phys. Rev. Lett. {\bf 68}, 3060 (1992)], studied branching annihilating random walks (BAW) with $n=1$-5 offspring. These models exhibit a continuous phase transition to an absorbing state. For odd $n$ the models belong to the universality class of directed percolation. For even $n$ the particle number is conserved modulo 2 and the critical behavior is not compatible with directed percolation. In this article I study the BAW with $n=4$ using time-dependent simulations and finite-size scaling obtaining precise estimates for various critical exponents. The results are consistent with the conjecture: $β/\nuh = {1/2}$, $\nuv/\nuh = {7/4}$, $γ= 0$, $δ= {2/7}$, $η= 0$, $z = {8/7}$, and $δ_{h} = {9/2}$.

cond-mat↗

arXiv1993

Critical behaviour of a surface reaction model with infinitely many absorbing states

In a recent letter [J. Phys. A26, L801 (1993)], Yaldram et al. studied the critical behaviour of a simple lattice gas model of the CO-NO catalytic reaction. The model exhibits a second order nonequilibrium phase transition from an active state into one out of infinitely many absorbing states. Estimates for the critical exponent $β$ suggested that the model belongs to a new universality class. The results reported in this article contradict this notion, as estimates for various critical exponents show that the model belongs to the universality class of directed percolation.

cond-mat↗