Transverse-size critical exponent of directed percolation from Yang-Lee zeros of survival probability

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

“…According to our numerical observations, the number of different branches is equal to n − 1 for a given n, meaning that the total number of branches should grow to infinity in the limit of a square lattice (L → ∞, n → ∞). In this respect, patterns of Y-L zeros for non-interacting lattice animals resemble those found recently in a similar study of directed percolation problem [22]. In addition to these n − 1 loci of zeros, the point z = 0 represents a locus of zeros and accumulation point of Y-L zeros; this is a simple consequence of the fact that the partition function allows the simple factorization:…”

“…By analogy with the case of directed percolation [22], we can assume that the first zeros z ∞ n of the grand canonical partition function for animals on a strip of infinite length, and finite but sufficiently large n, follow the power-law behavior…”

“…The procedure includes a free parameter ω, which should be adjusted self-consistently in order to minimize the errors and fluctuations in new sequences generated by repeated applications of the BST algorithm to an initial data sequence. It has been noticed [22,28] that quality of extrapolated values can be often significantly improved by using an accurate and sufficiently long sequence of input data. This can be explained by the fact that extrapolated values become more and more close to each other as calculation proceeds while the algorithm itself includes subtraction of these nearly the same values, which may lead to a significant loss of accuracy.…”

“…We have found that these zeros steadily approach corresponding BST values shown in table 1, and that they in fact match each other (if rounded to 14-digits) after about 10 8 iterations. It is interesting to note that a similar procedure for directed percolation requires much larger precision of input data, which makes iterative computation in that case hardly feasible for large L. Otherwise, convergence properties of BST algorithm in application to directed animals seem to be quite similar to those for directed percolation (interested reader can consult [22] for more detailed discussion and examples).…”

“…Recently, using an exactly solvable model of interacting self-avoiding walks [20], we have shown that study of grand canonical partition function zeros provides useful insights into geometrical critical behavior of the model, which allowed us to get very accurate estimates of size critical exponents in the swollen and compact polymer phases, as well as at the point of its collapse transition. Since the Y-L approach has also been very useful in studies of models of directed percolation [21,22], it is expected that similar analysis should be applicable to the case of directed animals as well.…”