Abstract. We calculate very long low-and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the five-particle contribution χ (5) and six-particle contribution χ (6) . These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. The series for χ (low-and high-temperature regime), χ (5) and χ (6) are now extended to 2000 terms. In addition, for χ (5) , 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by χ (5) modulo a prime.A diff-Padé analysis of the 2000 terms series for χ (5) and χ (6) confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n-particle components of the susceptibility, up to a small set of "additional" singularities. The exponents at all the singularities of the Fuchsian linear ODE of χ (5) and the (as yet unknown) ODE of χ (6) are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of χ (5) , and w 2 = 1/8 for the ODE of χ (6) , which are not singularities of the "physical" χ (5) and χ (6) , that is to say the series-solutions of the ODE's which are analytic at w = 0.Furthermore, analysis of the long series for χ (5) (and χ (6) ) combined with the corresponding long series for the full susceptibility χ yields previously conjectured singularities in some χ (n) , n ≥ 7. The exponents at all these singularities are also seen to be rational numbers.We also present a mechanism of resummation of the logarithmic singularities of the χ (n) leading to the known power-law critical behaviour occurring in the full χ, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility χ.

Abstract. We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (5) , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination ofχ (1) andχ (3) can be removed fromχ (5) and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the "depleted" differential operator and it is shown to be equivalent to the symmetric fourth power of L E , the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order termsχ (3) andχ (4) . We conjecture that a linear differential operator equivalent to a symmetric (n − 1)-th power of L E occurs as a left-most factor in the minimal order linear differential operators for allχ (n) 's.

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 < 8 √ 2, which is much smaller than the growth factor λ = 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.

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I study a nonequilibrium lattice model, the pair contact process, in which pairs of particles annihilate with probability p or else create a particle at a vacant nearest neighbor. The model exhibits a continuous phase transition from an active state, with an ongoing production of particles, to an absorbing state without pairs. The model has infinitely many absorbing states. Computer simulations in ID yielded critical exponents consistent with directed percolation, for the first time placing a model with infinitely many absorbing states firmly in the directed-percolation universality class.PACS numbers: 05.50.+q, 05.70.Ln The study of nonequilibrium many-particle systems is an important problem in many branches of physics, chemistry, biology, and even sociology [1,2]. Much attention has been given to nonequilibrium models exhibiting a continuous phase transition from an active steady state to a unique absorbing state (a state in which the system is trapped). One of the major achievements in the study of nonequilibrium phase transitions is the discovery that a wide variety of models exhibiting this kind of transition belong to the same universality class. Among these models the best known are probably the contact process [3-5], Schlogl's first and second models [6-8], directed percolation (DP) [9-12], Reggeon field theory (RFT) [13][14][15][16], and the ZGB (Ziflf-Gulari-Barshad) model [17][18][19]. The study of these and many other models [20][21][22][23][24][25][26] demonstrates the robustness of DP critical behavior in spite of quite dramatic differences in the evolution rules of the various models. Presently there is thus substantial evidence in favor of the hypothesis that models with a scalar order parameter exhibiting a continuous transition to a unique absorbing state generically belong to the universality class of directed percolation. This DP conjecture was first put forward by Grassberger [8] and Janssen [7] and later extended by Grinstein, Lai, and Browne [18] to multicomponent models such as the ZGB model.Whereas the universality of DP critical behavior for models with a single absorbing state seems well established, the study of models with more than one absorbing state is still in its very beginning. That models with more than one absorbing state can exhibit new critical behavior was first demonstrated by Grassberger, Krause, and von der Twer [27] in a study of a model involving the processes A'-• 3X and 2X-^ 0. This model is very similar to a class of models known as branching annihilating walks (BAW's) [24]. In the BAW a particle jumps, with probability p, to a nearest neighbor and if this site is occupied both particles annihilate. With probability 1 -p the particle produces n offspring which are placed on the neighboring sites. If an offspring is created on a site which is already occupied, it annihilates with the occupying particle leaving an empty site. For n even these models have non-DP behavior [25] whereas the behavior for n odd is compatible with DP [26]. Note that in both the mode...

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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.

We consider the Fuchsian linear differential equation obtained (modulo a prime) for˜χfor˜ for˜χ (5) , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of˜χof˜ of˜χ (1) and˜χand˜ and˜χ (3) can be removed from˜χfrom˜ from˜χ (5) and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the "depleted" differential operator and it is shown to be equivalent to the symmetric fourth power of L E , the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order terms˜χterms˜ terms˜χ (3) and˜χand˜ and˜χ (4). We conjecture that a linear differential operator equivalent to a symmetric (n − 1)-th power of L E occurs as a left-most factor in the minimal order linear differential operators for all˜χall˜ all˜χ (n) 's.