[KT95a]

Two-neighbour stochastic cellular automata (SCA) are the set of one-dimensional discrete time interacting particle systems with two parameters, which show non-equilibrium phase transitions from the extinction phase to the survival phase. The phase diagram was first studied by Kinzel using a numerical method called transfer-matrix scaling. For some parameter region the processes can be defined as directed percolation models on the spatio-temporal plane and the bond and site directed percolation models are included as special cases. Extending the argument of Dhar, Barma and Phani originally given for bond directed percolation, we introduce diode-resistor percolation models which are the planar lattice dual of the SCA and give rigorous lower bounds for the critical line. In special cases, our results give 0.6885 leq alpha_c and 0.6261 leq beta_c, where alpha_c and beta_c denote the critical probabilities of the site and bond directed percolation models on the square lattice. Combining the upper bound for the critical line recently proved by Liggett, we summarize the rigorous results for the phase diagram of the systems. Results of computer simulation are also shown.

[KK96a]

A new mean-field approximation is proposed for the sandpile model of Bak, Tang and Wiesenfeld and the distribution of heights in the self-organized critical state is calculated. We treat several series of successive topples to approximate various avalanches. In contrast to the previous mean-filed theory given by Tang and Bak, which assumes the stationary condition among topples in an avalanche, our theory takes into account long-term behavior of the system. The results are in good agreement with the values estimated by computer simulations even in low dimensions d=2 and 3. Higher approximations obtained by including local correlations or by imposing a reducibility condition, which is used in the cluster variation method, are also shown.

[IK96a]

We regard the bond directed percolation on a square lattice as a discrete-time Markov process of a one-dimensional interacting particle system. The coefficients in series expansion of the probability P_{n,m} of having m particles at time n-1 are studied. We derive the difference equations for the first and the second series of coefficients and prove that these coefficients are expressed using the ballot numbers, whose special cases are known as the Catalan numbers. As a corollary of our results, we prove a part of the conjecture by Baxter and Guttmann that the correction terms are expressed as rational functions of the Catalan numbers. We also give approximations for the percolation probability using the present formula.

[KI96a]

The asymmetric directed-bond percolation (ADBP) problem with an asymmetric parameter k is introduced and some rigorous results are given concerning a series expansion of the percolation probability on the square lattice. It is shown that the first correction term d_{n,1}(k) is expressed by Gauss's hypergeometric series with a variable k. Since the ADBP includes the ordinary directed bond-percolation as a special case with k=1, our results give another proof for Baxter-Guttmann's conjecture that d_{n,1}(1) is given by the Catalan number, which was recently proved by Bousquet-Melou. Direct calculations on finite lattices are performed and combining them with the present results determines the first 12 terms of the series expansion for the percolation probability of the ADBP on the square lattice.

[KI96b]

The directed site percolation probability has been studied by series expansions. Onody and Neves extended the series and conjectured that the so-called first correction term is given by the formula d_{n,1}=(3 n)!/n! (2n+1)! for the square lattice. Recently this conjecture was proved by Bousquet-Melou by identifying d_{n,1} with the total number of diagonally convex directed animals of height n. In the present paper we consider a more specified quantity, alpha_{n,m}= the number of diagonally convex directed animals with m sites at the height n. We give an explicit expression for alpha_{n,m} and show that it is identified with a coefficient appearing in the series expansion for the directed site percolation. As a corollary, the first correction-term formula of Onody and Neves is derived. The generating function of alpha_{n,m} is calculated and some summation formulae are given.

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