We consider the probability that a two-dimensional random walk starting at the origin never returns to the half-line (-∞,0]×{0} before time n. It is proved that for aperiodic random walk with mean zero and finite 2+∂(>2)-th absolute moment, this probability times n^<1/4> converges to some positive constant c^* as n→∞. Our investigation of this probability is motivated by the study of the DLA model. In the first half of this paper, we give an explanation of the DLA model and its related works. In the latter half, we outline the proof of our results.

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A class of one dimensional multi-state cellular automata is introduced. Their dynamics is described as a motion of particles and anti-particles that undergo pair-annihilation and creation through neutral bound states. The particles and anti-particles can form special configurations called solitons. Solitons remain stable under collisions exhibiting only phase shifts and exchange of internal degrees of freedom. The automata are integrable and associated with quantum groups and related mathematical structures such as crystal bases. They provide a unified perspective to understand the symmetry and integrability of the automata incorporating the earlier approach to the box-ball system by ultradiscretization.

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For cellular automata a brief survey from the view of ergodic theory and some entropy analysis is given. The notion of invariant measure, ergodicity and entropy is introduced and the euivalence between shift endomorphysms and cellular automata is shown. Though there are many literatures with respect to the surjective case, there is difficulty in the case of non surjective case, that has transient states. The author give some methods to study the transient dynamics.

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One of the main features of quantum probability(=noncommutative probability) is the diversity of notions of 'independence' for noncommutative random variables. Besides the three fundamental examples of universal independence (tensor, free and Boolean independence), there is another example called 'monotone independence' which was introduced and studied by the author. We give a brief review on 'monotone probability' which can be developed based on the notion of monotone independence. Especially we present the monotonic analogue of central limit theorem, law of small numbers, convolution, infinite divisibility and Levy-Hincin formula. Furthermore, we give a classification theorem for universal notions of independence.

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Flows consisted of more than two different phases of gas, liquid or solid are called multiphase flows. In this paper, the numerical methods of dispersed multiphase flows, which are the flows containing a large number of bubbles, drops or particles, are discussed. The fundamental characteristic of dispersed flows is its multi-scale structure. It contains the wide variety of scales, starting from the molecular scale to that of the large structure in the flow. For example, in the case of bubbly flow, it is known that very small amount of contaminant can drastically change the rising velocity of a bubble and it ends up with the change of macro-scale structure of the whole bubbly now through the micro-, meso- and macro- scale interactions. In this paper, the numerical methods to capture these multi-scale flow structures and the related problems are discussed.

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