On the number of ones in outcome sequence of extended Pohl generator

Formulas for distributions of number of ones (non-zeroes) in the cycle of the output sequence of generalized binary Pohl generator are obtained. Limit theorems for these distributions are derived in the case when the lengths of registers are coprime and tend to infinity, the contents of different registers are independent, but cell contents within each register may be dependent. The consequences of these theorems are given for the case when the contents of cells are independent random variables having equiprobable distribution on {0, 1}.

The Problem of Predecessors on Spanning Trees

We consider the equiprobable distribution of spanning trees on the square lattice. All bonds of each tree can be oriented uniquely with respect to an arbitrary chosen site called the root. The problem of predecessors is to find the probability that a path along the oriented bonds passes sequentially fixed sites i and j. The conformal field theory for the Potts model predicts the fractal dimension of the path to be 5/4. Using this result, we show that the probability in the predecessors problem for two sites separated by large distance r decreases as P(r) ∼ r −3/4. If sites i and j are nearest neighbors on the square lattice, the probability P(1) = 5/16 can be found from the analytical theory developed for the sandpile model. The known equivalence between the loop erased random walk (LERW) and the directed path on the spanning tree states that P(1) is the probability for the LERW started at i to reach the neighboring site j. By analogy with the self-avoiding walk, P(1) can be called the return probability. Extensive Monte-Carlo simulations confirm the theoretical predictions.

Many Entropies, Many Disorders

To overcome the deficits of entropy as a measure for disorder when the number of states available to a system can change, Landsberg defined “disorder” as the entropy normalized to the maximum entropy. In the simplest cases, the maximum entropy is that of the equiprobable distribution, corresponding to a completely random system. However, depending on the question being asked and on system constraints, this absolute maximum entropy may not be the proper maximum entropy. To assess the effects of interactions on the “disorder” of a 1-dimensional spin system, the correct maximum entropy is that of the paramagnet (no interactions) with the same net magnetization; for a non-equilibrium system the proper maximum entropy may be that of the corresponding equilibrium system; and for hierarchical structures, an appropriate maximum entropy for a given level of the hierarchy is that of the system which is maximally random, subject to constraints deriving from the next lower level. Considerations of these examples leads us to introduce the “equivalent random system”: that system which is maximally random consistent with any constraints and with the question being asked. It is the entropy of the “equivalent random system” which should be taken as the maximum entropy in Landsberg's “disorder”.

On asymmetries exhibiting a near-equiprobable distribution of directions

In his target article as well as in other writings, Denenberg presents a view of lateralization with which I fundamentally disagree: namely, that an affirmation of lateralization in a population is to be based primarily, if not exclusively, on observing a nonequiprobable distribution of asymmetric forms in that population.