Spatial linguistic surveys often reveal well defined geographical zones where certain linguistic forms are dominant over their alternatives. It has been suggested that these patterns may be understood by analogy with coarsening in models of two dimensional physical systems. Here we investigate this connection by comparing data from the Cambridge Online Survey of World Englishes to the behaviour of a generalised zero temperature Potts model with long range interactions. The relative displacements of linguistically similar population centres reveals enhanced east-west affinity. Cluster analysis reveals three distinct linguistic zones. We find that when the interaction kernel is made anisotropic by stretching along the east-west axis, the model can reproduce the three linguistic zones for all interaction parameters tested. The model results are consistent with a view held by some linguists that, in the USA, language use is, or has been, exchanged or transmitted to a greater extent along the east-west axis than the north-south.

We view random walks as the paths of foraging animals, perhaps searching for food or avoiding predators while forming a mental map of their surroundings. The formation of such maps requires them to memorise the locations they have visited. We model memory using a kernel, proportional to the number of locations recalled as a function of the time since they were first observed. We give exact analytic expressions relating the elongation of the memorised walk to the structure of the memory kernel, and confirm these by simulation. We find that more slowly decaying memories lead to less elongated mental maps.

We give a simple and direct treatment of the strong convergence of quantum random walks to quantum stochastic operator cocycles, via the semigroup decomposition of such cocycles. Our approach also delivers convergence of the pointwise product of quantum random walks to the quantum stochastic Trotter product of the respective limit cocycles, thereby revealing the algebraic structure of the limiting procedure. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.2010 Mathematics Subject Classification. 46L53 (primary); 46N50, 81S25, 82C10, 60F17 (secondary).

We study the spread of a persuasive new idea through a population of continuous-time random walkers in one dimension. The idea spreads via social gatherings involving groups of nearby walkers who act according to a biased "majority rule": After each gathering, the group takes on the new idea if more than a critical fraction 1-ɛ/2<1/2 of them already hold it; otherwise they all reject it. The boundary of a domain where the new idea has taken hold expands as a traveling wave in the density of new idea holders. Our walkers move by Lévy motion, and we compute the wave velocity analytically as a function of the frequency of social gatherings and the exponent of the jump distribution. When this distribution is sufficiently heavy tailed, then, counter to intuition, the idea can propagate faster if social gatherings are held less frequently. When jumps are truncated, a critical gathering frequency can emerge which maximizes propagation velocity. We explore our model by simulation, confirming our analytical results.

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