Exponential growth of ponds in invasion percolation on regular trees

In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. The tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster, with a logarithmic correction

Scaling limit of the invasion percolation cluster on a regular tree

We prove existence of the scaling limit of the invasion percolation cluster (IPC) on a regular tree. The limit is a random real tree with a single end. The contour and height functions of the limit are described as certain diffusive stochastic processes. This convergence allows us to recover and make precise certain asymptotic results for the IPC. In particular, we relate the limit of the rescaled level sets of the IPC to the local time of the scaled height function.Comment: Published in at http://dx.doi.org/10.1214/11-AOP731 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Metaphysics of the Moral Imagination: John Ruskin\u27s Realism, Revisited

The Victorian philosopher John Ruskin is primarily remembered for his political writing, as a forerunner of what we would today call Christian Socialist politics. In aesthetic circles, he is also often considered something of a punchline: a stuffy conservative who represents the worst vagaries of his day, an enemy of abstraction. Ruskin thus has a double-being in cultural memory: both an admired social reformer and a laughingstock art critic. These views of Ruskin can be potentially reconciled by showing how his critics have misunderstood his aesthetic philosophy. Ruskin is often described as an aesthetic realist, the view on which art must represent the world—i.e a painted tree must closely resemble its real-life counterpart. But Ruskin is not an aesthetic realist. He is a moral realist, who argues good art will be of service to its society by representing a rightly ordered ethical view of reality. In this way, art for Ruskin serves a reformist purpose just like his environmental and labor advocacy. Art is the wing of this social project manifested by the imagination, and requires rightly-ordered artists to perform it properly. Ruskin argues inspiration comes from a transcendent moral outside of the artist, which is then refracted through the artist’s own ethical temperament to create a work of varying moral quality. I conclude by arguing that because Ruskin’s moral realism is not an aesthetic realism, it leaves open new possibilities of understanding the relationship of social class to artistic production

Invasion percolation on regular trees

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its $r$-point function for any $r\geq2$ and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally. A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.Comment: Published in at http://dx.doi.org/10.1214/07-AOP346 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org