Classifying Cantor Sets by their Fractal Dimensions

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.Comment: 10 pages, revised version. To appear in Proceedings of the AMS

The random pinning model with correlated disorder given by a renewal set

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $\alpha$ > 0, when the correlated sequence is given by another independent renewal set with loop exponent $\alpha$ > 0. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case $\alpha$ > 2 (summable correlations), disorder is irrelevant if $\alpha$ 1/2, which extends the Harris criterion for independent disorder. The case $\alpha$ $\in$ (1, 2) (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha$ > 1/ $\alpha$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\alpha$ $\in$ (0, 1) is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition