Counting planar random walk holes

We study two variants of the notion of holes formed by planar simple random walk of time duration $2n$ and the areas associated with them. We prove in both cases that the number of holes of area greater than $A(n)$, where $\{A(n)\}$ is an increasing sequence, is, up to a logarithmic correction term, asymptotic to $n\cdot A(n)^{-1}$ for a range of large holes, thus confirming an observation by Mandelbrot. A consequence is that the largest hole has an area which is logarithmically asymptotic to $n$. We also discuss the different exponent of 5/3 observed by Mandelbrot for small holes.Comment: Published in at http://dx.doi.org/10.1214/009117907000000204 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

OCTOBER 28, 1918. REWRITING OR OVERLAYERING OF CZECH HISTORICAL MEMORY?

October 28, 1918 is the Czech Republic state holiday whose historical memory is a combination of Czech, Czechoslovak, and Central European 20th century history. On this date in Prague, the Czechoslovak Republic was proclaimed, and its fi rst law was passed. Th e events in Prague were part of the complex and long-lasting fall of the Habsburg monarchy and the creation of its successor states, in which national, state-forming, and ideological (e.g. Bolshevist) aspects were interwoven. Accordingly, we can speak of Czech, Slovak, (Czecho)-German, Hungarian, Polish and Rusinian October 28s. As the only state holiday (with an interruption in the period of the Nazi occupation), it was intended to act as the chief connecting and uniting holiday for the CSR state identifi cation; it was to strengthen ‘Czechoslovakism’. Its annual celebrations were associated with a series of rituals not only for the Czechs themselves but, over time and to varying degrees, also for the other nationalities living in the CSR: primarily the Slovaks and the Rusinians were seen to truly accept the ceremonial day. Th e Nazi occupying power was successful only insofar as it forced October 28, 1918 into private crypto-commemoration, while naturally it was celebrated by the resistance movement. Th e Communist regime tried to ‘rewrite’ October 28 in the spirit of social revolution, treating it as the precursor of its political victory aft er 1945 and in particular aft er 1948. It was to be commemorated as the Nationalization Day (of key industries in 1945) in direct relation with the liberation of the CSR by the Soviet Army (alone!) in 1945. Finally, the Communists att empted to force it out of the collective memory through its offi cial non-observance as a remembrance of 1918, and by designating it, in 1975-1988, as a signifi cant, but still a working, day. However, the memory of the Establishment of the Republic refused to be suppressed, as was evidenced in a particularly strong manner in the demonstrations of 1968, 1988 and, crucially, of 1989. All att empts at ‘rewriting’ this holiday in the spirit of ideologies failed in the end, although during the 1938/39 to 1989/92 period spontaneous public celebrations were successfully repressed to a signifi cant degree by means of the political manipulation of Czech/Czechoslovak history