Abstract:

Suppose that you pick a random name in the greater New York telephone directory and ask yourself the question: Is it possible that this person and I have an acquaintance in common? Or, if not: Does this person know somebody with whom I have an acquaintance in common? Or, in general: How many acquaintances are we typically away from each other? A famous experiment by S. Milgram from 1967 shows that six people are (typically) enough to connect two average Americans in an acquaintance chain. An unrelated problem---particularly suitable for the time spent on the Santa Monica beach---is as follows: Suppose you want to know everything about sandpiles. To get some insight you grab a handful of sand and start pouring it onto a flat surface. First, the sand spreads out evenly but then there comes a moment where you begin to create a visible pile. The sandpile grows gradually steeper until, suddenly, an avalanche sweeps along its side. The sandpile is now less steep but you keep pouring until the next avalanche comes and so on. And you wonder: Is there a critical slope for which the sandpile is stable and yet the insertion of a few more grains will trigger an avalanche? In my talk I will show how these ``real-world'' phenomena can be modeled by using the techniques of probability theory. A common link between the above problems will be provided by percolation---a problem interesting in its own right.