Exploring the Physical World

Random Walks in Random Environments

Weizmann Institute Annual Report • TAGS: Mathematics

Prof. Ofer Zeitouni is the incumbent of the Herman P. Taubman Professorial Chair of Mathematics.

Prof. Ofer Zeitouni

What happens when a person strolling along an intersecting path chooses directions with a roll of the dice?

A wanderer strolling along an unfamiliar branching path faces a dilemma each time he reaches an intersection. Which way should he turn? Where should he be the next moment? Tracing the path of this random wanderer, one can try to make sense of the basic, underlying rules that direct his course. For instance, what happens if he simply chooses his direction on the basis of rolling dice?

Is our wanderer assured of getting back to his starting point? Prof. Ofer Zeitouni of the Faculty of Mathematics and Computer Sciences says that if the system is a two-dimensional lattice, then rolling the dice will surely bring him, sooner or later, back to his starting point. But in systems of three or more dimensions, this is not true.

The situation becomes more complex if we weight the dice, so that the odds for the different outcomes are not equal. Now, the question of whether the wanderer is sure to eventually return to his starting point is open even for lower dimensions (two or higher).

Zeitouni: "What appears to be a purely theoretical question may, in fact, help us understand the pathways of electrons in crystals. The impurities in the crystals behave somewhat like the weighted dice." Another important aspect of the analysis relates to the amount of time our wanderer spends in a given area, which can reveal the nature of his movement over time.

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Prof. Ofer Zeitouni is the incumbent of the Herman P. Taubman Professorial Chair of Mathematics.