PHASE CHANGES IN RANDOM RECURSIVE STRUCTURES AND ALGORITHMS

Abstract

A brief survey, based mainly on my recent work with coauthors, is given of the different types of phase changes (or transitions) appearing in random discrete structures and in analysis of algorithms with a recursive character. Phase-transitions are important tools because they make it easy to see two things in one way or one thing in two ways. (quoted from the page A book called “n”). One of the most widely known phenomena of phase changes (of matters) is that water may change its state (to ice or to steam) when the underlying temperature varies. For mathematical functions (or structures or objects), we refer to “phase change” when there is a change of properties under varying parameters. When the phase change phenomenon is observed or discovered, the main problems are usually: – Where does the phase change? – How to describe the change or transition of phase? – Why does the change occur? Is there any intuitive interpretation? – Are there further phase changes in the transition range? and why? how? The simplest example of a phase change1 is the classical central limit theorem where the standard normal distribution is used to bridge the two extremes “event unlikely to happen” and “event happens almost always.” This viewpoint offers several advantages. First, it makes the usual statement of central limit theorems more concrete and physical; second, its quantitative refinement from the 0-1 law or the law of large numbers becomes transparent; third, it makes the notion of “scaling window” clearer since intuitively the higher the resolution of a telescope, the tinier image or object one can ∗Most materials of this paper appeared in my Chinese survey paper [24]. We use mostly the term “phase change” instead of the more common “phase transition” in this paper since there is an obvious notion of discreteness in our problems.