Distances in random structures

相变在自然界中无处不在——它们是如此普遍，以至于我们几乎看不见它们的奇怪之处。为什么水会在0摄氏度时突然变成冰，而不是随着温度的下降而逐渐变厚？为什么铁在770度时磁化而在769度时不磁化？为这些现象发展数学理论是这个时代最困难和最令人兴奋的数学挑战之一。

Phase transitions are ubiquitous in the natural world -- they are so commonplace that their strangeness is almost invisible to us. Why should water turn suddenly to ice at 0 degrees Celsius, rather than becoming gradually thicker as the temperature drops? Why will iron magnetize at 770 degrees but not at 769? Developing a mathematical theory for such phenomena is one of the most difficult and exciting mathematical challenges of the age. One of the classical mathematical models of phase transitions is known as percolation, which models defects in an ordered material. For example, if microscopic faults develop within a crystal, whose molecular bonds form a lattice structure, one may ask whether the material will remain structurally sound or whether macroscopic cracks will develop. It turns out that there is a critical density of faults, just below which the material remains sound, and just above which the crystal will disintegrate. The percolation phase transition -- and indeed, phase transitions in many other systems -- exhibits the intriguing phenomenon of universality, in which the precise macroscopic structure of the model does not seem to affect the qualitative macroscopic behaviour of the phase transition. Universality has been rigorously established in very few cases, and the mathematical treatment of universality is one of the major driving problems of the subject. One of my primary research interests, and a key focus of this proposal, is studying percolation phase transitions in high dimensional systems, with an ultimate aim of establishing universality. High dimensional systems appear throughout the physical and biological sciences: in this context the dimension of a system should be thought of as its number of degrees of freedom, or in other words the number of parameters that must be specified to fully describe the state of the system. For example, in classical Euclidean mechanics, to describe the state of a particle requires six parameters -- three to describe spatial position and three to specify velocity -- so one particle already forms a six-dimensional system.