Statistical mechanics and quantum fields

在仔细调节温度和压力下，水的液相和蒸汽相具有相同的密度，并且水在两相之间的所有长度尺度上随机波动。 在所有尺度上波动的系统被称为处于“临界点”。 由于猜想许多不同的系统可以统一在一个小的列表，这样的法律，统计力学的主要目标是计算概率定律的规模不变的波动。 1982年获得诺贝尔奖的肯·威尔逊开始了一个名为“重正化小组”的计划，旨在建立这些尺度不变概率定律的数学理论。 这项建议将推进威尔逊的计划。 由于临界点将许多系统结合在一起，因此从最简单的代表性情况开始是合理的，其中之一是自避行走。 自避行走是尺度不变的，因为从远处看去，很长一段自避行走的每一步都是不可见的，所能看到的只是一条在所有尺度上都有“摆动”的连续曲线。因此，我们首先关注的问题是，自我回避行走的端到端距离如何作为步数的函数而增长。

At carefully adjusted temperature and pressure liquid and steam phases of water have the same density, and water randomly fluctuates on all length scales between the two phases. Systems that fluctuate on all scales are said to be at a "critical point". Motivated by a conjecture that many different systems can be united under a small list of such laws, a primary goal of statistical mechanics is to calculate probability laws of scale invariant fluctuations. The Nobel prize winning work of Ken Wilson in 1982 began a program called the "Renormalisation Group" aimed at the creation of a mathematical theory of these scale invariant probability laws. This proposal will advance Wilson's program. Since critical points unite many systems it is reasonable to start with the simplest representative cases, one of which is self-avoiding walk. A self-avoiding walk is scale invariant in the sense that individual steps of a very long self-avoiding walk seen from far away are invisible and all that can be seen is a continuous curve that has "wiggles" on all scales. Thus we are initially focused on the question of how the end-to-end distance of a self-avoiding walk grows as a function of the number of steps.