Probability

A random nxm matrix gives a random linear transformation from $\mathbb{Z}^m$ to $\mathbb{Z}^n$ (between vectors with integral coordinates). Asking for the probability that such a map is injective is a question of the non-vanishing of determinants. In this talk, we discuss the probability that such a map is surjective, which is a more subtle integral question. We show that when $m=n+u$, for $u$ at least 1, as n goes to infinity, the surjectivity probability is a non-zero product of inverse values of the Riemann zeta function. This probability is universal, i.e. we prove that it does not depend on the distribution from which you choose independent entries of the matrix, and this probability also arises in the Cohen-Lenstra heuristics predicting the distribution of class groups of real quadratic fields. This talk is on joint work with Hoi Nguyen.

N.B. Due to a problem with the microphone, the audio for this recording is almost entirely missing. It is displayed here in the hope that the whiteboard material is still useful.

The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist. The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

• Lectures in this course: 1,
  2,
  3,
  4,
  5,
  6,
  7,
  8,
  9,
  10,
  11,
  12,
  13,
  14,
  15,
  16
• Notes

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by \u221a 2 + \u221a 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.

• Lectures in this course: 1,
  2,
  3,
  4,
  5,
  6,
  7,
  8,
  9,
  10,
  11,
  12,
  13,
  14,
  15,
  16
• Notes

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by √ 2 + √ 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.

• Lectures in this course: 1,
  2,
  3,
  4,
  5,
  6,
  7,
  8,
  9,
  10,
  11,
  12,
  13,
  14,
  15,
  16
• Notes