# Probability

## Extrema of 2D Discrete Gaussian Free Field - Lecture 13

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.

## Graphical approach to lattice spin models - Lecture 13

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.

## SPDEs on graphs: an asymptotic approach - Lecture 3

I will introduce a new class of SPDEs defined on graphs, obtained as the limit of suitable SPDEs, defined on two-dimensional domains and depending on some parameters. I will do this presenting two examples. The first example is given by some SPDEs defined on narrow channels with wings. As the width of the channel goes to zero the solutions converge to the solution of a suitable SPDE defined on the graph that can be obtained by identifying all points on the same cross section of the tubular domain. The second example is given by the analysis of the fast advection asymptotics for some stochastic reaction-diffusion-advection equations defined on the plane. To describe the asymptotics, I will consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.

## Extrema of 2D Discrete Gaussian Free Field - Lecture 12

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.

## Graphical approach to lattice spin models - Lecture 12

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.

## SPDEs on graphs: an asymptotic approach - Lecture 2

I will introduce a new class of SPDEs defined on graphs, obtained as the limit of suitable SPDEs, defined on two-dimensional domains and depending on some parameters. I will do this presenting two examples. The first example is given by some SPDEs defined on narrow channels with wings. As the width of the channel goes to zero the solutions converge to the solution of a suitable SPDE defined on the graph that can be obtained by identifying all points on the same cross section of the tubular domain. The second example is given by the analysis of the fast advection asymptotics for some stochastic reaction-diffusion-advection equations defined on the plane. To describe the asymptotics, I will consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.

## Graphical approach to lattice spin models - Lecture 11

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.

## Extrema of 2D Discrete Gaussian Free Field - Lecture 11

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.

## SPDEs on graphs: an asymptotic approach - Lecture 1

I will introduce a new class of SPDEs defined on graphs, obtained as the limit of suitable SPDEs, defined on two-dimensional domains and depending on some parameters. I will do this presenting two examples. The first example is given by some SPDEs defined on narrow channels with wings. As the width of the channel goes to zero the solutions converge to the solution of a suitable SPDE defined on the graph that can be obtained by identifying all points on the same cross section of the tubular domain. The second example is given by the analysis of the fast advection asymptotics for some stochastic reaction-diffusion-advection equations defined on the plane. To describe the asymptotics, I will consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.

## Graphical approach to lattice spin models - Lecture 10