Topics in Occupation Times and Gaussian Free Fields

@inproceedings{Sznitman2012TopicsIO,
  title={Topics in Occupation Times and Gaussian Free Fields},
  author={Alain-Sol Sznitman},
  year={2012}
}
Foreword The following notes grew out of the graduate course " Special topics in probability " , which I gave at ETH Zurich during the Spring term 2011. One of the objectives was to explore the links between occupation times, Gaussian free fields, Poisson gases of Markovian loops, and random interlacements. The stimulating atmosphere during the live lectures was an encouragement to write a fleshed-out version of the handwritten notes, which were handed out during the course. I am immensely… 

Figures from this paper

Lectures on Isomorphism Theorems

These notes originated in a series of lectures I gave in Marseille in May, 2013. I was invited to give an introduction to the isomorphism theorems, originating with Dynkin, which connect Markov local

Generating Galton-Watson trees using random walks and percolation for the Gaussian free field

The study of Gaussian free field level sets on supercritical Galton-Watson trees has been initiated by Ab¨acherli and Sznitman in Ann. Inst. Henri Poincar´e Probab. Stat., 54(1):173–201, 2018 . By

Covariant Symanzik identities

Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles

Inverting the coupling of the signed Gaussian free field with a loop-soup

Lupu introduced a coupling between a random walk loop-soup and a Gaussian free field, where the sign of the field is constant on each cluster of loops. This coupling is a signed version of

Brownian Loops and Conformal Fields

The main topic of these lecture notes is the continuum scaling limit of planar lattice models. One reason why this topic occupies an important place in the theory of probability and mathematical

First passage percolation with long-range correlations and applications to random Schr\"odinger operators

We consider first passage percolation (FPP) with passage times generated by a general class of models with long-range correlations on Z, d ≥ 2, including discrete Gaussian free fields,

Scaling limits, brownian loops, and conformal fields

The main topic of these lecture notes is the continuum scaling limit of planar lattice models. One reason why this topic occupies an important place in the theory of probability and mathematical

Bosonic loop soups and their occupation fields

We consider a model for random loops on graphs which is inspired by the Feynman–Kac formula for the grand canonical partition function of an ideal gas. We associate to this model a corresponding

Level-set percolation for the Gaussian free field on a transient tree

We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type

Topics and problems on favorite sites of random walks

In this article, we study special points of a simple random walk and a Gaussian free field, such as (nearly) favorite points, late points and high points. In section $2$, we extend results of [19]
...

References

SHOWING 1-10 OF 31 REFERENCES

Random walks and electric networks

The goal will be to interpret Polya’s beautiful theorem that a random walker on an infinite street network in d-dimensional space is bound to return to the starting point when d = 2, but has a positive probability of escaping to infinity without returning to the Starting Point when d ≥ 3, and to prove the theorem using techniques from classical electrical theory.

Vacant Set of Random Interlacements and Percolation

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter

Intersections of random walks

We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk.

Conformal loop ensembles: the Markovian characterization and the loop-soup construction

For random collections of self-avoiding loops in two-dimensional domains, we dene a simple and natural conformal restriction property that is conjecturally satised by the scaling limits of interfaces

Gaussian and non-Gaussian random fields associated with Markov processes

Interlacement percolation on transient weighted graphs

In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [14], to the more general setting of transient weighted graphs. We prove the Harris-FKG

An Introduction to Probability Theory and Its Applications

Thank you for reading an introduction to probability theory and its applications vol 2. As you may know, people have look numerous times for their favorite novels like this an introduction to

Markov processes as a tool in field theory

An isomorphism theorem for random interlacements

We consider continuous-time random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian

Continuous martingales and Brownian motion

0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.-