# 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…

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## References

SHOWING 1-10 OF 31 REFERENCES

### Random walks and electric networks

- Mathematics
- 1984

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

- Mathematics
- 2007

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

- Mathematics
- 2011

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

- Mathematics
- 2010

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…

### Interlacement percolation on transient weighted graphs

- Mathematics
- 2009

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

- Mathematics
- 1950

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…

### An isomorphism theorem for random interlacements

- Mathematics
- 2011

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

- Mathematics
- 1990

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