Topics in Occupation Times and Gaussian Free Fields pdf

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Alain-Sol Sznitman

Topics in

Occupation Times and

Gaussian Free Fields

Author:

Alain-Sol Sznitman

Departement Mathematik

ETH Zürich

Rämistrasse 101

8092 Zürich

Switzerland

2010 Mathematics Subject Classification: 60K35, 60J27, 60G15, 82B41

Key words: occupation times, Gaussian free field, Markovian loop, random interlacements

ISBN 978-3-03719-109-5

The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,

and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

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9 8 7 6 5 4 3 2 1

Preface

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 hand￾written notes, which were handed out during the course. I am immensely grateful to

Pierre-François Rodriguez, Artëm Sapozhnikov, Balázs Ráth, Alexander Drewitz,

and David Belius, for their numerous comments on the successive versions of these

notes. Support by the European Research Council through grant ERC-2009-AdG

245728-RWPERCRI is thankfully acknowledged.

Contents

Preface ...................................... v

Introduction ................................... 1

1 Generalities ................................. 5

1.1 The set-up ............................... 5

1.2 The Markov chain X: (with jump rate 1) ............... 7

1.3 Some potential theory ......................... 10

1.4 Feynman–Kac formula ......................... 23

1.5 Local times ............................... 25

1.6 The Markov chain Xx: (with variable jump rate) ............ 26

2 Isomorphism theorems ........................... 31

2.1 The Gaussian free field ........................ 31

2.2 The measures Px;y ........................... 35

2.3 Isomorphism theorems ......................... 41

2.4 Generalized Ray–Knight theorems .................. 45

3 The Markovian loop ............................. 61

3.1 Rooted loops and the measure r on rooted loops .......... 61

3.2 Pointed loops and the measure p on pointed loops ......... 70

3.3 Restriction property .......................... 74

3.4 Local times ............................... 75

3.5 Unrooted loops and the measure on unrooted loops ........ 82

4 Poisson gas of Markovian loops ...................... 85

4.1 Poisson point measures on unrooted loops .............. 85

4.2 Occupation field ............................ 87

4.3 Symanzik’s representation formula .................. 91

4.4 Some identities ............................. 95

4.5 Some links between Markovian loops and random interlacements . . 100

References .................................... 111

Index ...................................... 113

Introduction

This set of notes explores some of the links between occupation times and Gaussian

processes. Notably they bring into play certain isomorphism theorems going back to

Dynkin [4], [5] as well as certain Poisson point processes of Markovian loops, which

originated in physics through the work of Symanzik [26]. More recently such Poisson

gases of Markovian loops have reappeared in the context of the “Brownian loop soup”

of Lawler and Werner [16] and are related to the so-called “random interlacements”,

see Sznitman [27]. In particular they have been extensively investigated by Le Jan

[17], [18].

A convenient set-up to develop this circle of ideas consists in the consideration

of a finite connected graph E endowed with positive weights and a non-degenerate

killing measure. One can then associate to these data a continuous-time Markov chain

Xxt, t 0, on E, with variable jump rates, which dies after a finite time due to the

killing measure, as well as

the Green density g.x; y/, x; y 2 E, (0.1)

(which is positive and symmetric),

the local times Lxx

t D

Z t

0

1fXxs D xg ds; t 0, x 2 E. (0.2)

In fact g.; / is a positive definite function on E  E, and one can define a centered

Gaussian process 'x, x 2 E, such that

cov.'x; 'y/.D EŒ'x'y/ D g.x; y/; for x; y 2 E. (0.3)

This is the so-called Gaussian free field.

It turns out that 1

2 '2

z , z 2 E, and Lxz

1, z 2 E, have intricate relationships. For

instance Dynkin’s isomorphism theorem states in our context that for any x; y 2 E,

Lxz

1 C 1

2 '2

z

z2E under Px;y ˝ P G (0.4)

has the “same law” as

1

2 .'2

z /z2E under 'x'y P G, (0.5)

where Px;y stands for the (non-normalized) h-transform of our basic Markov chain,

with the choice h./ D g.;y/, starting from the point x, and P G for the law of the

Gaussian field 'z, z 2 E.

2 Introduction

Eisenbaum’s isomorphism theorem, which appeared in [7], does not involve h￾transforms and states in our context that for any x 2 E, s 6D 0,

Lxz

1 C 1

2 .'z C s/2

z2E under Px ˝ P G (0.6)

has the “same law” as

1

2 .'z C s/2



z2E

under 

1 C 'x

s



P G. (0.7)

The above isomorphism theorems are also closely linked to the topic of theorems of

Ray–Knight type, see Eisenbaum [6], and Chapters 2 and 8 of Marcus–Rosen [19].

Originally, see [13], [21], such theorems came as a description of the Markovian

character in the space variable of Brownian local times evaluated at certain random

times. More recently, the Gaussian aspects and the relation with the isomorphism

theorems have gained prominence, see [8], and [19].

Interestingly, Dynkin’s isomorphism theorem has its roots in mathematical physics.

It grew out of the investigation by Dynkin in [4] of a probabilistic representation for￾mula for the moments of certain random fields in terms of a Poissonian gas of loops

interacting with Markovian paths, which appeared in Brydges–Fröhlich–Spencer [2],

and was based on the work of Symanzik [26].

The Poisson point gas of loops in question is a Poisson point process on the state

space of loops on E modulo time-shift. Its intensity measure is a multiple ˛ of the

image of a certain measure rooted, under the canonical map for the equivalence

relation identifying rooted loops  that only differ by a time-shift. This measure

rooted is the -finite measure on rooted loops defined by

rooted.d / D P

x2E

Z 1

0

Qt

x;x.d / dt

t ; (0.8)

where Qt

x;x is the image of 1fXt D xgPx under .Xs/0st, if X: stands for the

Markov chain on E with jump rates equal to 1 attached to the weights and killing

measure we have chosen on E.

The random fields on E alluded to above, are motivated by models of Euclidean

quantum field theory, see [11], and are for instance of the following kind:

hF .'/i D Z

RE

F .'/ e 1

2E.';'/ Q

x2E

h

'2

x

2



d'x

. Z

RE

e 1

2E.';'/ Q

x2E

h

'2

x

2



d'x

(0.9)

with

h.u/ D

Z 1

0

evud.v/, u 0, with  a probability distribution on RC;

Introduction 3

and E.'; '/ the energy of the function ' corresponding to the weights and killing

measure on E (the matrix E.1x; 1y/, x; y 2 E is the inverse of the matrix g.x; y/,

x; y 2 E in (0.3)).

x2

x3 y1

w2 w3

w1 y2

x1

y3

Figure 0.1. The paths w1;:::;wk in E interact with the gas of loops through the random

potentials.

The typical representation formula for the moments of the random field in (0.9) looks

like this: for k 1, z1;:::;z2k 2 E,

h'z1 :::'z2k i D

P

pairings

of z1;:::;z2k

Px1;y1 ˝˝ Pxk;yk ˝ Q

eP

x2E vx.LxCLxx

1.w1/CCLxx

1.wk//

Q

eP

x2E vxLx

 ;

(0.10)

where the sum runs over the (non-ordered) pairings (i.e. partitions) of the symbols

z1; z2;:::;z2k into fx1; y1g;:::; fxk; ykg. Under Q the vx; x 2 E, are i.i.d. -

distributed (random potentials), independent of the Lx, x 2 E, which are distributed

as the total occupation times (properly scaled to take account of the weights and

killing measure) of the gas of loops with intensity 1

2 , and the Pxi ;yi , 1  i  k are

defined just as below (0.4), (0.5).

The Poisson point process of Markovian loops has many interesting properties.

We will for instance see that when ˛ D 1

2 (i.e. the intensity measure equals 1

2 ),

.Lx/x2E has the same distribution as 1

2 .'2

x/x2E , where

.'x/x2E stands for the Gaussian free field in (0.3). (0.11)

The Poisson gas of Markovian loops is also related to the model of random interlace￾ments [27], which loosely speaking corresponds to “loops going through infinity”. It

4 Introduction

appears as well in the recent developments concerning conformally invariant scaling

limits, see Lawler–Werner [16], Sheffield–Werner [24]. As for random interlace￾ments, interestingly, in place of (0.11), they satisfy an isomorphism theorem in the

spirit of the generalized second Ray–Knight theorem, see [28].