# Fredrickson-Andersen model on a Galton-Watson tree one example

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## Lower bound (proof of equation (3.2.2)).

Definition 3.2.16. A rectangle R (that is, a subset of Z2 of the form x + [L1] [L2]) is pre-internally spanned if there is a vertex x 2 R such that starting from entirely lled R, except for one empty site at x, R is internally spanned. We now x k = B , for some constant B that will be determined later on. Claim 3.2.17. Fix the rectangle [a] [b], for a 2 [k; 2k + 2] and b a. The probability that it is pre-internally spanned is at most e􀀀 1=.
Proof. In order to prevent the rectangle from being pre-internally spanned it suces to have two consequent columns with no easy vertices. This happens with probability h 1 􀀀 (1 􀀀 )2b ia 1 􀀀 (1 􀀀 )2aa h 1 􀀀 (1 􀀀 )2k ik e􀀀 1=: To obtain the second inequality we need to look at the function a 7! 1 􀀀 (1 􀀀 )2aa. It decreases to a minimum at some point / 1 , and then increases. We can x B such that this minimum is at 10B q , and the inequality follows. We will now bound the number of pre-internally spanned rectangles. Definition 3.2.18. Fix l 2 N. nl = nl (!) will be the number of pre-internally spanned rectangle inside [􀀀l; l]2 whose longest side is of length between k and 2k + 2. Claim 3.2.19. There exists l0 = l0 (!) such that for all l l0 nl 40k2 e􀀀 1=l2.

### Fredrickson-Andersen model on Z2 with threshold 1 or 2

We will now study the KCM on Z2 dened by the constraints in equations (3.1.1) and (3.1.2). We will see that the hitting time 0 scales polynomially with q, like in the bootstrap percolation. However, unlike the bootstrap percolation, this exponent will not be that of the FA1f constraint, but a random exponent changing from one realization of ! to the other. Theorem 3.3.1. Consider the KCM on Z2 with the constraints dened in equations (3.1.1) and (3.1.2). (1) There exists a constant c > 0 such that, -a.s., the relaxation time of the process is greater than ec=q. (2) -a.s. there exist and (with ) such that P 0 q􀀀 q!0 􀀀􀀀! 0.

#### Fredrickson-Andersen model on Zd with threshold 1 or d

The results of the previous section could be extended to Zd. In  it is explained how the proof in section 3.3 could be generalized, but here we will use a dierent strategy. We will see that 0 scales as a power law (like in the FA1f case ), with a power that may change from one realization to the other. We will not analyze the scaling of this power when is small as we did for the two dimensional case, but we can expect an iteretated exponent scaling that would t the FAdf model when is of order q (see Remark 3.3.3).
Theorem 3.4.1. Consider the KCM on Zd with the constraints dened in equations (3.1.1) and (3.1.2).
(1) There exists a constant c > 0 such that, -a.s., the relaxation time of the process is greater than exp(d􀀀1) c q , where exp() is the iterated exponential. (2) Recall Denition 2.2.4 and let = f0=0g. -a.s. there exist and (with ) such that for q small enough q􀀀.

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#### Mixed North-East and Fredrickson-Andersen 1 spin facilitated models on Z2

Unlike the models that have been studied in the previous sections, this KCM is not necessarily ergodic. For a xed environment ! there exists a critical value qc above which all sites are emptiable for the bootstrap percolation, and below which some sites remain occupied forever. Denote by pSP the critical probability for the Bernoulli site percolation on Z2 and by pOP the critical probability for the oriented Bernoulli percolation on Z2. Then qc 1 􀀀 1 􀀀 pSP 1 􀀀 ; since if we have an innite cluster of sites that are either easy or empty all sites are emptiable. In particular, if > pSP the critical probability is 0. On the other hand, if there is an innite up-right path of dicult sites that are all occupied, this path could never be emptied. This will imply that qc 1 􀀀 pOP 1􀀀 . We will see for this model that it is possible to have an innite relaxation time, and still the tail of the distribution of 0 decays exponentially, with a rate that scales polynomially with q. Theorem 3.5.1. Consider the kinetically constrained model described above, with > pSP and q < pOP.

Remerciements
Abstract
Résumé
Chapter 1. Introduction
1.1. Glasses and the liquid-glass transition
1.2. Kinetically constrained models
1.3. Ergodicity and the bootstrap percolation
1.4. Time scales
1.5. Random environments
1.6. Overview of the results
1.7. List of conventions and notation
Chapter 2. Variational tools
2.1. Spectral gap and relaxation time
2.2. Hitting times
2.3. The time spent in E
Chapter 3. Random constraints on Zd
3.1. Introduction of the models
3.2. Bootstrap percolation on Z2 with threshold 1 or 2
3.3. Fredrickson-Andersen model on Z2 with threshold 1 or 2
3.4. Fredrickson-Andersen model on Zd with threshold 1 or d
3.5. Mixed North-East and Fredrickson-Andersen 1 spin facilitated models on Z2
3.6. Fredrickson-Andersen 1 spin facilitated model on the polluted Z2
3.7. Fredrickson-Andersen 2 spin facilitated model on polluted Z2
Chapter 4. Models on the Galton-Watson tree
4.1. Model, notation, and preliminary results
4.2. Metastability of the bootstrap percolation
4.3. Fredrickson-Andersen model on a Galton-Watson tree one example
Chapter 5. The Kob-Andersen model on Zd
5.1. The Kob-Andersen model and the main result
5.2. Proof of the main theorem
Chapter 6. Questions
6.1. KCMs and bootstrap percolation in random environments
6.2. The Kob-Andersen model
Bibliography

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