Abstract
The Fredrickson-Andersen 2-spin facilitated model on \({{\mathbb {Z}}} ^d\) (FA-2f) is a paradigmatic interacting particle system with kinetic constraints (KCM) featuring dynamical facilitation, an important mechanism in condensed matter physics. In FA-2f a site may change its state only if at least two of its nearest neighbours are empty. Although the process is reversible w.r.t. a product Bernoulli measure, it is not attractive and features degenerate jump rates and anomalous divergence of characteristic time scales as the density q of empty sites tends to 0. A natural random variable encoding the above features is \(\tau _0\), the first time at which the origin becomes empty for the stationary process. Our main result is the sharp threshold
with \(\lambda (d,2)\) the sharp threshold constant for 2-neighbour bootstrap percolation on \({{\mathbb {Z}}} ^d\), the monotone deterministic automaton counterpart of FA-2f. This is the first sharp result for a critical KCM and it compares with Holroyd’s 2003 result on bootstrap percolation and its subsequent improvements. It also settles various controversies accumulated in the physics literature over the last four decades. Furthermore, our novel techniques enable completing the recent ambitious program on the universality phenomenon for critical KCM and establishing sharp thresholds for other two-dimensional KCM.
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Notes
If f and g are real-valued functions of q with g positive, we write \(f = O(g)\) if there exists a (deterministic absolute) constant
such that \(|f(q)|\leqslant C g(q)\) for every sufficiently small
. We also write \(f = \varOmega (g)\) if f is positive and \(g = O(f)\). We further write \(f = \varTheta (g)\) if both \(f = O(g)\) and \(f = \varOmega (g)\). Finally, we write \(f = o(g)\) if for all
for sufficiently small
we have \(|f(q)|\leqslant c g(q)\).This construction is inspired by one suggested by P. Balister in 2017, which he conjectured would remove the spurious log-corrections in the bound (1.8) available at that time.
This choice of geometrically increasing length scales is inspired by [19].
The non-standard convention that relaxation times are at least 1 is made for convenience.
Strictly speaking [26, Corollary 3.1] deals with the torus of cardinality \(\pi ({\mathcal {S}}_1)^{-1}\) but the same proof extends to our case of the box \(B_{i}\cup B_{j}\).
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Acknowledgements
We acknowledge enlightening discussions with P. Balister, B. Bollobás, J. Balogh, H. Duminil-Copin, R. Morris and P. Smith and the hospitality of IHES during the informal workshop in 2017 “Kinetically constrained spin models and bootstrap percolation”. On that occasion P. Balister suggested a flexible structure for the droplets, featuring freedom in the position of the internal core at all scales, which he conjectured would remove the spurious log-corrections in the bound (1.8) available at that time. We also thank the anonymous referees for the careful proofreading and helpful comments on the presentation of the paper. The authors declare that they have no conflict of interest. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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This work is supported by ERC Starting Grant 680275 “MALIG”, ANR-15-CE40-0020-01 and PRIN 20155PAWZB “Large Scale Random Structures”
Appendices
A Probability of super-good events
In this appendix we prove Proposition 4.6 and we gather several more technical and relatively standard bootstrap percolation estimates on the probability of super-good events used in Sect. 4.
For \(z>0\) we define
where \(\beta (u)=(u+\sqrt{u(4-3u)})/2\). It is known [28, Proposition 5(ii)] that \(\int _0^\infty g(z)\,dz=\pi ^2/18\). We next recall some straightforward properties of g.
Fact A.1
The function g is positive, decreasing, differentiable and convex on \((0,\infty )\). Moreover, the following asymptotic behaviour holds:
where \(x\sim y\) stands for \(x=(1+o(1))y\).
The relevance of this function comes from its link to the probability of traversability. Recalling Definition 4.1, for any positive integers a and b we set
where \({\mathbf {0}}\) stands for the fully infected configuration. Note that these probabilities are the same for left-traversability, while for up or down-traversability a and b are inverted in the r.h.s. The next lemma follows easily from [28, Lemma 8]. Let \(q'=-\log (1-q)=q+O(q^2)\).
Lemma A.2
For any positive integers a and b and \(\omega \in \{{\mathbf {0}},\mathbf{1}\}\) we have
Corollary A.3
For any positive integers a and b we have
Furthermore, for any boundary conditions \(\omega , \omega '\) and rectangle R of class \(1\leqslant n\leqslant 2N\) (recall Definitions 4.2 and 4.3), we have
uniformly over all possible values of s and boundary conditions \(\omega ,\omega '\).
Proof
Equation (A.1) follows immediately from Lemma A.2. To obtain (A.2) with n odd (the even case is treated identically), recall that
there are \(q^{-O(1)}\) possible values of \(s'\); by (A.1), for all s, \(s'\), \(\omega \) and \(\omega '\),
\(\square \)
We are now ready for the main result of this appendix.
Proof of Proposition 4.6
We will prove the same bound for the super-good event occurring with all \(s=0\) in Definition 4.3 on all scales, i.e. the initial infection \(\varLambda ^{(0)}\) being in the bottom-left corner of \(\varLambda ^{(n)}\). Once the offsets are fixed, it suffices to prove the bound on this probability for \(n=2N\), in which case it reads
by Lemma A.2 and symmetry. Since g is decreasing, the last sum is at most
The term for \(m=1\) is \(O(\log (1/q)/\sqrt{q})\) by Fact A.1. For the other terms we use that by convexity for any \(0<a<b\)
Using Fact A.1, we get
Finally, for \(m\geqslant 2\) we have \(\ell _m-\ell _{m-1}\leqslant 2\sqrt{q} \ell _{m-1}\) by (4.1), so
setting \(m_0=\max \{m,\ell _m\leqslant 1/q\}\). Putting these bounds together and recalling (4.3), we obtain that the r.h.s. of (A.3) is at least
This concludes the proof of Proposition 4.6. \(\square \)
We next turn to the event \({\overline{{\mathcal {S}}{\mathcal {G}}}}(V_2)\) from Definition 4.11 required in the proof of Lemma 4.10, so we fix \(n=2m\in [2,2N)\).
Lemma A.4
Recalling (4.17), we have
Proof
Recall that \(V_1\cup V_2=\varLambda ^{(n)}\) and assume \({\mathcal {S}}{\mathcal {G}}(\varLambda ^{(n)})\) occurs. For any \(0\leqslant s_1,s_2\leqslant \ell _{m}-\ell _{m-1}\) we write
Then by Corollary A.3 for any such \(s_1,s_2\) we have
so it suffices to show that
since \(\mu ({\mathcal {T}}_{\leftarrow }(I_1(\eta _{V_2})))\geqslant q\) for any \(\eta _{V_2}\in {\overline{{\mathcal {S}}{\mathcal {G}}}}(V_2)\).
However, by Definitions 4.3 and 4.11 and symmetry we have
the last equality following from Lemma A.2.
By convexity of g we get
By Fact A.1 we have that the r.h.s. of (A.4) is \(O(1/\ell _m)\). Putting this together we obtain
as desired, the second inequality coming from (4.1) as in the proof of Proposition 4.6. \(\square \)
B Proof of Proposition 5.2
Let \(({\mathcal {S}},{\mathcal {S}}_1,\pi )\) be the parameters of g-CBSEP on \({{\mathbb {T}}} _n^d\) and let \(\ell =\lceil \pi ({\mathcal {S}}_1)^{-1/d}\rceil \geqslant 2\). For simplicity we assume that \(n^{1/d}/\ell \in {{\mathbb {N}}} \) and we partition the torus \({{\mathbb {T}}} _n^d\) into \(M=(n/\ell )^d\) equal boxes \((B_j)_{j=1}^M\), where each \(B_j\) is a suitable lattice translation by a vector in \({{\mathbb {T}}} _n^d\) of the box \(B=[\ell ]^d\). The labels of the boxes can be thought of as belonging to \({{\mathbb {T}}} _M^d\) and we say that \(B_i,B_j\) are neighbouring boxes in \({{\mathbb {T}}} _n^d\) iff i, j are nearest neighbours in \({{\mathbb {T}}} _M^d\).
We then set \({\hat{{\mathcal {S}}}}= {\mathcal {S}}^B, {\hat{\pi }}((\sigma _x)_{x\in B})=\bigotimes _{x\in B}\pi (\sigma _x),{\hat{{\mathcal {S}}}}_1=\bigcup _{x\in B}\{\sigma _x\in {\mathcal {S}}_1\}\) and we consider the auxiliary renormalised g-CBSEP (in the sequel \({\hat{g}}\)-CBSEP) on the graph \({\hat{G}}= {{\mathbb {T}}} _M^d\) with parameters \(({\hat{{\mathcal {S}}}}, {\hat{{\mathcal {S}}}}_1, {\hat{\pi }})\). Using the assumption \(\lim _{n\rightarrow \infty }\pi ({\mathcal {S}}_1)=0\), we have that
Lemma B.1
Let
be the relaxation time of \({\hat{g}}\)-CBSEP on \({\widehat{G}}\). Then there exists a constant \(C=C(d)>0\) such that
.
Proof
We closely follow [32, Appendix A]. Write \({\hat{\varOmega }}_+\) for the space of \({\hat{g}}\)-CBSEP configurations with at least one particle and consider the projection \(\varphi :{\hat{\varOmega }}_+\mapsto \varOmega _+\) given by \(\varphi ({\hat{\omega }}):=\{{\mathbb {1}} _{\{{\hat{\omega }}_j\in {\hat{{\mathcal {S}}}}_1\}}\}_{ j\in {\hat{G}}}.\) As discussed in Remark 5.1, the projection of the \({\hat{g}}\)-CBSEP chain is the CBSEP chain on \({\hat{G}}\) reversible w.r.t. \(\pi ^+\), the product Bernoulli measure with parameter \(p={\hat{\pi }}({\hat{{\mathcal {S}}}}_1)\) conditioned on \(\varOmega _+\). For the latter, using \(p=\varTheta (1)\) as \(n\rightarrow \infty \), it was proved in [26, Theorem 1] that its relaxation time
. Hence, it is enough to prove that
for some constant \(C'\).
Let \({\hat{{{\mathbb {P}}} }}_{{\hat{\omega }}}(\cdot ), {\hat{{{\mathbb {E}}} }}_{{\hat{\omega }}}(\cdot )\) be the law and associated expectation of the \({\hat{g}}\)-CBSEP chain with initial condition \({\hat{\omega }}\in {\hat{\varOmega }}_+\) and let \({{\mathbb {P}}} _{\eta }(\cdot ), {{\mathbb {E}}} _{\eta }(\cdot )\) be the same objects for the projected chain (the CBSEP chain) with initial condition \(\eta \in \varOmega _+\).
In order to prove the lemma, it is sufficient to prove that for any function \(f:{\hat{\varOmega }}_+\mapsto {{\mathbb {R}}} \) with zero mean w.r.t. \({\hat{\pi }}^+\) and for any \({\hat{\omega }}\in {\hat{\varOmega }}_+\) the rate of exponential decay as \(t\rightarrow +\infty \) of \(|{\hat{{{\mathbb {E}}} }}_{{\hat{\omega }}}(f({\hat{\omega }}(t)))|\) is at least
for some \(c=c(p)>0\) independent of f and \({\hat{\omega }}\).
More formally,

For any such f write
where \(\tau _j\) is the first time such that \(\varphi ({\hat{\omega }}(t))_j\ne \varphi ({\hat{\omega }}(0))\), which is measurable w.r.t. the projected chain.
It follows from standard tools for finite reversible Markov chains (see e.g. [3, Section 5] that there exists \(K=K({\hat{\omega }})<+\infty \) such that \({\hat{{{\mathbb {P}}} }}_{{\hat{\omega }}}( \tau _j\geqslant t)\leqslant Ke^{-\lambda (j,{\hat{\omega }}) t} \) with

In particular, the rate of exponential decay as \(t\rightarrow +\infty \) of the second term of the r.h.s. of (B.1) satisfies our requirement.
In order to prove a similar result for the first term in the r.h.s. of (B.1), we observe that, conditionally on the event \(\bigcap _j \{\tau _j< t\}\) and on \(\varphi ({\hat{\omega }}(t))\), the variables \(({\hat{\omega }}_j(t))_{j\in {\hat{G}}}\) become independent with \({\hat{\omega }}_j(t)\sim {\hat{\pi }}(\cdot |\varphi ({\hat{\omega }}(t))_j)\). Hence, if we set \(g(\eta )={\hat{\pi }}\big (f({\hat{\omega }})|\varphi ({\hat{\omega }})=\eta \big ),\) we get
so that
The rate of exponential decay as \(t\rightarrow +\infty \) of both terms in the r.h.s. above is again at least
for some \(c>0\), since \(\pi ^+(g)={\hat{\pi }}^+(f)=0\). \(\square \)
Proof of Proposition 5.2
For any pair of neighbouring boxes \(B_i\) and \(B_j\) we write \({\hat{{\mathcal {E}}}}_{i,j}\) for the event \(\bigcup _{x\in B_i\cup B_j}\{\sigma _x\in {\mathcal {S}}_1\}\). Using Lemma B.1 and the definition of
we get that
where the sum in the r.h.s. is an equivalent way to express the Dirichlet form of \({\hat{g}}\)-CBSEP. Now fix a pair of adjacent boxes \(B_i,B_j\) and let
be the relaxation time of our original g-CBSEP with parameters \(({\mathcal {S}},{\mathcal {S}}_1,\pi )\) on \(B_i\cup B_j\). By symmetry
does not depend on i, j and the common value will be denoted by \({\widetilde{T}}_{\mathrm {rel}}\). If we plug the Poincaré inequality for g-CBSEP on \(B_i\cup B_j\)
into the r.h.s. above, we get
where the second inequality uses \({\mathbb {1}} _{{\hat{{\mathcal {E}}}}_{i,j}}{\mathbb {1}} _{{\mathcal {E}}_{x,y}}={\mathbb {1}} _{{\mathcal {E}}_{x,y}}\) and
Thus,
. It remains to bound \({\widetilde{T}}_{\mathrm {rel}}\) from above.
Let
denote the mixing time of g-CBSEP on \(B_{i}\cup B_{j}\) with parameters \({\mathcal {S}}'=\{0,1\},{\mathcal {S}}'_1=\{1\}\) and \(\pi '(1)=\pi ({\mathcal {S}}_1)=1-\pi '(0)\). Let \(T_{\mathrm{cov}}^{\mathrm{rw}}\) be the cover time of the continuous-time random walk on \(B_{i}\cup B_{j}\). Theorem 2 of [26] implies
. Moreover, it is well known (see e.g. [29]) that \(T_{\mathrm{cov}}^{\mathrm{rw}}\) is at most \(O\big (\ell ^d\log (\ell )\big )=O\big (\pi ({\mathcal {S}}_1)^{-1}\max (1,\log (1/\pi ({\mathcal {S}}_1)))\big )\) and [26, Corollary 3.1] provesFootnote 5 the same bound for
. In conclusion,
\(\square \)
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Hartarsky, I., Martinelli, F. & Toninelli, C. Sharp threshold for the FA-2f kinetically constrained model. Probab. Theory Relat. Fields 185, 993–1037 (2023). https://doi.org/10.1007/s00440-022-01169-2
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DOI: https://doi.org/10.1007/s00440-022-01169-2
Keywords
- Kinetically constrained models
- Interacting particle systems
- Sharp threshold
- Bootstrap percolation
- Glauber dynamics
- Poincaré inequality
Mathematics Subject Classification
- 60K35
- 82C22
- 60J27
- 60C05
such that
. We also write
for sufficiently small
we have