Kinetically constrained spin models on trees\thanksrefT1
Abstract
We analyze kinetically constrained – spin models (KCSM) on rooted and unrooted trees of finite connectivity. We focus in particular on the class of Friedrickson–Andersen models FAjf and on an oriented version of them. These tree models are particularly relevant in physics literature since some of them undergo an ergodicity breaking transition with the mixed firstsecond order character of the glass transition. Here we first identify the ergodicity regime and prove that the critical density for FAjf and OFAjf models coincide with that of a suitable bootstrap percolation model. Next we prove for the first time positivity of the spectral gap in the whole ergodic regime via a novel argument based on martingales ideas. Finally, we discuss how this new technique can be generalized to analyze KCSM on the regular lattice .
10.1214/12AAP891 \volume23 \issue5 2013 \firstpage1967 \lastpage1987 \newproclaimremark[Theorem]Remark \newproclaimdefinition[Theorem]Definition
Kinetically constrained spin models on trees
T1Supported by the European Research Council through the “Advanced Grant” PTRELSS 228032.
A]\fnmsF. \snmMartinelli\correflabel=e1]martin@mat.uniroma3.it and B]\fnmsC. \snmToninelli\thanksreft2label=e2]cristina.toninelli@upmc.fr
t2Supported in part by the French Ministry of Education through the ANR2010BLAN0108.
class=AMS] \kwd60K35 \kwd82C20 Kinetically constrained models \kwddynamical phase transitions \kwdglass transition \kwdbootstrap percolation \kwdstochastic models on trees \kwdinteracting particle systems
1 Introduction
Facilitated or kinetically constrained spin models(KCSM) are interacting particle systems which have been introduced in physics literature Fredrickson1 (), Fredrickson2 () to model liquid/glass transition and more generally “glassy dynamics” RitortSollich (), GST (). They are defined on a locally finite, bounded degree, connected graph with vertex set and edge set . Here we will focus on models for which the graph is an infinite, rooted or unrooted tree of finite connectivity , which we will denote by and , respectively. A configuration is given by assigning to each site its occupation variable which corresponds to an empty or filled site, respectively. The evolution is given by a Markovian stochastic dynamics of Glauber type. Each site waits an independent, mean one, exponential time and then, provided the current configuration around it satisfies an a priori specified constraint, its occupation variable is refreshed to an occupied or to an empty state with probability or , respectively. For each site the corresponding constraint does not involve , thus detailed balance w.r.t. Bernoulli() product measure can be easily verified and the latter is an invariant reversible measure for the process.
Among the most studied KCSM we recall FAjf models Fredrickson1 () for which the constraint requires at least (which is sometimes called “facilitating parameter”) empty sites among the nearest neighbors. FAjf models display a feature which is common to all KCSM introduced in physics literature: for each vertex the constraint imposes a maximal number of occupied sites in a proper neighborhood of in order to allow the moves. As a consequence the dynamics becomes slower at higher density and an ergodicity breaking transition may occur at a finite critical density . This threshold corresponds to the lowest density at which a site belongs with positive probability to an infinite cluster of particles which are mutually and forever blocked due to the constraints; see Section 3.
The FAjf models on do not display an ergodicity breaking transition at a nontrivial critical density, that is, for and otherwise CMRT (). On the other hand they do display such a transition on nonrooted trees when chalupa (), sellitto (), schwartz (). Furthermore if , this transition is expected to display a mixed first/second character and to share similar features to the mode coupling transition, a property which makes them particularly interesting from the point of view of the glass transition sellitto ().
Another key feature of KCSM is the existence of blocked configurations, namely configurations with all creation/destruction rates identically equal to zero. This implies the existence of several invariant measures and the occurrence of unusually long mixing times compared to hightemperature Ising models (see Section 7.1 of CMRT ()). Furthermore the constrained dynamics is usually not attractive so that monotonicity arguments valid, for example, ferromagnetic stochastic Ising models cannot be applied.
Due to the above properties the basic issues concerning the large time behavior of the process are nontrivial. The first rigorous results were derived in Aldous () for the East model which is defined on with the constraint requiring the nearest neighbor site to the right to be empty. In Aldous () it was proven that the spectral gap of East is positive for all and also that it shrinks faster than any polynomial in as . In CMRT () positivity of the spectral gap of KCSM inside the ergodicity region (i.e., for ) has been proved in much greater generality and (sometimes sharp) bounds for were established. These results include FAjf models on any for any choice of the facilitating parameter and of the spatial dimension .
The technique developed in CMRT () cannot be applied to models on trees because of the exponential growth of the number of vertices and, so far, very few rigorous results have been established. Indeed the only models for which results on the spectral gap are available are: (i) the FA1f model on and (actually on a generic connected graph) and (ii) the socalled East model on for which the root is unconstrained while, for any other vertex , the constraint requires the ancestor of to be empty. For these specific models and the positivity of the spectral gap has been proven in LNM () in the whole ergodicity region and for any choice of the graph connectivity.
Here we will study FAjf models on and for together with a new class of models that we call oriented FAjf models (OFAjf). In the OFAjf model the constraint at requires at least empty sites among the children of .
We first prove that the ergodicity threshold for the FAjf and OFAjf models, with the same choice for the parameter and the same graph connectivity , coincide and it is nontrivial (see Theorem 1). Then we prove positivity of the spectral gap in the whole ergodicity regime for the oriented OFAjf models. Finally, by combining the above results together with an appropriate comparison technique, we establish positivity of the spectral gap in the whole ergodicity regime for the FAjf models. The results concerning the spectral gap can be found in Theorem 2 and a simple application to the mixing time of finite system in Corollary 1. Finally, in the nonergodic regime, we prove that, for the oriented or nonoriented FAjf models, the spectral gap shrinks to zero exponentially fast in the system size; see Theorem 3.
The new technique devised to study constrained models on trees can be generalized to deal also with KCSM on other graphs. In Section 5 we discuss how one can recover the result of positivity of the spectral gap in the ergodic regime for models on . We detail in particular the case of the north–east model on (Theorem 4), a result which was already derived in CMRT () but with a completely different (and more lengthy) technique.
2 Models and main results
2.1 Setting and notation
The graphs
The models we consider are either defined on the infinite regular tree of connectivity , in the sequel denoted by or on the infinite, rooted ary tree . In the unrooted case each vertex has neighbors, while in the rooted case each vertex different from the root has children and one ancestor, and the root has only children. In the sequel we will denote by the set of vertices of either or of whenever no confusion arises, by the set of neighbors of a given vertex and, in the rooted case, by the set of its children. In the rooted case we denote by the depth of the vertex , that is, the graph distance between and the root .
The configuration spaces
For both oriented and nonoriented models we choose as configuration space the set whose elements will usually be assigned Greek letters. We will often write for the value at of the element . We will also write for the set , . With a slight abuse of notation, for any and any , we let to be the restriction of to the set and to be the configuration which equals on and on .
Probability measures
For any we denote by the product measure where each factor is the Bernoulli measure on with and with . If we abbreviate to .
Conditional expectations and conditional variances
Given a function depending on finitely many variables, in the sequel referred to as local function, and a set we define the function by the formula
Clearly coincides with the conditional expectation of given the configuration outside . Similarly we write for the conditional variance of given . Note that if and only if does not depend on the configuration inside . In case we abbreviate to .
2.2 Facilitated models
{definition}Fix and a facilitating parameter . The FAjf and OFAjf models at density are continuous time Glaubertype Markov processes on , reversible w.r.t. , with Markov semigroups and , respectively, whose infinitesimal generators act on local functions as follows:
(1)  
(2) 
The function (or ), in the sequel referred to as the constraint at , is defined by
(3)  
(4) 
It is easy to check by standard methods (see, e.g., Liggett ()) that the processes are well defined and that their generators can be extended to nonpositive selfadjoint operators on and , respectively.
Both processes can of course be defined also on finite regular trees, rooted or unrooted. In this case and in order to ensure irreducibility of the Markov chain the constraints must be suitably modified. {definition} Let be a finite subtree of either or of and let, for any , denote the extension of in given by
For any define the finite constraints by
We will then refer to the OFAjk model or the FAjk model on as the irreducible, continuous time Markov chains on with generators
(5)  
(6) 
respectively.
2.3 Ergodicity
Given with , it is natural to define (see CMRT ()) a critical density for each model as follows:
(7)  
(8) 
The regime or is called the ergodic region and we say that an ergodicity breaking transition occurs at the critical density. We will first establish the coincidence of the critical threshold for oriented and unoriented models.
Theorem 1
Given with , let and define
(9) 
Then and for any the value is a simple eigenvalue of the generators and . Moreover if and only if .
We then turn to the study of the relaxation to equilibrium in . A key object here is the spectral gap (or inverse of the relaxation time) of the generator (or ), defined as
(10) 
where the Dirichlet form is the quadratic form associated to . Indeed a positive spectral gap implies that the reversible measure is mixing for the semigroup with exponentially decaying correlations,
2.4 Main results on relaxation to equilibrium
For the reader’s convenience we split the presentation of our results into three subsections according to whether is below, above or equal to the critical value .
2.4.1 The subcritical case
Theorem 2
Given with , fix . Then and .
Exactly as in CMRT () (see Proposition 2.13 there), in order to prove positivity of the spectral gap for the infinite trees or , it is enough to prove a lower bound on the spectral gap of the corresponding models on finite balls which is uniform in the size of the ball. It is important to observe that in the oriented case the above result completes the proof of the exponential decay to equilibrium when and the initial distribution is either a Bernoulli product measure with density , or it is a measure on a deterministic configuration which does not contain blocked clusters. These results were indeed proven in CMST () (see Theorems 4.2 and 4.3) modulo the hypothesis of positivity of the spectral gap in the ergodic region.
We finally observe that the above result says nothing about the behavior of the spectral gap as a function of when . See, however, Section 2.4.3 below for some work in progress in this direction.
Our second result, a natural corollary of the spectral gap bounds of Theorem 2, concerns mixing times of the oriented model on finite subtrees of . In order to state it we need few extra notation.
Let be the finite rooted tree consisting of the first levels of . For any we denote by the law at time of the Markov chain with generator and by the relative density w.r.t. of , namely
Following Gine (), we define the family of mixing times by
Notice that coincides with the usual mixing time of the chain (see, e.g., Peres ()) and that, for any , .
Corollary 1
Given with , fix . Then there exists a constant such that
A key ingredient for the proof of the above Corollary will be the fact that the marginal of the law over is given by the product of the marginals over the individual subtrees rooted at the children of the root. Such a property is no longer true in the unoriented case. In this more complicate setting a possible route to get a (poorer) bound on the mixing time is the following.
Use a comparison between the Dirichlet forms of the FAjk and OFAjk models to get that the logarithmic Sobolev constant (see, e.g., Gine ()) of the FAjk model on a finite regular tree , with levels and centered at a vertex , is bounded from below by constantthe logarithmic Sobolev constant of the OFAjk model on the finite trees . Then use the left part of the wellknown bound (see Corollary 2.2.7 in Gine ())
where to infer that the logarithmic Sobolev constant of the OFAjk model is bounded from below by . Hence the logarithmic Sobolev constants of both the OFAjk and the FAjk models on are bounded from below by . Finally use the right part of the above bound to conclude that the mixing time for the FAjk model on is .
2.4.2 The supercritical phase
Our first result roughly says that, when , the occupation number for the process defined on the infinite tree does not equilibrate in .
Denote by either the root (in the oriented case) or an arbitrary vertex of (in the unoriented case).
Proposition 1
Given with , fix . Then
and the same inequality holds with instead of .
The second result concerns the spectral gap on finite balls. Given and , denote by either the ball in of radius and center (in the unoriented case) or the rooted tree consisting of the first levels of (in the oriented case).
Theorem 3
Given with , fix . Then there exists such that
2.4.3 The critical phase
The critical case is much more delicate and a detailed analysis is postponed to future work criticaltree (). We anticipate here that it is possible to show that the spectral gap on a ball of radius shrinks at least polynomially fast in . In the rooted case with one can also prove a converse poly() lower bound (a much harder task). These two results then imply that in the rooted case and for , there exist three positive constants , such that, for ,
If , the analysis of the lower bound on the spectral gap becomes much more difficult because of the discontinuous character of the bootstrap percolation transition. More precisely, and contrary to what happens for , for the root belongs to an infinite blocked cluster with positive probability. In this case it is still unclear whether a poly() lower bound on the spectral gap still holds.
3 Ergodicity threshold and blocked clusters: Proof of Theorem 1
{definition}Given with , the bootstrap map associated to the FAjf model is defined by
(11) 
with defined in (3). Analogously we define the bootstrap map for the OFAjf model by replacing with of (3). Having defined the bootstrap map it is natural to denote by the probability measure obtained by iterating times the map starting from . In other words, for any . As tends to infinity converge to a limiting measure Schonmann (), and it is natural to define the bootstrap percolation threshold as the supremum of the density of such that is concentrated on the empty configuration. Analogously we can define and in the oriented case.
It is quite clear that the two thresholds and must coincide. Choose in fact an arbitrary vertex and write the unrooted tree as where each is a copy of with root at . If , then a.s. each becomes eventually empty under the bootstrap map applied to and therefore also under the lessrestrictive bootstrap map . Thus . On the other hand, when the set
has positive probability and moreover for any . Hence .
That coincide with the third threshold given in (9) has been established in Proposition 1.2 of Peres () (see also sellitto (), FS () and STBT () for an extension to hyperbolic lattices). For completeness we shortly reprove this result by showing that .
We first observe that if and only if where . Second one easily checks that the nonincreasing sequence obeys the recursive equation with initial condition . Here has the expression
We now claim that if and only if . In order to prove the claim we first observe that is a fixed point of the map and that it is a nondecreasing function of . Hence .
To prove the converse we compute
where .
Therefore is strictly increasing in , and if it has a fixed point , then necessarily . Hence .
We finally check that if and only if . The Markov inequality implies that
Hence if and . When it is also clear that , where correspond to the extreme cases and , respectively. When the threshold coincides with the usual site percolation threshold (see Grimmett ()). When and an exact computation Peres () gives
It is not difficult to check that, for , the limit as of both sequences and is:

zero and attained at least exponentially fast if ;

zero and attained polynomially fast (in ) for and ;

strictly positive for and .
The proof of Theorem 1 now follows from the above discussion together with the following proposition which can be proved following exactly the same lines as Proposition 2.5 of CMRT ().
Proposition 2
and .
4 Relaxation to equilibrium: Proofs
4.1 The subcritical phase
In what follows we fix once and for all with , together with a density . {pf*}Proof of Theorem 2: the oriented case We begin by proving positivity of the spectral gap in the oriented case OFAjf at density .
We first fix some additional notation. We denote by the finite ary tree consisting of the first levels (counting the root ) of , where should be thought of as arbitrarily large compared to all other constants. For , will denote the ary subtree of rooted at and with levels, where is the level label of . We also set . In the sequel we shall refer to the number of levels as the depth of the tree .
The key idea for the proof is to introduce longrange constraints. {definition} For any , let be equal to in and equal to in . Then, for any integer we define
In what follows we will first consider an auxiliary longrange, kinetically constrained model on whose infinitesimal generator is as in (5) but with substituted by . We will show that this auxiliary model has a spectral gap which is bounded away from zero uniformly in the depth of , provided is large enough depending on . Then we will apply standard comparison arguments between the Dirichlet forms with constraints and to show that also the original model has a spectral gap which is uniformly positive in . By appealing to Remark 2.4.1 that completes the proof.
Let denote the new Dirichlet form corresponding to the generator
with the auxiliary constraints , that is,
Our aim is to establish the socalled Poincaré inequality
(12) 
for some constant independent of the depth of the tree . {remark} Notice that (12) is the natural analog of the renormalized Poincaré inequality in CMRT (); see formula (5.1) there. For the reader’s convenience we begin by recalling some elementary properties of the variance which will be applied in the sequel. Consider two probability spaces , , together with their product probability space . Then, for any ,
so that
(13) 
Clearly , and so forth. Moreover,
(14) 
Back to the proof and motivated by MSW () we first claim that
(15) 
To prove the claim we proceed recursively on the depth of . The claim is trivially true for . We now assume (15) when has depth , and using the formula for the conditional variance we write
(16) 
Notice that, given the spin at the root, is nothing but the variance of w.r.t. the product measure . Thus
and
Each one of the subtrees has depth , and therefore the inductive assumption implies that
By putting together the righthand side of (4.1) with the last term in (16), we get the claim for depth .
We now examine a generic term in the righthand side of (15). We write
so that
(18) 
The Cauchy–Schwarz inequality shows that
(19) 
because does not depend on the spin at . Notice that the righthand side in (19) is just the contribution of the root to the Dirichlet form .
We now turn to the analysis of the more complicated second term, in the nontrivial case . We write
where and we use the fact that does not depend on . Recall that the constraint depends only on the spin configuration in the first levels below , in the sequel denoted by . Then
where . Above we used Cauchy–Schwarz to obtain the second inequality. The last equality holds because does not depend on . Notice that coincides with where was defined at the beginning of the proof of Theorem 1.
Next we note that
(22) 
where we use the fact that by the definition of . Then by using (15), (4.1) and (22) we get
where we use the convexity of the variance to obtain the second inequality. In conclusion,
(24)  
where the factor accounts for the number of vertices such that a given vertex falls inside .
We now appeal to Remark 3 and conclude that for any there exists (which depends on and it diverges as ) such that for any . With this choice and recalling (15), the Poincaré inequality (12) with follows uniformly in the depth of . In other words the auxiliary long range model has a positive spectral gap greater than if .
We are now in a position to conclude the proof in the oriented case. Starting from (12) and using path arguments exactly as in Section 5 of CMRT (), we conclude that, for any we can find a constant independent of such that
Thus, thanks to Remark 2.4.1, we can conclude that the spectral gap of the oriented model on the infinite tree is bounded from below by .
The dependence on of comes from the fact that . Clearly the critical scale diverges as . \noqed {pf*}Proof of Theorem 2: the unoriented case For an arbitrary vertex we introduce an auxiliary block dynamics, reversible w.r.t. the measure , as follows. With rate one the block chain resamples the current configuration in from the equilibrium measure, and, always with rate one, it resamples the variable if and only if the constraint at the root is satisfied [i.e., ].
For such auxiliary block chain it is easy to prove a Poincaré inequality of the form (compare to Proposition 4.4 in CMRT ())
(25) 
for some constant .
Observe now that is the union of copies of the rooted tree so that
Thanks to the result in the oriented case and using , we get
(26) 
where . Thus
(27) 
Inserting (27) into (25) we conclude that the spectral gap of the FAjf model is bounded below by . {pf*}Proof of Corollary 1 We closely follow the proof of a similar result given in MartinelliWouts (). Recall that is the finite subtree consisting of the first levels of and that denotes the relative density w.r.t. of the law at time of the oriented chain started at . We can then write
together with
Above