Metastability in the reversible inclusion process

We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph $S$ with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its reversible measure takes maximum value on a subset of vertices $S_\star\subset S$. We consider initial conditions corresponding to a single condensate that is localized on one of those vertices and study the metastable (or tunneling) dynamics. We find that, if the random walk restricted to $S_\star$ is irreducible, then there exists a single time-scale for the condensate motion. In this case we compute this typical time-scale and characterize the law of the (properly rescaled) limiting process. If the restriction of the random walk to $S_\star$ has several connected components, a metastability scenario with multiple time-scales emerges. We prove such a scenario, involving two additional time-scales, in a one-dimensional setting with two metastable states and nearest-neighbor jumps.


Introduction
The inclusion process is an interacting particle system introduced in the context of non-equilibrium statistical mechanics, as a dual process of certain diffusion processes modeling heat conduction and Fourier's law [18,19,20].Besides, it is also related to models in mathematical population genetics [13], such as the Moran model, and to models of wealth distribution [15].In addition to this, the inclusion process is interesting in its own right as an interacting particle system belonging to the class of misanthrope processes [16].
In the inclusion process, particles jump over a set S of vertices, thus the total number of particles N is conserved by the dynamics.The transitions are driven by two competing contributions to the total jump rate.Denoting by η x the particle number at site x, and calling r : S × S → R + the jump rates of a continuous-time irreducible random walk on S, the process is defined by the following rules (see Section 2.1 for the process generator): i) firstly, particles move as continuous-time independent random walks: for a parameter d N > 0, each of the η x particles at site x waits a random time which is the minimum of exponential clocks of parameters d N r(x, z) for z ∈ S, and then jumps to site y with probability r(x, y)/( z∈S r(x, z)).
ii) secondly, particles jump because of an attractive interaction: each of the η x particles at site x waits a random time which is the minimum of exponential clocks of parameters η z r(x, z) for z ∈ S, and then jumps to site y with probability η y r(x, y)/( z∈S η z r(x, z)).
Whereas the first contribution leads to spreading of the particles over the sites of S, due to the second contribution particles have a preference to accumulate on a few sites.This is to be compared with the well-known exclusion process, where particles are subject to hard-core interactions that forbid more than one particle per site.The inclusion process is a bosonic system, whereas the exclusion process is fermionic.The relative strength of the two contributions is tuned by the parameter sequence d N .In the long time limit, the two contributions find a compromise into a (reversible) stationary measure that is shown explicitly in Section 2.2.As long as d N > 0, this measure has mass over all the configuration space.If the sequence d N approaches zero sufficiently fast as N → ∞, then the stationary measure concentrates on a small subset of configurations.This is the phenomenon of condensation in the inclusion process, first studied in [21].In the condensation regime essentially all particles accumulate on a single site of S ⋆ ⊆ S, the set over which the stationary measure of the random walk takes maximum value.The condensation phenomenon occurs in several other interacting particle systems [17], most notably the zero range process [23].
In this paper we consider the condensation regime of the inclusion process and study the dynamics of the condensate.This problem was previously considered in [22].There, however, the authors assumed a symmetric random walk kernel that therefore has a uniform reversible measure on S.Here we consider instead the case of a generic reversible measure, thus allowing the possibility that S ⋆ = S. Depending on the properties of the underlying random walk kernel, the following metastability scenario with possibly multiple time-scales emerges from our analysis.If the restriction of the random walk kernel to S ⋆ is still irreducible, then the system has only one time-scale.However, if such restriction is reducible into several connected components, then there exist up to three time-scales: a first time-scale over which the system moves within connected components; a second time-scale to see the jumps between components that are at graph distance equal to two; a third (even longer) time-scale for the jumps between components that are at graph distance larger than two.
Our results include the characterization of the single time-scale scenario in great generality.In particular, when the system has only one time-scale, we allow any geometry and we are able to derive the rates of the limiting Markovian dynamics.We give a rigorous proof of the multiple timescale scenario instead in the one-dimensional setting, i.e., for linear chains with nearest-neighbor jumps, whose end-points are the only maximal states of the reversible measure.In this case we fully characterize the second time-scale (together with the rates of the limiting dynamics) when |S| = 3, and we prove the existence of the third time-scale when |S| > 3. We conjecture that the same qualitative behavior of the motion of the condensate occurs in great generality.
The key ingredient of the proofs of our main results are potential theory methods.We refer in particular to the potential theoretic approach to metastability, introduced in a series of papers by Bovier, Eckhoff, Gayrard and Klein [8]- [10], and to the martingale approach, developed in some recent papers by Beltrán and Landim [3]- [6].A general treatment of metastable systems may be found in [28], where the pathwise approach to metastability is discussed, while we refer to [11] for a recent book on the potential theory approach to metastability.
In the next section we introduce the model and state our main results precisely.We also give an outline of the proofs.In Section 3 we analyze the metastable sets.The proofs for the three different time-scales are given in Sections 4-6, respectively.

Reversible inclusion process
Consider a set of sites S with κ := |S| < ∞ and let r : S × S → R + be the jump rates of a continuous-time irreducible random walk on S, reversible with respect to some probability measure m, i.e., m(x)r(x, y) = m(y)r(y, x), for all x, y ∈ S. ( Without loss of generality, we assume that r(x, x) = 0 for all x ∈ S.
Of special interest are the sites where m attains its maximum.Hence, we define and let which is a normalization of m such that m ⋆ (x) = 1 if and only if x ∈ S ⋆ .For a given underlying random walk we can now give the definition of the reversible inclusion process {η(t) : t ≥ 0}.For each N ≥ 1, the set of configurations E N correspond to all the possible arrangements of N particles on S, that is (2.4) The component η x of η has to interpreted as the number of particles at site x ∈ S.
To specify the possible transitions of the dynamics, for x, y ∈ S, x = y, and η ∈ E N such that η x > 0, let us denote by η x,y the configuration obtained from η by moving a particle from x to y: otherwise. (2.5) The inclusion process with N particles is then a Markov process {η(t) : t ≥ 0} on E N with generator L N , acting on functions F : E N → R, given by where {d N > 0 : N ∈ N} is a sequence of positive numbers that is specified later.

Condensation and metastability
The inclusion process has a stationary and reversible probability measure µ N (η), given by a product measure of negative binomials conditioned to a total number of particles N, i.e., where and (2.9) We abbreviate so that (2.7) becomes The stationary measure is unique, because the underlying random walk, and hence also the inclusion process, is irreducible.It can easily be checked that the measure in (2.7) satisfies the detailed balance, and thus is the reversible measure of the process.
If the parameter d N scales to zero fast enough in the limit N → ∞, then the inclusion process shows condensation, i.e., the invariant measure concentrates on disjoint sets of configurations (that we shall call metastable sets or condensates).To formalize this idea, let Moreover, for S 0 ⊂ S ⋆ , define E N (S 0 ) = x∈S 0 E x N and let ∆ = E N \ E N (S ⋆ ).The following result, proved in Section 3, shows that invariant measure asymptotically concentrates on the sets (in fact singletons) E x N , x ∈ S ⋆ , which turn out to be the metastable sets of the process:
The metastability problem we address in this paper is the following.Assume the process is started from a configuration corresponding to a single condensate.Then we determine the timescale(s) over which the condensate moves and characterize the law of the process describing the motion of the condensate.Remark 2.2.Notice that the metastable sets E x N , x ∈ S ⋆ , have equal µ N -measure.It may be worth to mention that in this situation some authors prefer to speak about tunneling behavior rather than metastability.However, this abuse of terminology is currently quite diffuse in the mathematical literature, and we just use the word metastability.

Results
In order to state our findings we introduce some notation.For a set A ⊂ E N , let τ A denote the hitting time of A: Moreover, with the identification E ⋆ N ≡ E N (S ⋆ ), let η E ⋆ N (t) denote the trace process on E ⋆ N , i.e., the process obtained from η(t) by cutting out all time periods where the system is not in Notice that this is still a Markov process (we refer to [3] for a proof of this result).
Finally, let us define the process where With the above notation, and the usual convention that E η (•) denotes the expectation when the process η(t) is started from η at time t = 0, we prove the following: Theorem 2.3 (First time-scale).If d N log N → 0 as N → ∞, then, for all x ∈ S ⋆ , (i) The average time to move the condensate at x to another site of S ⋆ is given by (ii) Assume that X N (0) = x.Then, the speeded-up process {X N (t/d N ) , t ≥ 0} converges weakly on the path space D(R + , S ⋆ ) to the Markov process {X(t) , t ≥ 0} on S ⋆ , with X(0) = x and generator Furthermore, the system spends negligible time outside the metastable states, i.e., ∀ T > 0 lim Remark 2.4.The weak convergence stated in item (ii) of Theorem 2.3 refers to the path space endowed with the Skorokhod topology.We stress the fact that from this result, together with condition (2.20), one can also infer the weak convergence of the speeded-up projected process {ψ N (η(t/d N )) , t ≥ 0} to the Markov process {X(t) , t ≥ 0} as defined above, though with a topology on the path space, called soft topology, that is weaker than the Skorokhod one.We refer to [24] for the details.
From Theorem 2.3, we conclude that on this first time-scale the condensate can only jump between sites in S ⋆ that are connected in the graph induced by the underlying dynamics.In particular, if x, y ∈ S ⋆ are not connected by a path in S ⋆ , then the condensate will not move from x to y on the time-scale 1/d N .Since the inclusion process is irreducible, we therefore expect that this movement occurs on a longer time-scale.
We formalize these ideas focusing on a specific one-dimensional setting.For an integer κ ≥ 2, let that is indeed an example of dynamics that is not irreducible when restricted to S ⋆ .
For such systems we have the following, where we say that d N decays subexponentially if, for all δ > 0, lim N →∞ d N e δN = ∞.Theorem 2.5 (Second time-scale).Consider an underlying random walk as in (2.21), with κ = 3. Assume that d N decays subexponentially and d N log N → 0 as N → ∞.Then (i) The average time to move the condensate between the sites of S ⋆ is given by Then, the speeded-up process X N (tN/d 2 N ) converges weakly on the path space D(R + , S ⋆ ) to the Markov process {X(t) , t ≥ 0} on S ⋆ , starting at X(0) = x and jumping back and forth between x and S ⋆ \ {x} at rate .
(2.23) Furthermore, the system spends negligible time outside the metastable states, i.e., ∀ T > 0 and x ∈ S ⋆ , lim As will be clear from the proof of Theorem 2.5 (see Section 5) the explanation for the presence of this second time-scale is that, in order to move the condensate between sites 1 and 3, the system is forced to bring particles through site 2. The presence of particles on sites of S \ S ⋆ is however an unlikely event, that slows down the motion of the condensate through sites of S ⋆ and yields a much longer transition time-scale.In this sense, we may consider the sites of S \ S ⋆ as traps for the dynamics of the system.
Following this idea, the natural further question is about the presence of possibly many timescales related to the length of these traps, that is to the graph-distance between disconnected sites of S ⋆ .We answer this question in the affirmative for linear systems as those defined in (2.21), proving that an even longer time-scale is required to move the condensate between the disconnected sites {1, κ} ∈ S ⋆ at arbitrary (but finite) graph-distance greater than 2. We have the following: Theorem 2.6 (Third time-scale).Consider an underlying random walk as in (2.21), with κ > 3. Furthermore, assume that d N decays subexponentially and d N log N → 0 as N → ∞.Then, the average time to move the condensate between sites x, y ∈ S ⋆ , x = y, satisfies the bounds ) Notice that the upper bound in (2.25) excludes the possibility of the presence of longer transition time-scales (or deeper traps).As is shown in Section 6.1, the proof of the upper bound in (2.25) (which corresponds to the lower bound on capacities) can actually be extended to more general setting beyond the one-dimensional case.Thus we conjecture that the inclusion process has always at most three time-scales for the motion of the condensate, although we cannot exclude the possibility of the presence of intermediate time-scales.

Discussion
Symmetric inclusion process.The paper [22] proves results similar to those of item (ii) of Theorem 2.3 in the case where the underlying random walk is symmetric, i.e., r(x, y) = r(y, x), and under the assumption that d N → 0 and d N N → ∞ as N → ∞.In this case the underlying random walk is reversible with respect to the measure m ⋆ ≡ 1, so that S = S ⋆ .In particular, all the sites of S ⋆ belong to the same connected component and the motion of the condensate involves only the first time-scale, of order 1/d N .Let us mention that the results of [22] were obtained by completely different techniques, namely by a direct scaling and expansion of the generator (2.6), that was shown to converge to the generator (2.19) of the limiting Markov process.
Multiple time-scales.Our analysis yields a metastable behavior characterised -in generalby multiple time-scales.Though we prove the existence of the second and third time-scale given in Theorems 2.5 and 2.6 only for the one-dimensional setting in (2.21), we conjecture that the same time-scales show up for general underlying dynamics and that no further time-scales can occur.In fact, we expect that the leading mechanism beyond the motion of the condensate can be reduced to a train of particles moving along single paths between metastable sets.In this sense, each path can be seen as a one-dimensional system, and the results should be proved in a similar way.
However, to formalize this idea one has first to define, for each time-scale, a new family of metastable sets obtained by merging together the metastable states that are connected on a lower time-scale (a formalization of this merging can for example be found in [4]).Then, one has to show that the reduction to one-dimensional paths is correct, or in other words, that flows of particles other than that described above, are unlikely to happen.Because of the complex geometry that may appear in general situations, this may be a rather difficult task.
For other systems with multi-scale metastable behavior see, for example, the Blume-Capel model [14,26] and the random field Curie-Weiss model [7].
Comparison with the zero range process.The zero range process (ZRP) is a well known interacting particle system that under suitable hypotheses displays the condensation phenomenon (see e.g.[23,2], and reference therein).The dynamics of a condensate for the ZRP has been studied in the finite reversible case in [5], as a first application of the martingale approach to metastability that was proposed by the same authors [4,6].The results have then been generalized to the case of a diverging number of sites [12,1], and to the totally asymmetric case [25].The quite complete picture of metastability obtained in the ZRP, allows for a comparison with the results obtained in the reversible inclusion process.In both cases: (i) the condensate is present only on sites of S ⋆ ; (ii) the metastable sets are equally probable w.r.t. the equilibrium measure, and thus they are equally stable; (iii) the energetic barriers that separate the metastable sets are (at most) logarithmic with the number of particles, thus yielding transition times that are at most polynomial in N.
More interesting are instead differences between the two processes: (a) the ZRP has only one relevant time-scale, at which the condensate can jump directly between any sites in S ⋆ .This is due to the fact that the rates of the scaling process on S ⋆ are given by the capacities of the underlying random walk, that are all positive by irreducibility, thus making irreducible also the condensate dynamics; (b) the condensate of the ZRP does not consist of N particles, but only of N − ℓ N particles, for some ℓ N such that ℓ N → ∞ and ℓ N /N → 0 as N → ∞.It is exactly due to that small number of particles wandering around the graph, that the condensate of the ZRP is able to jump to all sites of S ⋆ on the same time-scale.

Outline of the proof
As mentioned in the introduction, to prove our theorems we will use potential theory methods.In potential theory, crucial quantities (at least in the case of reversible dynamics) are capacities between sets.Let D N denote the Dirichlet form associated to the generator L N , that for F : E N → R, is given by (2.26) For two disjoint subsets A, B ⊂ E N , the capacity between A and B can be defined through the Dirichlet variational principle where F N (A, B) = {F : F (η) = 1 for all η ∈ A and F (η) = 0 for all η ∈ B}. (2.28) The unique minimizer of the Dirichlet principle is the equilibrium potential, i.e., the harmonic function h A,B that solves the Dirichlet problem (2.29)

It can be easily checked that
(2.30) As pointed out in [8]- [10], one main fact about capacities in the framework of metastability, is that they are related to the mean hitting time between sets through the formula where ν A,B is a probability measure on A such that, for all η ∈ A, and τ + A is the return time to A, i.e., τ + A = inf{t > 0 : η(t) ∈ A, η(s) = η(0) for some s ∈ (0, t)} . (2.33) Notice in particular, that when A is just a singleton, as in the situations we are dealing with, the measure ν A,B is just a Dirac delta over the singleton.The results stated in Theorems 2.3(i), 2.5(i) and 2.6 are based on (2.31) for A = E x N and B = E N (S ⋆ \ {x}).Capacities also play an important rôle in [3], where potential theory ideas and martingale methods have been combined in order to prove the scaling limit of suitably speeded-up processes, as the one that we have defined in (2.16).In our setting, where metastable sets are given by singletons, the approach of [3] to prove the convergence stated in Theorems 2.3(ii) and 2.5(ii), amounts to verifying the existence of a sequence (θ N , N ≥ 1) of positive numbers, that corresponds to the chosen time-scale, such that, for any x, y ∈ S ⋆ , x = y, the following limit exists p(x, y) := lim where r E⋆ N ( • , • ) are the jump rates of the trace process η E ⋆ N (t) .The set of asymptotic rates (p(x, y)) x,y∈S⋆ identifies the limiting dynamics.By Lemma 6.8. in [3], (2.35) so that, once more, the main tool to prove our main results turns out to be the computation of the asymptotic capacities.
The computation of the capacities in the first time-scale is performed in Section 4, while the capacities in the second and in the third time-scale are analysed in Sections 5 and 6, respectively.In all the three cases, we first provide a lower bound by restricting the Dirichlet form to a suitable subset of E N (or flow of configurations).We then use the obtained insights to construct an approximated equilibrium potential and deduce, via the Dirichlet principle, a matching upper bound.
In our lower bounds we repeatedly use the following lemma, which uniformly bounds (parts of) the Dirichlet form from below by the effective resistance of a linear electrical network.Lemma 2.7.Let R i,i+1 > 0, i = 1, . . ., k − 1.Then, for any function F : {1, . . ., k} → R, (2.36) Proof.Define the function so that g(1) = 0 and g(k) = 1.Then, where the last equality follows using the series law for the effective capacity of a linear chain (see, e.g., [27]).

Metastable sets
In this section we study the partition function Z N,S and characterize its asymptotic behavior in the limit N → ∞.This result is used to prove that the configurations in ∆ = E N \ E ⋆ N are very unlikely in equilibrium and that E x N , x ∈ S ⋆ are the metastable sets (Proposition 2.1).That in turn is the main ingredient for the proof of (2.20) and (2.24) in Theorems 2.3 and 2.5, respectively.
We start analyzing the weight function w N (ℓ).
Lemma 3.1.For d N log N → 0 as N → ∞, and 0 ≤ k < N, Proof.First note that so that indeed the two limits are the same.We rewrite and The upper bound follows from Wendel's inequality [29]: which indeed converges to 1 by our assumption on d N .
We can now compute the limiting behavior of the partition function: Proof.Since Z N,S includes the κ ⋆ configurations where all particles are on one of the sites in S ⋆ , it is clear that by Lemma 3.1.
To prove the upper bound we proceed by induction.We label the sites by 1, . . ., κ, and let E n,k be the set of configurations with n particles on the first k sites, i.e., E n,k = E n,{1,...,k} .Let us define, for 1 By induction over k, we aim to prove that, for all 1 ≤ n ≤ N , 1 ≤ k ≤ κ, and N large enough, where C k < ∞ is a constant that only depends on k and may change from line to line.We start the induction with k = 1, for which clearly, by Lemma 3.1, Assume that (3.10) holds true for k − 1 and for all 1 ≤ n ≤ N.Then, using the induction hypothesis and Lemma 3.1, Using that m ⋆ (k) ≤ 1 and d N log(n − ℓ) = o(1) by assumption, and for N large enough, the sum in ℓ can be bounded from above by Since ℓ ≤ n/2 we have that (1 − ℓ/n) ≥ 1 2 .Hence, we can bound (3.13) from above by This proves the induction step.Thus, The proposition follows by combining this upper bound with the lower bound in (3.8).
Combining these results, Proposition 2.1 follows trivially: Proof of Proposition 2.1.For all x ∈ S ⋆ , by Lemma 3.1 and Proposition 3.2, lim As a consequence, lim 4 Dynamics of the condensate on the first time-scale In this section we analyze capacities on the time-scale 1/d N and prove Theorem 2.3.We prove the lower bound on capacities in Section 4.1, the upper bound in Section 4.2, and we give the proof of Theorem 2.3 in Section 4.3.

Lower bound on capacities
Proposition 4.1.For a nonempty subset S 1 by reversibility.Note that the set A x,y N can be parameterized by the number of particles at x, and is thus a one-dimensional set.For a fixed couple x, y ∈ S ⋆ , let G be the restriction of F to the set A x,y N , i.e., for η ∈ A x,y N such that η x = ℓ, define G(ℓ) := F (η). Then we can rewrite where we used that m so that (4.4) equals Note that G(0) = 0 and G(N) = 1, so that it follows from Lemma 2.7 that Hence, and the proposition follows from Proposition 3.2.

Upper bound on capacities
Proposition 4.2.For a nonempty subset S 1 The strategy of the proof is to provide a suitable test function We first describe how we construct the test function, and then study the corresponding Dirichlet form by splitting it into several parts, and analyzing each of them separately.We conclude the section collecting all the results and providing the proof of the above proposition.
Construction of the test function.Inspired by the lower bound derived in the previous section, we want F (η) to be approximately equal to G * (η x ) , where G * is the minimizer of , which is given by To avoid difficulties for small and large values of η x , we choose an arbitrary small ε > 0 and set the function equal to 0 if η x /N ≤ ε, and equal to 1 if strictly bigger than 1 for ε small enough.All together, for any x ∈ S, we define the functions and similarly, for S 1 ⊂ S, the functions F S 1 : E N → R as Split of the Dirichlet form.To split the Dirichlet form, define, for a set A ⊆ E N , We can then write By definition , then either x 1 = x 2 and η x 1 = N, or y 1 = y 2 and η y 1 = N.In both cases, F S 1 (η z,w ) = F S 1 (η) for all z, w ∈ S because of the definition of φ ε .Therefore, we can write where S 2 = S \ S 1 .
Dirichlet form inside tubes.The main contribution to the Dirichlet form comes from configurations inside tubes between sites x ∈ S 1 , y ∈ S 2 , as the next lemma shows.
Note also that for configurations such that η x < εN, or η x > (1−ε)N , we have that F x (η z,w ) = F x (η).Hence, we can restrict the sum to configurations η such that εN ≤ η x ≤ (1 − ε)N and get Since F x (η) = φ ε (η x ) does not change if the number of particles on x stays the same, we can further rewrite this, also using reversibility, as Because of the bound on φ ′ ε (t), we have that Thus, also using Lemma 3.1, where in the second equality we used Proposition 3.2.Hence, lim sup and the lemma follows by taking the limit ε → 0.
The contribution to the Dirichlet form coming from configurations inside a tube between sites x, y ∈ S 1 is negligible, as the next lemma shows.
If a particle moves from x to y, or viceversa, the total number of particles on sites x and y stays equal to N and hence since by definition, φ ε (x) + φ ε (1 − x) = 1 for all x ∈ [0, 1].We can use again that F v does not change if the number of particles on v stays the same, and restrict the sum to configurations with Proof.As in [5], by the Cauchy-Schwarz inequality we get and then Again, we can restrict the sum to configurations with εN ≤ η x ≤ (1 − ε)N.Furthermore, if η ∈ E N \ A x N and η x = j, all sites besides x have at most N − j − 1 particles, and thus Note that if z, w = x, then F x (η z,w ) = F x (η), since F x only depends on the number of particles on x.Hence, Since |S| < ∞, we can bound r(x, y), r(y, x) ≤ max z,w∈S r(z, w) =: R and m ⋆ (x) ≤ 1, and also bound max{j(d N + N − j), (N − j)(d N + j)} ≤ j(N − j)(1 + o(1)).Combining this with (4.23), we get Notice that the last sum can be written as where the inequality follows from (3.10).Hence, also using Lemma 3.1 and Proposition 3.2, from which it follows that 1 and that together with (4.34) proves the lemma.
To prove (2.20) observe that by the stationarity of µ N we have . 5 Dynamics of the condensate on the second time-scale This section is organized similarly to the previous one.We first provide a lower bound on capacities, then a matching upper bound, and finally we give the proof of Theorem 2.5.

Lower bound on capacities
Proposition 5.1.Let the underlying random walk be as in (2.21), with κ = 3.Then, for Proof.Fix a function F ∈ F (E N (1), E N (3)).Using reversibility, we can write the Dirichlet form of F as where ξ + ∂ z denotes a configuration ξ with N − 1 particles, and with one extra particle on z.

.13)
The first sum consists of all configurations with ℓ particles on at most 2 adjacent sites in {2, . . ., κ − 1}, and with the rest of the particles only on sites 1 and κ, while the second sum consists of all other configurations.This latter sum turns out to have a negligible contribution, as we show later.Let us first analyze the first sum: η q (η q+1 + d N )r(q, q + 1) F (η q,q+1 ) − F (η) q=p η q (η q+1 + d N )r(q, q + 1) F (η q,q+1 ) − F (η) 2 , (6.14) since all other q give a 0 contribution, because then η q = 0.
Using Lemma 3.1, the above equals To bound the contribution of the last sum in (6.13), we set M = max v / ∈S⋆ m ⋆ (v) and observe that, for all such configurations and all q, m η ⋆ w N (η)η q (η q+1 + d N ) ≤ M ℓ because either at least 5 sites are occupied, or 4 sites are occupied but η q+1 = 0. Then one can show, as above, that this contribution is also negligible compared to (6.17).
To show that the sum over ℓ ≥ εN in (6.12) is negligible, we write κ−2 q=1 η q (η q+1 + d N )r(q, q + 1) ≤ (κ − 3)RN 2 , ( and hence, without loss of generality, we can optimize over functions g such that g(2) = 1 and g(κ − 1) = 0.Then, taking the infimum over all such functions g, it follows from (6.25) that lim sup because this is again the effective capacity of a linear chain.The proposition now follows by taking ε → 0.

Proof of Theorem 2.6
Proof.As a consequence of Propositions 6.1 and 6.3, if d N decays subexponentially and d N log N → 0 as N → ∞, for some constant 0 < C 1 , C 2 < ∞.In view of (2.31), recalling that Cap N (A, B) = Cap N (B, A), and applying (4.45), this provides formula (2.25) and conclude the proof of the theorem.
Remark 6.4.The constants in (6.1) and (6.6) do not match, so that we do not obtain results as in Theorems 2.3(ii) and 2.5(ii).We expect that the lower bound can be improved by not restricting the Dirichlet form to configurations with just one particle outside of S ⋆ , but also allowing a small number of particles to make the transition together.Indeed, these are the configurations that contribute to the upper bound.Including these jumps, however, does not result in a linear chain, and therefore the problem is hard to analyze.Computing the capacity in general systems is an even harder open problem, since several (possibly intersecting) paths of varying lengths can give a significant contribution to the Dirichlet form.

Lemma 4 . 5 .
Dirichlet form outside tubes We finally show in the next lemma, that the configurations outside the collections of tubes A z N gives a negligible contribution to the Dirichlet form.For d N log N → 0 as N → ∞, lim N →∞ 20) follows from Proposition 2.1.This concludes the proof of theorem.