The Parabolic Anderson Model on a Galton-Watson tree revisited

In [1] a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton-Watson random tree with an i.i.d. random potential whose marginal distribution is double-exponential. Under the assumption that the degree distribution has bounded support, two terms in the asymptotic expansion were identified under the quenched law, i.e., conditional on the realisation of the random tree and the random potential. The second term contains a variational formula indicating that the solution concentrates on a subtree with minimal degree according to a computable profile. The present paper extends the analysis to degree distributions with unbounded support. We identify the weakest condition on the tail of the degree distribution under which the arguments in [1] can be pushed through. To do so we need to control the occurrence of large degrees uniformly in large subtrees of the Galton-Watson tree.


Introduction and main results
Section 1.1 provides a brief introduction to the parabolic Anderson model. Section 1.2 introduces basic notation and key assumptions. Section 1.3 states the main theorem and gives an outline of the remainder of the paper.

The PAM and intermittency
The parabolic Anderson model (PAM) is the Cauchy problem where X is an ambient space, ∆ X is a Laplace operator acting on functions on X , and ξ is a random potential on X . Most of the literature considers the setting where X is either Z d or R d with d ≥ 1 (for mathematical surveys we refer the reader to [3], [9]). More recently, other choices for X have been considered as well: the complete graph [5], the hypercube [2], Galton-Watson trees [1], and random graphs with prescribed degrees [1].
The main target for the PAM is a description of intermittency: for large t the solution u(·, t) of (1.1) concentrates on well-separated regions in X , called intermittent islands. Much of the literature has focussed on a detailed description of the size, shape and location of these islands, and the profiles of the potential ξ(·) and the solution u(·, t) on them. A special role is played by the case where ξ is an i.i.d. random potential with a double-exponential marginal distribution P(ξ(0) > u) = e −e u/̺ , u ∈ R, (1.2) where ̺ ∈ (0, ∞) is a parameter. This distribution turns out to be critical, in the sense that the intermittent islands neither grow nor shrink with time, and therefore represents a class of its own.
The analysis of intermittency typically starts with a computation of the large-time asymptotics of the total mass, encapsulated in what are called Lyapunov exponents. There is an important distinction between the annealed setting (i.e., averaged over the random potential) and the quenched setting (i.e., almost surely with respect to the random potential). Often both types of Lyapunov exponents admit explicit descriptions in terms of characteristic variational formulas that contain information about where and how the mass concentrates in X . These variational formulas contain a spatial part (identifying where the concentration on islands takes place) and a profile part (identifying what the size and shape of both the potential and the solution are on the islands).
In the present paper we focus on the case where X is a Galton-Watson tree, in the quenched setting (i.e., almost surely with respect to the random tree and the random potential). In [1] the large-time asymptotics of the total mass was derived under the assumption that the degree distribution has bounded support. The goal of the present paper is to relax this assumption to unbounded degree distributions. In particular, we identify the weakest condition on the tail of the degree distribution under which the arguments in [1] can be pushed through. To do so we need to control the occurrence of large degrees uniformly in large subtrees of the Galton-Watson tree.

The PAM on a graph
We begin with some basic definitions and notations (and refer the reader to [3], [9] for more background).
Let G = (V, E) be a simple connected undirected graph, either finite or countably infinite. Let ∆ G be the Laplacian on G, i.e., x ∈ V, f : V → R. (1.3) Our object of interest is the non-negative solution of the Cauchy problem with localised initial condition, where O ∈ V is referred to as the root of G. We say that G is rooted at O and call G = (V, E, O) a rooted graph. The quantity u(x, t) can be interpreted as the amount of mass present at time t at site x when initially there is unit mass at O. Criteria for existence and uniqueness of the non-negative solution to (1.4) are well-known (see [6], [7] for the case G = Z d ), and rely on the Feynman-Kac formula u(x, t) = E O e t 0 ξ(Xs)ds 1l{X t = x} , (1.5) where X = (X t ) t≥0 is the continuous-time random walk on the vertices V with jump rate 1 along the edges E, and P O denotes the law of X given X 0 = O. We are interested in the total mass of the solution, U (t) := Often we suppress the dependence on G, ξ from the notation. Note that, by time reversal and the linearity of (1.4), U (t) =û(0, t) withû the solution of (1.4) with a different initial condition, namely,û(x, 0) = 1 for all x ∈ V . As in [1], throughout the paper we assume that the random potential ξ = (ξ(x)) x∈V consists of i.i.d. random variables with marginal distribution satisfying: The restrictions in (1.7) are helpful to avoid certain technicalities that require no new ideas. In particular, (1.7) is enough to guarantee existence and uniqueness of the non-negative solution to (1.4) on any discrete graph with at most exponential growth (as can be inferred from the proof in [7] for the case G = Z d ). All our results remain valid under milder restrictions (e.g. [7, Assumption (F)] plus an integrability condition on the lower tail of ξ(0)).
The following characteristic variational formula is important for the description of the asymptotics of U (t) when ξ has a double-exponential tail. Denote by P(V ) the set of probability measures on V . For p ∈ P(V ), define and set The first term in (1.9) is the quadratic form associated with the Laplacian, describing the solution u(·, t) in the intermittent islands, while the second term in (1.9) is the Legendre transform of the rate function for the potential, describing the highest peaks of ξ(·) in the intermittent islands.

The PAM on a Galton-Watson tree
Let D be a random variable taking values in N. Start with a root vertex O, and attach edges from O to D first-generation vertices. Proceed recursively: after having attached the n-th generation of vertices, attach to each one of them independently a number of vertices that has distribution D, and declare the union of these vertices to be the (n + 1)-th generation of vertices. Denote by GW = (V, E) the graph thus obtained and by P its probability law. Write P and E to denote probability and expectation for D, and supp(D) to denote the support of P. The law of D is the offspring distribution of GW, the law of D is the degree distribution of GW.
Throughout the paper, we assume that the degree distribution satisfies: Under this assumption, GW is P-a.s. an infinite tree. Moreover, where B r (O) ⊂ V is the ball of radius r around O in the graph distance (see e.g. [10, pp. 134-135]). Note that this ball depends on GW and therefore is random. For our main result we need an assumption that is much stronger than Assumption 1.2(2). To state our main result, we define the constant (1.14) With Theorem 1.4 we have completed our task to relax the main result in [1] to degree distributions with unbounded support. The extension comes at the price of having to assume a tail that decays faster than double-exponential as shown in (1.11). This property is needed to control the occurrence of large degrees uniformly in large subtrees of GW. No doubt Assumption 1.3 is stronger than is needed, but to go beyond would require a major overhaul of the methods developed in [1], which remains a challenge.
In (1.4) the initial mass is located at the root. The asymptotics in (1.14) is robust against different choices.
A heuristic explanation where the terms in (1.14) come from was given in [1, Section 1.5]. The asymptotics of U (t) is controlled by random walk paths in the Feynman-Kac formula in (1.6) that run within time r t /̺ log r t to an intermittent island at distance r t from O, and afterwards stay near that island for the rest of the time. The intermittent island turns out to consist of a subtree with degree d min where the potential has a height ̺ log(ϑr t ) and a shape that is the solution of a variational formula restricted to that subtree. The first and third term in (1.14) are the contribution of the path after it has reached the island, the second term is the cost for reaching the island.
For d ∈ N \ {1}, let T d be the infinite homogeneous tree in which every node has downward degree d. It was shown in [1] that if ̺ ≥ 1/ log(d min + 1), then Presumably T d min is the unique minimizer of (1.12), but proving so would require more work.
Outline. The remainder of the paper is organised as follows. Section 2 collects some structural properties of Galton-Watson trees. Section 3 contains several preparatory lemmas, which identify the maximum size of the islands where the potential is suitably high, estimate the contribution to the total mass in (1.6) by the random walk until it exits a subset of GW, bound the principal eigenvalue associated with the islands, and estimate the number of locations where the potential is intermediate. Section 4 uses these preparatory lemmas to find the contribution to the Feynman-Kac formula in (1.6) coming from various sets of paths. Section 5 uses these contributions to prove Theorem 1.4. Appendices A-B contain some facts about variational formulas and largest eigenvalues that are needed in Section 3.
Assumptions 1.1-1.2 are needed throughout the paper. Only in Sections 4-5 do we need Assumption 1.3.

Structural properties of the Galton-Watson tree
In the section we collect a few structural properties of GW that play an important role throughout the paper. None of these properties was needed in [1]. Section 2.1 looks at volumes, Section 2.2 at degrees, Section 2.3 at tree animals.
The tail behaviour in (2.4) requires that d max < ∞. In our setting we have d max = ∞, which corresponds to γ + = ∞, and so we expect exponential tail behaviour. The following lemma provides a rough bound. and consequently, because µ > 1, Put a n := c exp(−bc n−1 k=0 µ −(k+2) ), which satisfies 0 < a n ≤ c. From the last inequality in (2.9) it follows that E e a n+1 W n+1 ≤ E e anWn . (2.10) Since n → a n is decreasing with lim n→∞ a n = a * > 0, Fatou's lemma gives Because E[e a 0 W 0 ] = e a 0 < ∞, we get the claim.
The following lemma says that P-a.s. a ball of radius R r centred anywhere in B r (O) has volume e ϑRr+o(Rr ) as r → ∞, provided R r is large compared to log r. |B Rr (x)| = ϑ P − a.s.

(2.12)
Proof. We first prove the claim for lower balls. Afterwards we use a sandwich argument to get the claim for balls.
For y ∈ GW that lies k generations below O, let y[−i], 0 ≤ i ≤ k be the vertex that lies i generations above y. Define the lower ball of radius around y as . Let Z k denote the vertices in the k-th generation. To get the upper bound, pick δ > 0 and estimate P sup (2.14) = O(e ϑr ), and so in order to be able to apply the Borel-Cantelli lemma, it suffices to show that the probability in the last line decays faster than exponentially in r for any δ > 0. To that end, estimate where we use (2.3) with µ = e ϑ . This produces the desired estimate.
To get the lower bound, pick 0 < δ < 1 and estimate P inf (2.16) It again suffices to show that the probability in the last line decays faster than exponentially in r for any δ > 0. To that end, estimate where we use (2.5), (2.3) with µ = e ϑ , and put c − := inf L − ∈ (0, ∞). For δ small enough this produces the desired estimate. This completes the proof of (2.12) for lower balls.
To get the claim for balls, we observe that and therefore Hence we get (2.12).

Degrees
Write D x to denote the degree of vertex x. The following lemma implies that, P-a.s. and for r → ∞, D x is bounded by a vanishing power of log r for all x ∈ B 2r (O).

Lemma 2.3. [Maximal degree in a ball around the root]
(a) Subject to Assumption 1.2(2), for every δ > 0, , it suffices to show that P(D > δr) = O(e −cr ) for some c > 2ϑ. Since P(D > δr) ≤ e −aδr E(e aD ), the latter is immediate from Assumption 1.2(2) when we choose a > 2ϑ/δ. (b) The only change is that in the last line P(D > δr) must be replaced by P(D > (log r) δr ). To see that the latter is O(e −cr ) for some c > 2ϑ, we use the tail condition in (1.11) with δ r = f (s) and s = log r.

Tree animals
For n ∈ N 0 and x ∈ B r (O), let be the set of tree animals of size n + 1 that contain x. Put a n (x) = |A n (x)|.
Lemma 2.4. [Number of tree animals] Subject to Assumption 1.2(2), P-a.s. there exists an r 0 ∈ N such that a n (x) ≤ r n for all r ≥ r 0 , x ∈ B r (O) and 0 ≤ n ≤ r.
Proof. We first prove the claim for lower tree animals. Afterwards we us a sandwich argument to get the claim for tree animals.
be the set of lower tree animals of size n + 1 that contain x. Put a ↓ n (x) = |A ↓ n (x)|. Fix δ > 0. By Lemma 2.3(a) and the Borel-Cantelli lemma, P-a.s. there exists an . Any lower tree animal of size n + 1 containing a vertex in . Any lower tree animal of size n + 1 can be created by adding a vertex to the outer boundary of a lower tree animal of size n. This leads to the recursive inequality Pick δ = 1 to get the claim for lower tree animals.
To get the claim for tree animals, note that a n (x) ≤ n k=0 a ↓ n (x[−k]) (compare with (2.19)), and so a n (x) ≤ (n + 1)r n for all x ∈ B r (O) and all 0 ≤ n ≤ r.

Preliminaries
In this section we extend the lemmas in [1, Section 2]. Section 3.1 identifies the maximum size of the islands where the potential is suitably high. Section 3.2 estimates the contribution to the total mass in (1.6) by the random walk until it exits a subset of GW. Section 3.3 gives a bound on the principal eigenvalue associated with the islands. Section 3.5 estimates the number of locations where the potential is intermediate.

Maximum size of the islands
For every r ∈ N there is a unique a r such that By Assumption 1.1, for r large enough For r ∈ N and A > 0, let be the set of vertices in B r (O) where the potential is close to maximal, be the S r -neighbourhood of Π r,A , and C r,A be the set of connected components of D r,A in GW, which we think of as islands. For M A ∈ N, define the event Note that Π r,A , D r,A , B r,A depend on GW and therefore are random.
Proof. We follow [4, Lemma 6.6]. By Assumption 1.1, for every x ∈ V and r large enough, with c A = e −2A/̺ . By Lemma 2.2, P-a.s. for every y ∈ B r (O) and r large enough, where we use that S r = o(log r) = o(r), and hence for every m ∈ N, Consequently, P-a.s.
By choosing m > 1/c A , we see that the above probability becomes summable in r, and so we have proved the claim with M A = ⌈1/c A ⌉.
Lemma 3.1 implies that (P × P)-a.s. B r,A does not occur eventually as r → ∞. Note that P-a.s. on the event [B r,A ] c , where the last inequality follows from Lemma 2.2.

Mass up to an exit time
Proof. We follow the proof of [7, Lemma 2.18] and [12,Lemma 4.2]. Define This is the solution to the boundary value problem Via the substitution u =: 1 + v, this turns into It is readily checked that for γ > λ Λ the solution exists and is given by where R γ denotes the resolvent of ∆ + ξ in ℓ 2 (Λ) with Dirichlet boundary condition. Hence where 1 denotes the constant function equal to 1, and ·, · Λ denotes the inner product in ℓ 2 (Λ). To get the first inequality, we combine Lemma 2.3(a) with the lower bound in (B.2) from Lemma B.1, to get ξ − γ ≤ λ Λ + δr − γ ≤ δr on Λ. The positivity of the resolvent gives To get the second inequality, we write To get the third inequality, we use the Fourier expansion of the resolvent with respect to the orthonormal basis of eigenfunctions of ∆ + ξ in ℓ 2 (Λ).

Principal eigenvalue of the islands
The following lemma provides a spectral bound. all C ∈ C r,A satisfy : λ C (ξ; GW) ≤ a Lr − χ C (GW) + ε.  ∞). Next, by Lemma 2.4, for any x ∈ B r (O) and 1 ≤ n ≤ r, the number of connected subsets Λ ⊂ V with x ∈ Λ and |Λ| = n + 1 is P-a.s. at most (n + 1)r n ≤ e 2n log r for r ≥ r 0 . Noting that e Sr ≤ r, we use a union bound and that by Lemma 2.2 log L r = ϑr + o(r) as r → ∞ P-a.s., to estimate for r large enough,

Maximum of the potential
The next lemma shows that a Lr is the leading order of the maximum of ξ in B r (O).

Path expansions
In this section we extend [1, Section 3]. Section 4.1 proves three lemmas that concern the contribution the total mass in (1.6) coming from various sets of paths. Section 4.2 proves a key proposition that controls the entropy associated with a key set of paths. The proof is based on the three lemmas in Section 4.1. Proof. The proof is identical to that of Lemma 3.2, with δr replaced by (log r) δr (recall Lemma 2.3).
Recall the definitions from Section 3.1. For π ∈ P and A > 0, define with the convention sup ∅ = −∞. This is the largest principal eigenvalue among the components of C r,A in GW that have a point of high exceedance visited by the path π.

Mass of the solution along excursions
Proof. The proof is identical to that of [1, Lemma 3.2]. The left-hand side of (4.7) can be evaluated by using the fact that T ℓ is the sum of ℓ independent Exp(deg(π i )) random variables that are independent of π (ℓ) (X). The condition on γ ensures that all ℓ integrals are finite.
Proof. The proof is identical to that of [1,Lemma 3.3], with d max replaced by (log r) δr (recall Lemma 2.3).
We follow [1,Definition 3.4] and [11,Section 6.2]. Note that the distance between Π r,A and D c r,A in GW is at least S r = (log L r ) α (recall (3.4)-(3.5)).
Whenever supp(π) ∩ Π r,A = ∅ and ε > 0, we define to be the total time spent in exterior excursions, respectively, on moderately low points of the potential visited by exterior excursions (without their last point).

Key proposition
The main result of this section is the following proposition.
and  Fix A ≥ A 0 (r), β ∈ (0, α) and ε ∈ (0, 1 2 β) as in Lemma 3.6. Let r 0 ∈ N be as given in Lemma 4.5, and take r ≥ r 0 so large that the conclusions of Lemmas 2.3, 3.1, 3.3 and 3.6 hold, i.e., assume that the events B r and B r,A in these lemmas do not occur. Fix x ∈ B r (O). Recall the definitions of C r,A and P (m,s) . Note that the relation ∼ is an equivalence relation in P (m,s) , and define P (m,s) x := equivalence classes of the paths in P(x, V ) ∩ P (m,s) . (4.23) The following bounded on the cardinality of this set is needed. Proof. We can copy the proof of [1, Lemma 3.6], replacing d max by (log r) δr .
Note that q Sr r,A < e −3c 0 log r , so Thus the proof will be finished once we show that, for some ε ′ > 0 and whp, respectively, a.s. eventually as r → ∞, We can copy the argument at the end of [1,Section 3.4]. For each π ∈ N define an auxiliary path π ⋆ as follows. First note that by using our assumptions we can find points z ′ , z ′′ ∈ supp(π) (not necessarily distinct) such that where the latter holds by (3.12). Write {z 1 , z 2 } = {z ′ , z ′′ } with z 1 , z 2 ordered according to their hitting times by π, i.e., inf{ℓ : π ℓ = z 1 } ≤ inf{ℓ : π ℓ = z 2 }. Define π e as the concatenation of the loop erasure of π between x and z 1 and the loop erasure of π between z 1 and z 2 . Since π e is the concatenation of two self-avoiding paths, it visits each point at most twice. Finally, define π ⋆ ∼ π e by replacing the excursions of π e from Π r,A to D c r,A by direct paths between the corresponding endpoints, i.e., replace eachπ (i) e by |π (i) e | = ℓ i , (π (i) e ) 0 = x i ∈ Π r,A , and (π (i) e ) ℓ i = y i ∈ D c r,A by a shortest-distance path π (i) ⋆ with the same endpoints and | π (i) ⋆ | = dist G (x i , y i ). Since π ⋆ visits each x ∈ Π r,A at most 2 times, Applying Lemmas 3.6-3.7 and using (3.1) and L r > r, we obtain, for r large enough, On the other hand, since | supp(π ⋆ )| ≥ (log L r ) κ , by (4.32) we have where the first inequality uses that the distance between two points on π ⋆ is less than the total length of π ⋆ . Now (4.31) follows from (4.34)-(4.35).

Upper bound
We follow [1,Section 4.2]. The proof of the upper bound in (5.2) relies on two lemmas showing that paths staying inside a ball of radius ⌈t γ ⌉ for some γ ∈ (0, 1) or leaving a ball of radius t log t have a negligible contribution to (1.6), the total mass of the solution. where J t is the number of jumps of X up to time t, and we use that |B r (O)| ≤ (log r) δrr . Next, J t is stochastically dominated by a Poisson random variable with parameter t(log r) δr . Hence for large r. Using that ℓ t ≥ t log t, we can easily check that, for r ≥ ℓ t and t large enough, Since lim t→∞ ℓ t = ∞ and lim t→∞ U * (t) = ∞, this settles the claim. Hence with F t (r) := ̺ log(ϑr) − r t log(ε̺ log(ϑr)) − δ r log(log r) , r > 0. (5.22) The function F t is maximized at any point r t satisfying ̺t = r t log(ε̺ log(ϑr t )) − (δ r + r d dr δ r ) log log r + Inserting (5.24) into (5.21), we obtain which is the desired upper bound because ε > 0 is arbitrary. (5.2). To avoid repetition, all statements hold (P × P)-a.s. eventually as t → ∞. Set

Proof of the upper bound in
Since ε > 0 is arbitrary, this completes the proof of the upper bound in (1.14).

Lower bound
We follow [1, Section 4.1]. Fix ε > 0. By the definition of χ, there exists an infinite rooted tree T = (V ′ , E ′ , Y) with degrees in supp(D g ) such that χ T (̺) < χ(̺) + 1 4 ε. Let Q r = B T r (Y) be the ball of radius r around Y in T . By Proposition A.1 and (A.2), there exist a radius R ∈ N and a potential profile q : For ℓ ∈ N, let B ℓ = B ℓ (O) denote the ball of radius ℓ around O in GW. We will show next that, (P × P)-a.s. eventually as ℓ → ∞, B ℓ contains a copy of the ball Q R where the potentail ξ is bounded from below by ̺ log log |B ℓ | + q.
Proposition 5.4. [Balls with high exceedances] (P × P)-almost surely eventually as ℓ → ∞, there exists a vertex z ∈ B ℓ with B R+1 (z) ⊂ B ℓ and an isomorphism ϕ : In particular, Any such z necessarily satisfies |z| ≥ cℓ (P × P)-a.s. eventually as ℓ → ∞ for some constant c = c(̺, ϑ, χ(̺), ε) > 0. Proof of the lower bound in (1.14). Let z be as in Proposition 5.4. Write τ z for the hitting time of z by the random walk X. For s ∈ (0, t), we estimate where we use the strong Markov property at time τ z . We first bound the last term in the integrand in (5.30). Since ξ ≥ ̺ log log |B ℓ | + q in B R (z), for large v, where we used that B R+1 (z) is isomorphic to Q R+1 for the indicators in the first inequality, and applied Lemma B.2 and (5.28) to obtain the second and third inequalities, respectively. On the other hand, since ξ ≥ 0, and we can bound the latter probability from below by the probability that the random walk runs along a shortest path from the root O to z within a time at most s. Such a path (y i ) |z| i=0 has y 0 = O, y |z| = z, y i ∼ y i−1 for i = 1, . . . , |z|, has at each step from y i precisely deg(y i ) choices for the next step with equal probability, and the step is carried out after an exponential time E i with parameter deg(y i ). This gives where Poi γ is the Poisson distribution with parameter γ, and P is the generic symbol for probability. Summarising, we obtain where in the last inequality we use that s ≤ |z| and ℓ ≥ |z|. Further assuming that ℓ = o(t), we see that the optimum over s is obtained at Note that, by Proposition 5.4, this s indeed satisfies s ≤ |z|. Applying (1.10) we get, after a straightforward computation, (P × P)-a.s. eventually as t → ∞, Inserting log |B ℓ | ∼ ϑℓ, we get The optimal ℓ for F ℓ satisfies i.e., ℓ = r t [1 + o (1)]. For this choice we obtain Since ε > 0 is arbitrary, this completes the proof of the lower bound in (1.14).
REMARK: It is clear from (5.23) and (5.39) that, in order to get the correct asymptotics, it is crucial that both δ r and r d dr δ r tend to zero as r → ∞. This is why Assumption 1.3 is the weakest condition on the tail of the degree distribution under which the arguments in [1] can be pushed through.

A Dual variational formula
We introduce alternative representations for χ in (1.9) in terms of a 'dual' variational formula. withz = arg max z∈Γ q(z) and Dz the degree ofz.
(2) The eigenfunction corresponding to λ Λ (q; G) can be taken to be non-negative.
(3) If q is real-valued and Γ Λ is finite and connected in G, then the second inequality in (B.2) is strict and the eigenfunction corresponding to λ Λ (q; G) is strictly positive.
Proof. Write which settles the claim in (1). The claims in (2) and (3) are standard.