Proposition 4.7. Under the assumptions of Theorem 1.8 we have limτ→∞Trγτ,p = Trγp for all
p∈N.
This subsection is devoted to the proof of Proposition 4.7. This is done in several steps, using arguments similar to those used in the proof of Proposition 4.1. The main difference is that we have to work with a different type of observable, and, consequently, with a different graph structure from the one used in Subsection 4.1. More precisely, we fix throughout this sectionp ∈Nand we defineξ ≡ξp to be the identity operator onH(p), with integral kernel
ξ(x1, . . . , xp;y1, . . . , yp) ..= p
Y
j=1
δ(xj −yj). (4.15)
Note that ξ /∈Bp. Nevertheless, we can extend the definitions (2.9) and (3.6) of Aξτ and Aξ forξ
as in (4.15). Note that here we again takeη = 0. With this definition, we have
Trγτ,p = Aξτ(1), Trγp = Aξ(1). (4.16)
With the same notation as in the preceding subsection, we set up the Duhamel expansion ofAξτ,p(z)
and Aξτ(z) in Rez >0. We first focus on the quantum case. For fixed m∈N we work again with
the setX ≡ X(m, p) from Definition2.5and the set of partitionsR≡R(m, p) from Definition 4.2. We need to modify the graph structure in Definition 2.11for the current setting.
Definition 4.8. Fix m, p∈N. To each Π∈Rwe assign an edge-coloured undirected multigraph ( ˜VΠ,E˜Π,σ˜Π)≡( ˜V,E˜,σ˜), with a colouring ˜σ ..E → ±1, as follows.
(i) OnX we introduce the equivalence relationα∼βif and only ifiα=iβandrα=rβ. We define
the vertex set ˜V ..={[α] ..α∈ X }as the set of equivalence classes ofX. We use the notation ˜
(ii) The set ˜V carries a total order 6 inherited from X: [α] 6 [β] whenever α 6 β. Note that here we use the lexicographic order onX introduced in Section4.1.
(iii) For a pairing Π ∈ R, each edge (α, β) ∈ Π gives rise to an edge e = {[α],[β]} of ˜E with ˜
σ(e) ..=δβ.
(iv) We denote by conn( ˜E) the set of connected components of ˜E, so that ˜E =F
P∈conn( ˜E)P. We
call the connected components P of ˜E paths.
Note that in point (i) of Definition (2.11), we in particular view α and β as equivalent when
iα=iβ =m+ 1 andrα =rβ. This is the main difference with the graph structure from Definition
2.11. By a slight abuse of notation, we denote both the equivalence relation from Definition 2.11
and from Definition 4.8 by ∼. From context, it will be clear to which equivalence relation we are referring. From Definitions4.2and4.8, we deduce that each vertex of ˜V has degree 2. Therefore ˜V
factorizes as a product of closed paths. In particular, these closed paths can be loops. Moreover, all elements of conn( ˜E) are closed paths.
For the following we fix m, p∈N and a pairing Π∈R, and let ( ˜V,E˜,σ˜) denote the associated graph from Definition4.8. Givent∈A≡A(m), we appropriately modify Definition2.7and define the value Iτ,ξΠ(t) of Π at t by the right-hand side of (2.38) with V and E replaced by ˜V and ˜E
respectively. Note that (4.3) holds in this setting.
With each x = (xα)α∈X ∈ ΛX and t = (tα)α∈X ∈ A we associate integration labels y =
(ya)a∈V˜ ∈Λ ˜
V and time labels s = (s
a)a∈V˜ ∈R ˜
V as in (2.31), except that we now take the vertex
set ˜V and the equivalence relation ∼ to be the one from Definition 4.8. In addition we adapt the splitting (2.36) to this context, where nowyi ..= (ya)a∈V˜i. Given P ∈conn( ˜E) we denote by ˜V(P)
the set of vertices of P. Moreover, we let ˜Vi(P) ..= ˜V(P)∩V˜i fori= 1,2. Given e∈E˜, we define
Jτ,e as in Definition2.14. (Here we replace each occurrence ofV and E in Definition2.14by ˜V and
˜
E respectively.)
Lemma 4.9. Suppose that P ∈conn( ˜E). Then Z ΛV˜(P) Y a∈V(P)˜ dya Y e∈P Jτ,e(ye,s) 6 C |V(P)|˜ . (4.17)
Proof. We consider two cases: (i) ˜V(P)⊂V˜2 and (ii) ˜V1(P)6=∅. In case (i), all of the vertices in
P belong to ˜V2 and the claim follows by arguing as in the proof of Lemma4.4.
Let us therefore focus on case (ii). If ˜V(P) = 1, thenP is a loop in ˜V1 and so the left-hand side
of (4.17) equals kGτkS1, which satisfies the claimed bound.
We henceforth assume ˜V(P) > 2. Since P is a closed path, there exists l ∈ N and distinct
elements b1, . . . , bl ∈ V˜1 such that P = Flj=1Pj, where for each j = 1, . . . , l the path Pj is of
the form Pj = {ej1, ej2, . . . , ejqj} for some qj ∈ N and edges e j
k such that bj ∈ e j
1, bj+1 ∈ ejqj and
ejk∩ejk+1 ∈ V˜2, for all k= 1, . . . , qj−1. Here we set bl+1 ..= b1. (It is possible to haveqj = 1 in
which casePj is the path of length 1 joiningbj and bj+1.)
We now write the left-hand side of (4.17) as
Z Λl dyb1· · ·dybl l Y j=1 Z ΛV˜2(Pj) Y a∈V˜2(Pj) dya Y e∈Pj Jτ,e(ye,s) ! . (4.18)
Arguing as in (2.55)-(2.56) we can get rid of all the time evolutions in thej-th factor of the integrand in (4.18) for all j= 1, . . . , l. In particular, the proof of (2.57) (in the proof of Lemma2.18) implies that the j-th factor is
6 C|V˜2(Pj)| 1 +kG τkS2(H) |V˜2(Pj)|k Gτ(ybj;·)kH kGτ(·;ybj+1)kH+Gτ(ybj;ybj+1) .
The estimate (4.17) now follows by applying the Cauchy-Schwarz inequality inyb1, . . . , ybland using
LemmaC.1.
From Lemma4.9and (4.1) we deduce that, for any Π∈R,t∈A, the estimate (4.4) holds when
ξ is given by (4.15). In particular, it follows that Corollary 2.21 holds in this setting. Next, for Π∈R, we defineIΠξ as in Definition2.24withV andEreplaced by ˜V and ˜E respectively. We define
aξ∞,m as in (4.6). With this notation, (2.79) still holds in this setting by using the same telescoping
proof adapted to the one-dimensional setting as in Section 4.1. For the remainder term, we note that for R(t, z) by (4.8) we have that (4.9) holds even if we take ξ as in (4.15). The proof is a minor modification of the proof of Proposition4.5. More precisely, we note that for S(t) given by (4.10), (4.13) holds if we take ˜ξ ..=ξ. Namely, in this case Θτ(ξ)
(n)
>0 for alln∈Nin the sense of operator kernels. The proof now proceeds as in Section4.1. We hence deduce that Corollary4.6
holds in this setting. As in the previous subsection, we deduce that the function Aξτ is analytic in
{z.. Rez >0}
We now consider the classical case. The identity (3.7) holds whenξ is given by (4.15). Arguing as earlier, we obtain (3.8) in this setting. As in the proof of Lemma3.3, we reduce the estimate of the remainder term to that of the explicit term by the trivial estimate |RξM(z)|6 |zM|M! R
Θ(ξ)WMdµ. We deduce that Lemma3.3holds in this setting. Note that now we cannot use Cauchy-Schwarz as earlier. Moreover, the same proof allows us to deduce that the functionAξ is analytic for Rez >0. We now conclude the proof of Proposition4.7. Combining all of the results of this subsection, it follows that the assumptions of PropositionA.1are satisfied withν= (Cp)p, σ=CkwkL∞. Hence,
it follows that, for Rez >0 we have Aξτ(z) → Aξ(z) as τ → ∞. In particular, settingz = 1 and
recalling (4.16), we obtain Proposition 4.7.