Continuum directed random polymers on disordered hierarchical diamond lattices

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Continuum directed random polymers on disordered hierarchical diamond lattices Jeremy Thane Clark University of Mississippi, Department of Mathematics, United States Received 1 March 2018; received in revised form 6 February 2019; accepted 16 May 2019 Available online xxxx

Abstract I discuss models for a continuum directed random polymer in a disordered environment in which the polymer lives on a fractal called the diamond hierarchical lattice, a self-similar metric space forming a network of interweaving pathways. This fractal depends on a branching parameter b ∈ N and a segmenting number s ∈ N. For s > b my focus is on random measures on the set of directed paths that can be formulated as a subcritical Gaussian multiplicative chaos. This path measure is analogous to the continuum directed random polymer introduced by Alberts et al. (2014). c 2019 Elsevier B.V. All rights reserved. ⃝ Keywords: Gaussian multiplicative chaos; Diamond hierarchical lattice; Random branching graphs

1. Introduction Alberts, Khanin, and Quastel [2,3] introduced a continuum directed random polymer (CDRP) model for a one-dimensional Wiener motion (the polymer) over a time interval [0, 1] whose law is randomly transformed through a field of impurities spread throughout the medium of the polymer. The polymer’s disordered environment is generated by a time–space Gaussian white noise {W(x)}x∈D where D := [0, 1] × R (in other terms, W is a δ-correlated Gaussian field). For an inverse temperature parameter β > 0, the CDRP is a random probability measure QW β on the set of trajectories Γ := C([0, 1]) that is formally expressed as β2 1 2 M(d p) for M(d p) = eβW p − 2 E[W p ] µ(d p) , (1.1) M(Γ ) where µ refers to the standard Wiener measure on Γ and {W p } p∈Γ is a Gaussian ∫ 1 (field )formally defined by integrating the white noise over a Brownian trajectory: W p = 0 W p(r ) dr . The

Q W (d p) =

E-mail address: [email protected]. https://doi.org/10.1016/j.spa.2019.05.008 c 2019 Elsevier B.V. All rights reserved. 0304-4149/⃝

Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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random measure M ≡ M(W) is a function of the field such that E[M] = µ and yet M is a.s. singular with respect to µ. The rigorous mathematical meaning of the random measure M in (1.1) requires special consideration since exponentials of the field W p do not have an immediately clear meaning, and, indeed, if the measure M is singular with respect to µ the expression exp{βW p − (β 2 /2)E[W2p ]} cannot define a Radon–Nikodym derivative d M/dµ anyway. The construction approach of M in [2] involves an analysis of the finite-dimensional distributions through Wiener chaos expansions. Another point-of-view is that the random measure M has the form of a Gaussian multiplicative chaos (GMC) measure over the Gaussian field {W p } p∈Γ . GMC theory began with an article by Kahane [19] and much of the progress on this topic has been motivated by the demands of quantum gravity theory [4,9,10,23] although GMC theory arises in many other fields, including random matrix theory [27] and number theory [25]. A GMC is classified as subcritical or critical, respectively, depending on whether the expectation measure, E[M], is σ -finite or not. Because of its relevance to quantum gravity, the relatively unwieldy case of critical GMC has attracted the most attention, with recent results in [5,18,22]. The random measure M in (1.1) is subcritical since E[M] = µ is a probability measure. Shamov [26] has formulated subcritical GMC measure theory in a particularly complete and accessible way. In this article, I will study a GMC measure analogous to (1.1) for a CDRP living on a fractal D b,s referred to as the diamond hierarchical lattice. Given a branching parameter b ∈ {2, 3, . . .} and a segmenting parameter s ∈ {2, 3, . . .}, the diamond hierarchical lattice is a compact metric space that embeds bs shrunken copies of itself, which are arranged through b branches that each have s copies running in series; see the construction outline below in (A) and (B) of Section 1.1. Diamond hierarchical lattices provide a useful setting for formulating toy statistical mechanical models; for example, [12–16,20,24]. Lacoin and Moreno [21] considered (discrete) directed polymers on disordered diamond hierarchical lattices, classifying the disorder behavior based on the cases b < s, b = s, and b > s, which are combinatorially analogous, respectively, to the d = 1, d = 2, and d > 3 cases of directed polymers on the (1 + d)-rectangular lattice. In [1], we studied a functional limit theorem for the partition function of the b < s diamond lattice polymer in a scaling limit in which the temperature grows as a power law of the length of the polymer. This limit result is analogous to the intermediate disorder regime in [3] for directed polymers on the (1 + 1)-rectangular lattice. My focus here will be on developing the theory for a CDRP corresponding to the limiting partition function obtained in [1]. A similar CDRP model on the b = s diamond lattice, if it exists, will require a different approach for its construction; see [8] for computations relevant to the continuum limit of discrete polymers. It is interesting to compare this question with results [6,7] by Caravenna, Sun, and Zygouras on scaling limits of the partition function for (1 + 2)-rectangular lattice polymers. 1.1. Overview of the continuum directed random polymer on the diamond lattice In this section I will sketch the construction of the continuum directed random polymer on the diamond hierarchical lattice and explore some of its properties. I discuss the diamond lattice fractal and its relevant substructures in more detail in Section 2. Proofs of propositions are placed in Section 3. (A). The sequence of diamond hierarchical graphs For a branching number b ∈ {2, 3, 4, . . .} and a segmenting number s ∈ {2, 3, 4, . . .}, the first diamond graph D1b,s is defined by b parallel branches connecting two root nodes A and Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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b,s B wherein each branch is formed by s bonds running in series. The graphs Dn+1 , n ≥ 1 are b,s then constructed inductively by replacing each bond on D1 by a nested copy of Dnb,s ; see the illustration below of the (b, s) = (2, 3) case.

The set of bonds (edges) on the graph Dnb,s is denoted by E nb,s . A directed path is a oneto-one function p : {1, . . . , s n } → E nb,s such that the bonds p( j), p( j + 1) are adjacent for 1 ≤ j ≤ s n − 1 and the bonds p(1) and p(s n ) connect to A and B, respectively. I will use the following notations: Vnb,s

Set of vertex points on Dnb,s

E nb,s

Set of bonds on the graph Dnb,s

Γnb,s

Set of directed paths on Dnb,s

[p]n

The path in Γnb,s determined by p ∈ Γ Nb,s for N > n

Remark 1.1. The hierarchical structure of the sequence of diamond graphs implies that Vnb,s is b,s canonically embedded in VNb,s for N > n. The vertices in Vnb,s \Vn−1 are referred to as the nth generation vertices. In the same vein, E nb,s and Γnb,s define equivalence relations on E Nb,s and Γ Nb,s , respectively. For instance, p, q ∈ Γ Nb,s are equivalent up to generation n if [ p]n = [q]n . (B). The diamond hierarchical lattice Intuitively, the diamond hierarchical lattice, D b,s , is a fractal that emerges as the “limit” of the diamond graphs, Dnb,s , as n → ∞. My convention is to view D b,s as a metric space embedding a family of interweaving copies of the interval [0, 1] for which the endpoints 0 and 1 are identified with the root vertices A and B, respectively. In this framework directed paths are isometric maps p : [0, 1] → D b,s with p(0) = A and p(1) = B. In Section 2 and Appendix A, I will give precise definitions for D b,s and for the sets V b,s , E b,s , Γ b,s and the measures µ, ν described in the notation list below. V b,s

Set of vertex points on D b,s

b,s

Complement of V b,s in D b,s

E

Γ b,s

Set of directed paths on D b,s

Di,b,sj

First generation embedded copies of D b,s on the jth segment of the ith branch

ν

Uniform probability measure on D b,s

µ

Uniform probability measure on Γ b,s

[ p]n

The path in Γnb,s determined by p ∈ Γ b,s

Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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Remark 1.2. V b,s is a countable, dense subset of D b,s . b,s Remark 1.3. In analogy to Remark 1.1, V b,s is canonically identifiable with ∪∞ n=1 Vn . Also, b,s b,s b,s b,s E n and Γn define equivalence relations on E and Γ for each n ∈ N.

Remark 1.4. The measure ν is defined such that ν(V b,s ) = 0 and, under the interpretation of Remark 1.3, ν(e) = 1/|E nb,s | for each n ∈ N and e ∈ E nb,s . Similarly µ is defined so that µ(p) = 1/|Γnb,s | for any p ∈ Γnb,s . ( ) Remark 1.5. Let Γi,b,sj , µ(i, j) be copies of (Γ b,s , µ) corresponding to the embedded subcopies, Di,b,sj , of D b,s . The path space (Γ b,s , µ) can be decomposed as Γ b,s =

b ⋃

b

s

×Γ

b,s i, j

and

µ =

i=1 j=1

s

1 ∑ ∏ (i, j) µ b i=1 j=1

(1.2)

by way of s-fold concatenation of the paths. I will assume that b < s throughout the remainder of the text except where specified otherwise. Proposition 1.6. Fix some p ∈ Γ b,s and let q ∈ Γ b,s be chosen uniformly at random, i.e., according to the measure µ(dq). Define the set of intersection times I p,q = {r ∈ [0, 1] | ∑n (n) p(r ) = q(r )}, and define N p,q := sk=1 1[ p]n (k)=[q]n (k) , in other terms, as the number of bonds s−log b . shared by [ p]n , [q]n ∈ Γnb,s . Finally, let h ∈ (0, 1) be defined as h := log log s 1 h (n) (n) (i). As n → ∞ the sequence of random variables T p,q p,q [= ( s{n ) N }] converges a.s. to a limit (n) T p,q . The moment generating function ϕn (t) = E exp t T p,q converges pointwise to (n) the moment generating function of T p,q , and, in particular, the second moment of T p,q converges to the second moment of T p,q . [ (ii). I p,q is a finite set with probability 1 − pb,s for pb,s ∈ (0, 1) satisfying pb,s = b1 1 − (1 − ] pb,s )s . In this case, the intersections occur only at vertex points. (iii). In the event that I p,q is infinite, the Hausdorff dimension of I p,q is a.s. h.

Definition 1.7. The intersection time T p,q of two paths p, q ∈ Γ b,s is defined as ( 1 )h (n) . T p,q := lim n N p,q n→∞ s Remark 1.8. The intersection time satisfies the formal identity ∫ 1∫ 1 ( ) δ D p(r ), q(t) dr dt = T p,q , 0

(1.3)

0

∫ where δ D is the δ-distribution on D b,s satisfying f (x) = Db,s δ D (x, y) f (y)ν(y) for a test function f : D b,s → R. The above identity can be understood in terms of the discrete graphs Dnb,s for which any two directed paths p, q : {1, . . . , s n } → E nb,s satisfy ( 1 )h ∑ ( 1 )h 1 ∑ 1 ∑ 1p j =qk (n) = 1 = N p,q . (1.4) p =q s n 1≤ j≤s n s n 1≤k≤s n (bs)−n s n 1≤ j≤s n j j sn Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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(C). A Gaussian field on directed paths Let W denote a Gaussian white noise on (D b,s , ν) defined within some probability space (Ω , F, P), i.e., a linear map from H := L 2 (D b,s , ν) into a Gaussian subspace of L 2 (Ω , F, P) such that ) ( ψ ∈H ↦−→ W(ψ) ∼ N 0, ∥ψ∥2H . I also use the alternative notations ∫ W(ψ) ≡ ⟨W, ψ⟩ ≡ W(x)ψ(x)ν(d x) , D b,s

[ ] where the field W(x), x ∈ D formally satisfies the δ-correlation E W(x)W(y) = δ D (x, y). Next I discuss a field W p , p ∈ Γ b,s formally defined by integrating the white noise along paths: ∫ 1 ( ) W p(r ) dr. Wp = b,s

0

The kernel K Γ ( p, q) is equal to the intersection time, T p,q , since (1.3) yields that ∫ 1∫ 1 ( ) [ ] δ D p(r ), q(t) dr dt = T p,q . K Γ ( p, q) = E W p Wq = 0

0

Define the linear map Y : H −→ L 2 (Γ b,s , µ) as ∫ 1 ( ) ψ p(r ) dr . (Y ψ)( p) = 0

To see that Y is a bounded operator, notice that by Jensen’s inequality ∫ ∫ 1 ∫ ⏐ ⏐ ( ⏐ )⏐ ⏐ψ p(r ) ⏐2 dr µ(d p) ⏐(Y ψ)( p)⏐2 µ(d p) ≤ Γ b,s 0 Γ b,s ∫ ⏐2 ⏐ ⏐ψ(x)⏐ ν(d x) = ∥ψ∥2 , = H D b,s

where the first equality holds because the weight assigned by ν to a set R ⊂ D b,s is equal to the expected amount of time that a path p ∈ Γ b,s chosen uniformly at random spends in R; see Proposition 2.9 and Remark 2.10. In particular, Y is continuous when the topology of the codomain is identified with L 0 (Γ b,s , µ), i.e., convergence in measure with respect to µ. In the terminology of [26], a continuous function Y from H to L 0 (Γ b,s , µ) is referred to as a generalized H-valued function over the measure space (Γ b,s , µ). The pair (W, Y ) encodes the Gaussian field W p on Γ b,s by defining ∫ ( ) W p f ( p)µ(d p) := ⟨W, Y ∗ f ⟩ for a test function f ∈ L 2 Γ b,s , µ . (1.5) Γ b,s

∫ ( ) The adjoint Y ∗ : L 2 Γ b,s , µ → H can be expressed in the form (Y ∗ f )(x) = Γ b,s f ( p) ∫1 0 δ D ( p(r ), x). I will use the notation (Y ψ)( p) ≡ ⟨Y p , ψ⟩ and summarize (1.5) as W p = ⟨Y p , W⟩. (D). Gaussian multiplicative chaos on paths In this section, I discuss a random measure Mβ on Γ b,s formally related to µ and the field W p through Mβ (d p) = eβW p −

β2 2 2 E[W p ]

µ(d p) .

(1.6)

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} { 2 The above does not define exp βW p − β2 E[W2p ] as a Radon–Nikodym derivative since {W p } p∈Γ b,s is a Gaussian field rather than just an indexed family of centered Gaussian random variables. I state the proposition below using Shamov’s formulation of subcritical GMC measures [26]. Proposition 1.9. There exists a unique random measure, Mβ (d p), on (Γ b,s , µ) satisfying the properties (I)–(III) below. (I). E[Mβ ] = µ (II). Mβ is adapted to the white noise W. Thus I can write Mβ (d p) ≡ Mβ (W, d p). (III). For ψ ∈ H and a.e. realization of the field W, Mβ (W + ψ, d p) = eβ(Y ψ)( p) Mβ (W, d p) . Remark 1.10. Mβ is the subcritical GMC on (Γ b,s , µ) over the field (W, Y ) with expectation µ. { } Remark 1.11. I prove Proposition 1.9 by constructing a sequence of GMC measures Mβ(n) n∈N that form a martingale and have well-defined Radon–Nikodym derivatives d Mβ(n) /dµ. By [26] the existence and uniqueness of the subcritical GMC measure Mβ in Proposition 1.9 is equivalent to the operator βY defining a random shift of the field W. In other terms, the law ˜ Pβ determined by L˜Pβ [W] := LP×µ [W + βY p ] is absolutely continuous with respect to the law P. Theorem 1.12.

βY defines a random shift of the field W.

Theorem 1.13. For β > 0 let the random measure Mβ (d p) be defined as in Proposition 1.9. (i). Mβ is a.s. mutually singular to µ. (ii). The product measure Mβ × Mβ is a.s. supported on pairs ( p, q) ∈ Γ b,s × Γ b,s such that the intersection set I p,q = {r ∈ [0, 1] | p(r ) = q(r )} is either finite or has Hausdorff dimension h = (log s − log b)/ log s. (i, j) (iii). Let (Γi,b,sj , Mβ ) be independent copies of (Γ b,s , Mβ ) corresponding to the firstgeneration embedded copies, Di,b,sj , of D b,s . Then there is equality in distribution of random measures b

M√ s

bβ

s

1 ∑ ∏ (i, j) = M b i=1 j=1 β d

under the identification

Γ

b,s

≡

b ⋃

s

×Γ

b,s i, j

.

i=1 j=1

Remark 1.14. Part (ii) of Theorem 1.13 implies that the random measure Mβ a.s. has no atoms. The next theorem states two strong disorder properties in the β ≫ 1 regime. Analogous results were obtained in [21] for discrete polymers on diamond graphs. Theorem 1.15. Let the random measure Mβ (d p) be defined as in Proposition 1.9, and define the random probability measure Q β (d p) = Mβ (d p)/Mβ (Γ b,s ). As β → ∞, Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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(i). the random variable Mβ (Γ b,s ) converges in probability to 0, and (ii). the random variable max Q β (p) converges in probability to 1 for each fixed n. p∈Γnb,s

Remark 1.16. In particular, part (ii) implies that when β ≫ 1 most of the weight of the measure Mβ (d p) is concentrated on a single coarse-grained path p ∈ Γnb,s . (E). A Gaussian multiplicative chaos martingale I will expand on the structure and properties of the map Y : H → L 2 (Γ b,s , µ). Proposition 1.17.

The linear operator Y is compact and has the following properties:

(i). Y can be decomposed as Y = U D where U : H → L 2 (Γ b,s , µ) is an isometry and n−1 D : H → H is a self-adjoint operator with eigenvalues λn = s − 2 for n ∈ N ∪ {∞}. (ii). The null space of Y , which I denote by H∞ (λ∞ = 0), is infinite dimensional. For 2 ≤ n < ∞, the eigenspace Hn corresponding to the eigenvalue λn has dimension (bs)n−1 (b − 1). The eigenspace corresponding to λ0 = 1 has dimension b, which I decompose into the one-dimensional space of constant functions, H0 , and an orthogonal complement, H1 . (iii). For 1 ≤ n < ∞ the space Hn has an orthogonal basis f (e,ℓ) labeled by (e, ℓ) ∈ b,s E n−1 × {1,( . . . , b − the function f (e,ℓ) is supported on e ⊂ D b,s . ) 1}, where ( b,s ) ∗ 2 b,s 2 (iv). Y Y : L Γ , µ → L Γ , µ is a self-adjoint operator with eigenvalues ˆ λn = s −n+1 for n ∈ N∪{∞} with eigenspaces having dimension b for n =(1 and) (bs)n−1 (b−1) s−1 1/2 for n ≥ 2. Hence Y Y ∗ has Hilbert–Schmidt norm ∥Y Y ∗ ∥ H S = b s−b but is not traceclass. Remark 1.18. ⊕∞ H is the orthogonal complement to the space of all ψ ∈ H = L 2 (D b,s , ν) ∫ 1 ( k=0) k such that 0 ψ p(r ) dr is zero for every path p ∈ Γ b,s . ⏐ ] ( ) [ Definition 1.19. Define Y (n) : H → L 2 Γ b,s , µ to act as Y (n) ψ = E Y ψ ⏐ Fn , where Fn is the σ -algebra on Γ b,s generated by the map p ↦→ [ p]n . Remark 1.20. Y (n) can also be written in the following forms: ⏐ ⏐∫ ) ( ) ( ˜ p), ˜ where the right side can be understood as the • Y (n) ψ ( p) = ⏐Γnb,s ⏐ p∈[ ˜ p]n Y ψ ( p)µ(d ( ) b,s average of Y ψ ( p) ˜ over all p ˜ ∈ Γ in the same generation-n equivalence class of p. ) ⟨ ⟩ ( ) ( • Y (n) ψ ( p) = Y p(n) , ψ for Y p(n) ∈ H defined as Y p(n) := χT (n) /ν T p(n) , where T p(n) = p

n

∪sk=1 [ p]n (k), i.e., the generation-n coarse-grained trace of the path p through the space D b,s . Proposition 1.21.

( ) The maps Y (n) : H → L 2 Γ b,s , µ satisfy the properties below.

(i). Let Pn : H → H be the orthogonal projection onto ⊕nk=0 Hk for Hk defined as in part (ii) of Proposition 1.17. For any ψ ∈ H, Y (n) ψ = Y Pn ψ . (ii). As n → ∞, the map Y (n) converges in operator norm to Y . (iii). As n → ∞, Y (n) (Y (n) )∗ converges in Hilbert–Schmidt norm to Y Y ∗ , which has integral kernel K Γ ( p, q) = T p,q , i.e., the intersection time of the paths. Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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(iv). For any k ∈ N and p ∈ Γ b,s , Y p(k) − Y p(k−1) ∈ Hk . In particular, the following sequence of vectors in H are orthogonal: Y p(0) , Y p(1) − Y p(0) , Y p(2) − Y p(1) , . . .

(1.7)

Proposition 1.22. Define Fn to be the σ -algebra on Ω generated by the field variables ⟨W, ψ⟩ for ψ ∈ ⊕nk=0 Hk . Let Mβ(n) be the GMC measure over the finite-dimensional field (W, βY (n) ), i.e., with Radon–Nikodym derivative { } d Mβ(n) β2 = exp β⟨W, Y p(n) ⟩ − ∥Y p(n) ∥2H . dµ 2 { (n) } The sequence of measures Mβ n∈N forms a martingale with respect to the filtration Fn and a.s. converges vaguely to the GMC measure Mβ . (F). Chaos expansion construction of the GMC measure The Gaussian multiplicative chaos Mβ can also be constructed through chaos expansions analogous to those in [2]. The chaos decomposition generated by the field W is ∞ ⨁ ) :k: HW , L Ω , F(W), P = 2

(

(1.8)

k=0 :k: where HW is the orthogonal complement of the set P k−1 (H) within P k (H) for { ( )⏐ Pk (H) := p ⟨W, ψ1 ⟩, . . . , ⟨W, ψk ⟩ ⏐ p : Rk → R is a degree-k polynomial and } ψ1 , . . . , ψ k ∈ H . :k: and the k-fold symmetric tensor, H⊙k , of There is a canonical isometry between HW H.( This isometry as a map from symmetric functions f (x1 , . . . , xk ) in ) can be expressed :k: expressed as stochastic integrals: L 2 (D b,s )k , k!1 ν k to elements of HW ∫ 1 f (x1 , . . . , xk )W(x1 ) · · · W(xk )ν(d x1 ) · · · ν(d xk ) . (1.9) k! (Db,s )k

Intuitively, the integral above is to be understood as over points (x1 , . . . , xk ) ∈ (D b,s )k for which the components x j are distinct. See [17, Section 7.2] for the general theory of stochastic integrals. Let S be a finite subset of E b,s and define Γ Sb,s as the collection of paths p ∈ Γ b,s such that S ⊂ Range( p). If Γ Sb,s is nonempty, define µ S as the probability measure uniformly distributed over paths in Γ Sb,s (and thus supported on Γ Sb,s ). If Γ Sb,s = ∅, i.e., there is no path containing all the points in S, then µ S is defined as zero. Definition 1.23. For a Borel set A ⊂ Γ b,s , define ρk (x1 , . . . , xk ; A) as a map from (D b,s )k to [0, ∞) with { bγ ({x1 ,...,xk }) µ{x1 ,...,xk } (A) x1 , . . . , xk ∈ E b,s are distinct, ρk (x1 , . . . , xk ; A) := 0 otherwise, where γ (S) is defined for a finite subset of E b,s as ∞ ( ∑ ⏐{ ⏐ }⏐) γ (S) := |S| − ⏐ e ∈ E kb,s ⏐ e ∩ S ̸= ∅ ⏐ . k=0 Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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In the above formula for γ (S), elements of E kb,s are to be understood as subsets of E b,s , and the k = 0 term of the sum is always interpreted as |S| − 1. ⏐{ ⏐ }⏐ Remark 1.24. The term ⏐ e ∈ E kb,s ⏐ e ∩ S ̸= ∅ ⏐ counts the number of distinct equivalence classes from E kb,s corresponding to elements in S. Theorem 1.25. For any Borel set A ⊂ Γ b,s , the random variable Mβ (W, A) is equal to the chaos expansion ∫ ∞ ∑ βk µ(A) + ρk (x1 , . . . , xk ; A)W(x1 ) · · · W(xk )ν(d x1 ) · · · ν(d xk ). k! (Db,s )k k=1 The sequence of symmetric functions {ρk (x1 , . . . , xk ; A)}k∈N satisfies ∫ ( ) (i). ∫ Db,s ρk x(1 , . . . , xk ; A ν(d ) xk ) = ρk−1 (x1 , . . . , xk−1 ; A) and (ii). (Db,s )k ρk x1 , . . . , xk ; A ν(d x1 ) · · · ν(d xk ) = µ(A). (G). Scaling limits from { non } Gaussian variables For each n ∈ N let ωa(n) a∈E b,s be a family of i.i.d. random variables indexed by the edge n set of the diamond lattice Dnb,s[. Assume that ( )] the variables have mean zero, variance one, and finite exponential moments: E exp βωa(n) < ∞ for β ∈ R. Define a random measure M(n) β on Γnb,s as follows: { } (n) sn exp βωp(k) ∑ ∏ 1 [ { }] M(n) for A ⊂ Γnb,s . (1.10) β (A) = ⏐⏐ b,s ⏐⏐ Γn p∈A k=1 E exp βω(n) p(k) The theorem below follows as a consequence of Theorem 4.6 of [1]. Theorem 1.26. Fix N ∈ N and let subsets of Γ Nb,s be identified with the canonically corresponding subsets of Γ b,s and of Γnb,s for n > N . For βn := β(b/s)n/2 , the family of b,s random variables M(n) βn (A) labeled by A ⊂ Γ N has joint convergence in law as n → ∞ given by { (n) } { } L Mβn (A) A⊂Γ b,s H⇒ Mβ (A) A⊂Γ b,s . (1.11) N

N

2. Diamond hierarchical lattice In this section I will provide a path-based construction of the diamond hierarchical lattice as a compact metric space. This discussion is applicable to any choice of b, s ∈ {2, 3, . . .}. The proofs of propositions are in the appendix. 2.1. Construction of the diamond lattice as a metric space The hierarchical formulation of the diamond graph Dnb,s in terms of b ·s embedded copies of carries with it a canonical one-to-one correspondence between ({1, . . . , b} × {1, . . . , s})n b,s and the set of bonds, E nb,s . I will construct the diamond hierarchical lattice, ( )∞ D , as an b,s equivalence relation on the set of sequences D := {1, . . . , b} × {1, . . . , s} determined by a semi-metric d D : Db,s × Db,s → [0, 1] defined below. b,s Dn−1

Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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Let ˜ π : Db,s → [0, 1] be the “projective” map sending a sequence x = {(bkx , skx )}k∈N to ˜ π (x) :=

∞ ∑ skx − 1 . sk k=1

Of course, the right side above is the base s generalized decimal expansion of the number ˜ π (x) ∈ [0, 1] having kth digit skx −1. The root vertices of the continuum lattice will be identified with the sets A := {x ∈ Db,s | ˜ π (x) = 0} and B := {x ∈ Db,s | ˜ π (x) = 1}. y y x x For two points x = {(bk , sk )}k∈N and y = {(bk , sk )}k∈N in Db,s , I write x ↕ y if x or y is contained in A ∪ B or for some n ∈ N y

y

(bkx , skx ) = (bk , sk ) for 1 ≤ k < n − 1

and

bnx = bny

but

snx ̸= sny .

In other terms the sequence of pairs defining x and y disagrees for the first time at an scomponent value. Intuitively, this means that there exists a directed path going through both x and y. We then define the semi-metric d D on Db,s as the traveling distance ⎧ ⏐ ⏐ ⏐˜ ⏐ ⎨ π (x) − ˜ π (y) if x ↕ y, ( ) d D (x, y) := d D (x, z) + d D (z, y) otherwise. inf ⎩ b,s z∈D

, z↕x, z↕y

The semi-metric is bounded by 1 since by choosing appropriate z ∈ A or z)∈ B in the infimum ( above, I can conclude that d D (x, y) ≤ min ˜ π (x) + ˜ π (y), 2 − ˜ π (x) − ˜ π (y) . Definition 2.1. D b,s

The diamond hierarchical lattice is defined as ( ) := Db,s / x, y ∈ Db,s with d D (x, y) = 0 .

In future, I will treat the metric d D (x, y) and the map ˜ π as acting on D b,s . Remark 2.2. The vertex set, Vnb,s , on the diamond graph Dnb,s is canonically embedded on D b,s ; see Appendix A. The representation of elements in D b,s ⋃ by sequences {(b j , s j )} j∈N is unique except for the countable collection of vertices V b,s := n Vnb,s . Remark 2.3. The self-similar structure of the fractal D b,s can be understood through a family of contractive shift maps Si, j : Db,s → Db,s for (i, j) ∈ {1, . . . , b} × {1, . . . , s} y y y y that send x = {(bkx , skx )}k∈N to Si, j (x) = y = {(bk , sk )}k∈N with (b1 , s1 ) = (i, j) and y y x x (bk , sk ) = (bk−1 , sk−1 ) for k ≥ 2. The Si, j ’s are well-defined as functions on D b,s , and map D b,s onto the shrunken subcopies Di,b,sj with ( ) 1 d D Si, j (x), Si, j (y) = d D (x, y) , s

x, y ∈ D b,s .

Proposition 2.4. (D b,s , d D ) is a compact metric space with Hausdorff dimension 1 + The vertex set V b,s is dense in D b,s .

log b . log s

The import of the next proposition is that a probability measure ν can be placed on D b,s such that subsets identifiable with elements of E nb,s are assigned measure (bs)−n . Proposition 2.5. Let B D be the Borel σ -algebra on D b,s generated by the metric d D . There is a unique measure ν on (D b,s , B D ) such that ν(V b,s ) = 0 and for (b j , s j ) ∈ {1, . . . , b}×{1, . . . , s} Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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the cylinder sets ⏐ ⏐ := x ∈ E ⏐⏐ x = {(b xj , s xj )} j∈N with b xj = b j and } s xj = s j for 1 ≤ j ≤ n {

C(b1 ,s1 )×···×(bn ,sn )

b,s

( ) −1 (identifiable with elements in E nb,s ) have measure ν C(b1 ,s1 )×···×(bn ,sn ) = |E nb,s | = (bs)−n . 2.2. Directed paths on the diamond hierarchical lattice b,s b,s ( I define ) a directed path on D to be a continuous function p : [0, 1] → D such that ˜ π p(r ) = r for all r ∈ [0, 1]. I will use the uniform metric on the set of directed paths: ( ) ( ) dΓ p1 , p2 = max d D p1 (r ), p2 (r ) p1 , p2 ∈ Γ b,s . 0≤r ≤1

( ) Remark 2.6. Note that dΓ p1 , p2 = s −(n−1) , where n ∈ N is the lowest generation such that there is a vertex v ∈ Vnb,s in the range of p1 but not of p2 . Remark 2.7. Γnb,s is canonically identified with an equivalence relation of Γ b,s in which q ≡n p iff [ p]n = [q]n , or, equivalently, dΓ ( p, q) ≤ s −n . Remark 2.8. The metric d D on D b,s can be reformulated in terms of the space of directed paths, Γ b,s , as ( ) d D (x, y) = inf |˜ π (x) − ˜ π (z)| + |˜ π (z) − ˜ π (y)| . p,q∈Γ b,s ,z∈D b,s z∈Range( p)∩Range(q)

( ) The uniform measure on Γ b,s refers to the triple Γ b,s , BΓ , µ in the proposition below. Proposition 2.9. Let BΓ be the Borel σ -algebra generated byn the metric dΓ . There is a unique s −1 −1 measure µ on (Γ b,s , BΓ ) satisfying µ(p) = |Γnb,s | = b− s−1 for all n ∈ N and p ∈ Γnb,s . Moreover, µ is related to ν through the following identity: for any R ∈ B D ∫ ∫ ν(R) = 1 p(r )∈R dr µ(d p) . (2.1) Γ b,s

[0,1]

Remark 2.10. The intuitive meaning of (2.1) is that ν(R) equals the expected amount of time that a random path p ∈ Γ b,s chosen according to the measure µ will spend in R ∈ B D . This implies, more generally, that for any f ∈ L 1 (D b,s , ν) ∫ ∫ ∫ ( ) f (x)ν(d x) = f p(r ) dr µ(d p) . D b,s

Γ b,s

[0,1]

3. Proofs 3.1. Intersection time between random directed paths The intersections between randomly chosen paths p, q ∈ Γ b,s can be encoded into realizations of a discrete-time branching process that begins with a single node and for which Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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each generation n node has exactly s children independently with probability 1/b, or has no children at all. (n) Given p, q ∈ Γ b,s recall that N p,q is the number of bonds shared by the coarse-grained b,s b,s paths [ p]n , [q]n ∈ Γn . For q ∈ Γ chosen at random (i.e., according to µ), let Fn(q) be the σ -algebra of subsets of Γ b,s generated by [q]n . Proof of Proposition 1.6. We can write I p,q as I p,q =

∞ ⋂

(n) I p,q

for

⋃

(n) I p,q := [0, 1] −

1≤k≤s n [ p]n (k)̸=[q]n (k)

n=1

(k − 1 k ) , . sn sn

(n) Let p(n) b,s be the probability that the number, N p,q , of bonds shared by [ p]n and [q]n is not zero. (n) Then the probability that N p,q never becomes zero is the limit p(n) b,s ↘ pb,s . The probabilities (n) pb,s satisfy the recursive relation ( )s ] ( ) 1[ 1 − 1−x p(n+1) = G b,s p(n) for G b,s (x) := b,s b,s b

and have initial value p(0) b,s = 1. When s is larger than b, the probability pb,s ∈ (0, 1) is the unique attractive fixed point of the map G b,s : [0, 1] → [0, 1]. ( )n (n) form a nonzero, mean-one martingale with respect to Part (i): The variables mn := bs N p,q , generated by [q] . Hence, there is an a.s. limit m∞ = lim mn . The moment the filtration, F(q) n n n→∞ tmn generating functions ϕn (t) := E[e ] satisfy the recursive relation b−1 1 ( b,s ( b ))s b,s ϕn+1 (t) = with ϕ0b,s (t) = et . (3.1) + t ϕ b b n s b,s The sequence ϕnb,s (t) converges pointwise to a nontrivial limit which is the moment ( b,s (ϕb∞))(t), s b−1 1 b,s generating function of m∞ , satisfying ϕ∞ (t) = b + ϕ t . Note that the limit of ∞ b s [ ] 1 s b,s + x , and thus P m = 0 = 1 − p . ϕ∞ (t) as t → −∞ solves x = b−1 ∞ b,s b b (n) Parts (ii) and (iii): In the event that N p,q is zero for some n ∈ N, I p,q is finite and p(t) = b,s (n) q(t) ∈ V for t ∈ I p,q . Conversely, I will show below that if N p,q is never zero, then the set I p,q is a.s. infinite since its dimension-h Hausdorff measure is infinite for any 0 < h < h. This (n) would suffice to prove the proposition since the above remarks show that N p,q becomes zero for large enough n ∈ N with probability 1 − pb,s . I will split up the analysis between proving dim H (I p,q ) ≤ h and dim H (I p,q ) ≥ h. To show that dim H (I p,q ) ≤ h, I will argue that the Hausdorff measure Hh (I p,q ) is a.s. finite. For a given (n) (n) δ > 0 pick n with ( 1s )n < δ. Since I p,q ⊂ I (n) p,q and I p,q is covered by N p,q intervals of length 1/s n , ( )hn ∑ (n) 1 Hh,δ (I p,q ) = inf |Ik |h ≤ N p,q = mn . I p,q ⊂∪k Ik s k |Ik |≤δ

Thus, Hh (I p,q ) = lim Hh,δ (I p,q ) ≤ lim inf mn = m∞ . δ→0

n→∞

Therefore Hh (I p,q ) is a.s. finite and dim H (I p,q ) ≤ h. (n) Next I will condition on the event that N p,q is not zero for any n ∈ N and show that dim H (I p,q ) ≥ h a.s. It suffices to show that Hh (I p,q ) > 0 for any 0 < h < h. Let S (n) p,q be the Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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(n) (N ) collection of intervals [ k−1 , skn ] ⊂ I p,q , skn ] ∩ I p,q such that [ k−1 is not finite for any N > n (in sn sn other terms, ancestors of the interval do not go extinct). For a Borel set A ⊂ [0, 1], let C(A) (n) ˜ be the set of coverings of A by elements in ∪∞ n=1 S p,q . Define the Hausdorff-like measure Hh as ∑ ˜h,δ (A) = inf ˜h (A) = lim H ˜ 1 (A) for H |Ik |h . (3.2) H h, n n→∞

{Ik }∈C(A) |Ik |≤δ

s

k

For any Borel A ⊂ [0, 1] we have that 1 ˜ ˜h (A) . Hh (A) ≤ Hh (A) ≤ H 2s h

(3.3)

˜h,1/s n (A) is defined as an infimum over a smaller The second inequality above holds since H collection of coverings than Hh,1/s n (A). The first inequality holds since any interval I ⊂ [0, 1] [ ] k is covered by two adjacent intervals of the form k−1 , for n := ⌊log1/s |I |⌋. Thus, if sn sn ˜h,1 (I p,q ) > 0 holds a.s. then Hh (I p,q ) > 0 holds a.s. H (n) (n) ˜p,q Let N be the number elements in S (n) p,q . Conditioned on the event that N p,q is never zero, (n) (0) ˜p,q ˜p,q N forms a Markov chain taking values in N with initial value N = 1 and satisfying the distributional equality ˜(n)

(n+1) d ˜p,q N =

N p,q ∑

[ ] n j for i.i.d. variables n j ∈ {1, . . . , s} with P n j = ℓ

j=1

( ) ℓ s pb,s (1 − pb,s )s−ℓ . = ℓ 1 − (1 − pb,s )s Fix some 0 < h < h. Define the variables ∑ L p,q,n := inf |Ik |h , {Ik }∈C(I p,q ) |Ik |≥ 1n s

(3.4)

k

˜h,1 (I p,q ). The hierarchical symmetry of which have the a.s. convergence L p,q,n ↘ L p,q,∞ := H the model implies that the L p,q,n ’s satisfy the distributional recursion relation ( ∑ ) n ( ) 1 h ( j) d L p,q,n+1 = min 1, L p,q,n , (3.5) s j=1 ( j)

where the L p,q,n ’s are independent copies of L p,q,n and n ∈ {1, . . . , s} is independent of the [ ] ( ) pℓ (1−pb,s )s−ℓ ( j) . The distribution of L p,q,∞ is a fixed point of (3.5). L p,q,n ’s with P n = ℓ = ℓs b,s 1−(1−p [ ] b,s )s The probability x = P L p,q,∞ = 0 satisfies x =

(xpb,s + 1 − pb,s )s − (1 − pb,s )s , 1 − (1 − pb,s )s

which has solutions only for x = 0 and x = 1. However, x = 1 is not possible since a.s. convergence L p,q,n ↘ 0 as n → ∞ contradicts (3.5). To see the rough idea for this, notice that if 0 < L p,q,n ≪ 1 with high probability when n ≫ 1 then the expectation of (3.5) yields [∑ ] n ( ) ( 1 )h [ ] [ ] [ ] 1 h ( j) E L p,q,n+1 ≈ E L p,q,n = E[n] E L p,q,n = s h−h E L p,q,n s s j=1 Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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[ ] because E[n] = bs = s h . The above shows that the expectations of E L p,q,n will contract away from 0 since h − h > 0. □ 3.2. The compact operator Y In this section I will prove Propositions 1.17 and 1.21. Definition 3.1. For ℓ ∈ {1, . . . , b}, let v (ℓ) = (v1(ℓ) , . . . , vb(ℓ) ) be orthonormal vectors in Rb where v (1) = √1b (1, . . . , 1). Let p1 , . . . , pb be an enumeration of the elements in Γ1b,s , i.e., the branches of D1b,s . • Define f (ℓ) ∈ H and ˆ f (ℓ) ∈ L 2 (Γ b,s , µ) for ℓ ∈ {1, . . . , b} as f

(ℓ)

b √ ∑ vi(ℓ) χ∪sk=1 [pi ](k) (x) (x) = b

and

i=1

• For (e, ℓ) ∈

b,s ∪∞ n=0 E n

ˆ f (ℓ) ( p) =

b √ ∑ b vi(ℓ) χpi ( p) . i=1

× {1, . . . , b}, define f (e,ℓ) ∈ H as n

f (e,ℓ) (x) = (sb) 2 χe (x) f (ℓ) (xe ) , where for x ∈ e the point xe ∈ D b,s refers to the position of x in the shrunken copy of D b,s corresponding to e. b,s 2 b,s ˆ • For (e, ℓ) ∈ ∪∞ , µ) as n=0 E n × {1, . . . , b}, define f (e,ℓ) ∈ L (Γ n ˆ f (e,ℓ) ( p) = b 2 χe∩Range( p)̸=∅ ˆ f (ℓ) ( pe ),

where if e ∩ Range( p) ̸= ∅ the path pe ∈ Γ b,s refers to a magnification of the portion of the path p in the shrunken copy of D b,s corresponding to e. Proof of Proposition 1.17. It suffices to show that the operator Y : H → L 2 (Γ b,s , µ) has the form ∞ ∑ ∑ ⏐ ⏐ ⏐ ⟩⟨ ⟩⟨ k⏐ Y = ⏐ˆ f (Db,s ,1) f (Db,s ,1) ⏐ + s− 2 ⏐ ˆ f (e,ℓ) f (e,ℓ) ⏐ . (3.6) k=0

b,s e∈E k ℓ∈{2,...,b}

Clearly Y maps f (Db,s ,1) = 1 Db,s to ˆ f (Db,s ,1) = 1Γ b,s . Pick e ∈ E kb,s and ℓ ∈ {2, . . . , b}. For b,s p∈Γ , ∫ 1 ( ) ( ) Y f (e,ℓ) ( p) = f (e,ℓ) p(r ) dr 0 ∫ 1 ( ) ( ) k = (sb) 2 χe p(r ) f (ℓ) ( p(r ))e dr 0

( ) 1 k k = k (sb) 2 χe∩Range( p)̸=∅ f (ℓ) pe = s − 2 ˆ f (e,ℓ) ( p) . s The third equality holds since the path p(r ) is in e for a time interval of length 1/s k in the event that e ∩ Range( p) is nonempty. The orthogonal complement in H of the space spanned by the vectors f (Db,s ,1) and f (e,ℓ) ∫1 with (e, ℓ) ∈ E kb,s × {2, . . . , b} is comprised of all vectors ψ for which 0 = 0 ψ( p(r ))dr for all p ∈ Γ b,s , which is, by definition, the null space of Y . □ Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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[ ⏐ ] Proof of Proposition 1.21. Part (i): The conditional expectation E · ⏐ Fn satisfies { ⋃n−1 b,s ˆ ⏐ ] [ f e ∈ Ek , (e,ℓ) ⏐ E ˆ f (e,ℓ) Fn = ⋃k=0 ∞ 0 e ∈ k=n E kb,s and ℓ ∈ {2, . . . , b} . The result then follows from the form (3.6) of Y since Y (n) has the form n−1 ∑ ∑ ⏐ ⏐ ⏐ ⟩⟨ ⟩⟨ k⏐ Y (n) = ⏐ ˆ f (Db,s ,1) f (Db,s ,1) ⏐ + s− 2 ⏐ ˆ f (e,ℓ) f (e,ℓ) ⏐ = Y Pn . k=0

b,s e∈E k ℓ∈{2,...,b}

Part (ii): As a consequence of (i), the operator norm of the difference between Y (n) and Y has the form ∥Y (n) − Y ∥∞ = s −n/2 . Part (iii): The operator Y (n) (Y (n) )∗ can be written in the form n−1 ∑ ∑ ⏐ ⏐ ⏐ ⏐ ⟩⟨ ⟩⟨ Y (n) (Y (n) )∗ = ⏐ ˆ f (Db,s ,1) ˆ f (Db,s ,1) ⏐ + s −k ⏐ ˆ f (e,ℓ) ˆ f (e,ℓ) ⏐ . k=0

b,s e∈E k ℓ∈{2,...,b}

(n) (n) ∗ ∗ It follows √ that the Hilbert–Schmidt norm of the difference between Y (Y ) and Y Y is ( bs )n/2 s b−1 . s−b ( ) Next I show that Y) Y ∗ has integral kernel T ( p, q). The map (Y (n) )∗ : L 2 Γ b,s , µ → H ( sends f ∈ L 2 Γ b,s , µ to ∫ ∫ (Y (n) )∗ f = f ( p)Y p(n) d p = bn f ( p)χT (n) d p . Γ b,s

p

Γ b,s

) ( ) Thus, Y (n) (Y (n) )∗ : L Γ b,s , µ → L 2 Γ b,s , µ has kernel K (n) ( p, q) ( 2

n

s bn ∑ K ( p, q) = = n 1[ p]n (k)=[q]n (k) := mn . s k=1 ( ) However, mn ≡ mn ( p, q) converges to m∞ ( p, q) = T p,q in L 2 Γ b,s × Γ b,s , µ × µ by part (i) of Proposition 1.6. (n)

⟨

Y p(n) , Yq(n)

⟩

Part (iv): The vectors Y p(n) ∈ H satisfy Y p(k) = Pk Y p(n) for k ≤ n. Thus, Y p(n) − Y p(n−1) = (Pn − Pn−1 )Y p(n) ∈ Hn .

□

3.3. Existence and uniqueness of the GMC measure for the CDRP 3.3.1. GMC martingale Proposition 3.2. Define Fn to be the σ -algebra on Ω generated by variables ⟨W, ψ⟩ for ψ ∈ ⊕nk=0 Hk where Hk is defined as in part (ii) of Proposition 1.17. { (n) (n) 2 } β2 (i). The sequence of random variables eβ⟨W,Y p ⟩− 2 ∥Y p ∥H n∈N forms a mean-one martingale with respect to Fn . Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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{∫ (n) (n) 2 β2 (ii). For any Borel set A ⊂ Γ b,s , the sequence of random variables eβ⟨W,Y p ⟩− 2 ∥Y p ∥H A } µ(d p) is a mean-µ(A), square-integrable martingale with respect to Fn that n∈N converges a.s. to a nonzero limit. Proof. Part (i): If N > n, then ⟨W, Y p(N ) − Y p(n) ⟩ and ⟨W, Y p(n) ⟩ are independent normal random variables since Y p(N ) − Y p(n) and Y p(n) are orthogonal elements in H by part (iv) of Proposition 1.21. Thus, we can write (N )

eβ⟨W,Y p

2

⟩− β2 ∥Y p ∥2H (N )

(N )

= eβ⟨W,Y p

2

−Y p ⟩− β2 ∥Y p −Y p ∥2H (n)

(N )

(n)

(n)

(n)

eβ⟨W,Y p β2

2

⟩− β2 ∥Y p ∥2H . (n)

(n) 2

The conditional expectation with respect to Fn is eβ⟨W,Y p ⟩− 2 ∥Y p ∥H . } {∫ (n) 2 (n) β2 is a mean-µ(A) martingale. We can Part (ii): The sequence eβ⟨W,Y p ⟩− 2 ∥Y p ∥H µ(d p) n∈N

A

write ∫

2

β 2 eβ⟨W,Y p ⟩− 2 ∥Y p ∥H µ(d p) (n)

A

(n)

=

∫ ∏ sn

eβ⟨W,b

β2 n 2 nχ [ p]n (k) ⟩− 2 ∥b χ[ p]n (k) ∥H

µ(d p)

A k=1

Since the ⟨W, bn χ[ p]n (k) ⟩’s are independent, centered normal variables with variance (b/s)n , the second moment of the above is equal to ∫ 2 b n (n) eβ ( s ) N p,q µ(d p)µ(dq) , (3.7) A×A (n) where N p,q is the number of bonds shared by the coarse-grained paths [ p]n and [q]n . By (n) converges µ × µ-a.e. as n → ∞ to the part (i) of Proposition 1.6, the sequence ( bs )n N p,q b,s intersection time T p,q , and when A = Γ the expression (3.7) converges to the moment ∫ 2 generating function, E[exp(β 2 T p,q )]. It follows that (3.7) converges to A×A eβ T p,q µ(d p)µ(dq) for an arbitrary Borel set A ∈ BΓ . □

3.3.2. Martingale limit construction of the GMC measure Proof of Proposition 1.9 and 1.22. Recall that Mβ(n) is defined as the GMC measure on (Γ b,s , µ) over the finite-dimensional field (W, βY (n) ): ∫ (n) (n) 2 β2 Mβ(n) (A) = eβ⟨W,Y p ⟩− 2 ∥Y p ∥H µ(d p) , A ∈ BΓ . (3.8) A

The space HY = ⊕∞ k=0 Hk is the orthogonal complement of the null space of Y . Define F∞ as the σ -algebra generated by the variables ⟨W, φ⟩ for φ ∈ HY . Existence: Let D be a countable subcollection of continuous functions on Γ b,s that are dense b,s with respect to the supremum{∫norm (where, recall, } dΓ is the underlying metric on Γ ). For (n) each ψ ∈ D, the sequence Γ b,s ψ( p)Mβ (d p) n∈N is a martingale w.r.t. the filtration Fn having uniformly bounded second moments as a consequence of Proposition 3.2. Consequently, {∫ } (n) ψ( p)M (d p) converges a.s. to a limit in L 2 (Ω , F, P). Thus, the measures Mβ(n) a.s. β Γ b,s n∈N converge vaguely to a limit measure Mβ adapted to the field W, i.e., Mβ ≡ Mβ (W). Properties (I)–(III) For instance, to verify property ⋃ followN easily from this construction. (n) (III) fix some φ ∈ ∞ N =1 ⊕k=0 Hk . Notice that since Mβ (W, d p) a.s. converges vaguely to Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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Mβ (W, d p), I have the a.s. equality Mβ (W + φ, d p) = lim Mβ(n) (W + φ, d p) n→∞

(n)

= lim eβ⟨Y p

,φ⟩

n→∞

Mβ(n) (W, d p) .

N However, since φ ∈ ⊕k=0 Hk for some N , I have ⟨Y p(n) , φ⟩ = ⟨Y p , φ⟩ when n ≥ N , and thus,

= eβ⟨Y p ,φ⟩ lim Mβ(n) (W, d p) n→∞

= eβ⟨Y p ,φ⟩ Mβ (W, d p) .

(3.9)

⋃∞

N Hk is dense in the complement of the null space of Y , and it follows The space N =1 ⊕k=0 that the above extends to all elements in H.

Uniqueness: Next I argue that Mβ is the unique GMC measure over the field (W, βY ). I will reduce the uniqueness of Mβ to the uniqueness of the GMC measures Mβ(n) over the finite˜β be a random measure satisfying (I)–(III), and define dimensional fields (W, βY (n) ). Let M (n) ˜ ˜β w.r.t. Fn Mβ as the conditional expectation of M ( ) [ ( )⏐ ] ˜β(n) W(n) , A = E M ˜β W, A ⏐ Fn , M (3.10) n where W(n) refers to the finite-dimensional field ( ) of variables ⟨W, ψ⟩ for ψ ∈ ⊕k=0 Hk . Since ∞ ˜ H = ⊕k=0 Hk , the random variable Mβ W, A is recovered as the a.s. limit ( ) [ ( )⏐ ] ˜β W, A = lim E M ˜β W, A ⏐ Fn . M (3.11) n→∞

˜β(n) ] = E[ M ˜β ] = µ. For φ ∈ ⊕n Hk , Obviously (3.10) implies that E[ M k=0 ( (n) ) [ ( )⏐ ] [ ( )⏐ ] (n) ˜β W + φ, A = E M ˜β W + φ, A ⏐ Fn = eβ⟨Y p ,φ⟩ E M ˜β W, A ⏐ Fn M ( ) ˜β(n) W(n) , A . = eβ⟨Y p ,φ⟩ M ˜β(n) ≡ M ˜β(n) (W) is viewed as a function of the entire field W, then If M (n)

˜β(n) (W + φ, d p) = eβ⟨Y p M

,φ⟩

˜β(n) (W, d p) M

˜β(n) is a GMC measure over since Y p(n) = Pn Y p for the projection Pn : H → ⊕nk=0 Hk . Thus, M the trivial field (W, βY (n) ) and must be equal to Mβ(n) . From the construction analysis above, Mβ(n) a.s. converges vaguely to a GMC measure Mβ over the field (W, βY ). From (3.11) it ˜β . □ follows that Mβ = M 3.4. Properties of the GMC measure In this section I will prove Theorem 1.13. Proof of Theorem 1.13. Part (i): First I will show that for β > 0 the GMC measure Mβ is mutually singular to µ with probability 0 or 1. Let Mβ(c) and Mβ(s) be the continuous and singular components in the Hahn decomposition of Mβ w.r.t. µ. Since the white noise underlying Mβ has a uniform covariance measure ν over D b,s , there is statistical uniformity in how the Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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random measure Mβ weighs the space of directed paths, Γ b,s . This, in particular, that [ implies ] the components Mβ(c) and Mβ(s) must have uniform expectations, and hence E Mβ(c) = λµ [ ] and E Mβ(s) = (1 − λ)µ for some value λ ∈ [0, 1]. However, if λ ∈ (0, 1) then the random measures λ−1 Mβ(c) and (1−λ)−1 Mβ(s) must both be GMC measures satisfying properties (I)–(III) of Proposition 1.9. This, however, contradicts the uniqueness of that GMC measure. Therefore, λ ∈ {0, 1}. Suppose to reach a contradiction that λ = 1, i.e., Mβ = Mβ(c) . Let G β (W, p) be the Radon–Nikodym derivative of Mβ with respect to µ. For almost every p, G β (W, p) is a random variable with finite second moment and for all φ ∈ H G β (W + φ, p) = eβ⟨φ,Y ( p)⟩ G β (W, p) .

(3.12)

However, the above is only possible if Y ( p) ∈ H, which is a contradiction. Part (ii): The intersection time, T p,q , of two random paths p, q ∈ Γ b,s chosen independently ∫ according to the measure Mβ has moment generating function Γ b,s ×Γ b,s eαT p,q Mβ (d p)Mβ (dq), which has expectation ] [∫ ∫ 1 2 αT p,q e(α+ 2 β )T p,q µ(d p)µ(dq) e Mβ (d p)Mβ (dq) = E b,s b,s b,s b,s Γ ×Γ Γ ×Γ ( ) b,s α + β 2 /2 < ∞ . (3.13) = ϕ∞ It follows that the set of pairs ( p, q) for which the intersection set I p,q has Hausdorff dimension > h, and thus for which T p,q = ∞, is a.s. of measure zero with respect to Mβ × Mβ . The set of pairs ( p, q) for which the intersection set I p,q has Hausdorff dimension < h satisfy T p,q = 0, and ∫ ({ ⏐ }) lim eαT p,q Mβ (d p)Mβ (dq) = Mβ × Mβ ( p, q) ⏐ T p,q = 0 . α→−∞ Γ b,s ×Γ b,s

Taking the expectation [ ({ ⏐ })] E Mβ × Mβ ( p, q) ⏐ T p,q = 0 =

b,s lim ϕ∞ (γ ) = pb,s .

γ →−∞

(3.14)

⏐ { } However, ( p, q) ⏐ T p,q = 0 contains the set of pairs such that the intersection set I p,q is finite and [ ({ ({ ⏐ ⏐ })] }) E Mβ × Mβ ( p, q) ⏐ |I p,q | < ∞ = µ × µ ( p, q) ⏐ |I p,q | < ∞ = pb,s . (3.15) The second equality above holds by part (ii) of Proposition 1.6. To see the first equality ⏐ { } in (3.15), notice that ( p, q) ⏐ |I p,q | < ∞ can be written as the n → ∞ limit ⋃ ⏐ { } p × q ↗ ( p, q) ⏐ |I p,q | < ∞ . b,s p,q∈Γn ∀(k) p(k)̸=q(k)

Moreover, the random measure Mβ is independent over sets p, q ∈ Γnb,s for which p(k) ̸= q(k) for 1 ≤ k ≤ s n . Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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It follows from (3.14) and (3.15) that [ ({ ⏐ })] = 0. E Mβ × Mβ ( p, q) ⏐ T p,q = 0 and |I p,q | = ∞

(3.16)

Therefore, Mβ × Mβ is supported on pairs ( p, q) for which either I p,q is finite or has Hausdorff dimension h. (i, j)

Part (iii): Let Mβ be measurable with respect to independent copies W(i, j) of the field W, ˜ such that and define the field W b ∑ s ∑ ˜ φ⟩ := √1 ⟨W(i, j) , φ (i, j) ⟩ ⟨W, bs i=1 j=1

˜= φ∈H

for

b ⨁ s ⨁

H(i, j) ,

i=1 j=1

˜b,s , ˜ where H(i, j) are copies of the Hilbert space H = L D b,s , B D , ν . Define (Γ µ) for ( 2

˜b,s := Γ

b ⋃

b

s

×Γ

b,s i, j

˜ µ :=

and

i=1 j=1

)

s

1 ∑ ∏ (i, j) µ . b i=1 j=1

(3.17)

( b,s ) ˜ ,˜ ˜ : H −→ L 2 Γ Finally, define Y µ as ( )∑ b s s ⟨ (i, j) (i, j) ⟩( ( j) ) 1∑ b,s ˜ ⟨Y , φ⟩( p) = χ p∈ Γi, j Y ,φ p . s i=1 j=1 j=1

×

×s

In the above p j ∈ Γi,b,sj are components of p ∈ j=1 Γi,b,sj . ∑b ∏ s (i, j) ˜ = 1 i=1 The computation below shows that M defines a GMC measure over j=1 Mβ b √s ˜ ˜): β Y the field (W, b ( ( )∏ ) b s s ∑ ( ) φ (i, j) (i, j) ˜ + φ, d p = 1 ˜ W χ p∈ Mβ W(i, j) + √ , d p ( j) M Γi,b,sj b i=1 bs j=1 j=1 ( ) b s s ∑s ⟨ (i, j) (i, j) ⟩( ( j) ) ∏ 1∑ Y ,φ p β √1 (i, j) = χ p∈ Γi,b,sj e bs j=1 Mβ b i=1 j=1 j=1 ( (i, j) ) ( j) × W , dp ( ) √ b s s ∏ ) s ˜ 1∑ (i, j) ( Γi,b,sj e b β⟨Y ,φ⟩( p) χ p∈ Mβ W(i, j) , d p ( j) = b i=1 j=1 j=1 √s ˜ ( ) β⟨ Y ,φ⟩( p) ˜ dp ˜ W, =e b M

× × ×

˜ is equal in law to M√ s . □ Therefore, the GMC measure M β b

3.5. Strong disorder behavior as β → ∞ ( ) In this section I will prove Theorem 1.15. The proof below that Mβ W, Γ b,s converges in probability to zero is a straightforward adaption of the argument of Lacoin and Moreno for discrete polymers on diamond lattices in [21]. Proof ] of Theorem 1.15. It suffices to show that the fractional moment [√ of part (i) E Mβ (W, Γ b,s ) converges to zero as n → ∞. Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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Let h : D b,s → R be the constant function h(x) = λ for some λ ∈ R, and ˆ Pλ be the measure on W with derivative dˆ Pλ 1 2 1 2 b,s = e⟨W,h⟩− 2 ∥h∥H = eλW(D )− 2 λ . dP Let ˆ Eλ denote the expectation with respect to ˆ Pλ . The Cauchy–Schwarz inequality yields that [( ] [ ] ( ( ( )) 12 )) 12 −λW(Db,s )+ 1 λ2 b,s b,s ˆ 2 E Mβ W, Γ = Eλ Mβ W, Γ e ]1 [ [ ( )] 12 ( −λW(Db,s )+ 1 λ2 )2 2 ˆ 2 ≤ˆ Eλ Mβ W, Γ b,s E e . [ ] [ ] ( Since)ˆ Eλ F(W) =( E F(W + h) for any integrable function F of the field and Mβ W + ) h, dp = eβ⟨Y p |h⟩ Mβ W, d p where ⟨Y p |h⟩ = λ, the above is equal to [ ( )] 12 [ −λW(Db,s )+ 1 λ2 ] 21 2 = E eλβ Mβ W, Γ b,s E e 1

1 2

= e 2 λβ+ 2 λ .

(3.18)

The above is minimized as exp{− 18 β 2 } when λ = − 12 β, and thus tends to zero as β grows.

□

The proof below also borrows ideas from [21]. Proof of part (ii) of Theorem 1.15. Fix n ∈ N, q, p ∈ Γnb,s with q ̸= p, and α > 1. It suffices to show that as n → ∞ ) [ ( ] ( ] νβ W, p ( ) ∈ λ−1 , λ −→ 0 . P (3.19) νβ W, q The analysis below shows that there exist c, C > 0 such that for all β > 1 √ [ ( ) ) [ ( ] ] ( −1 ] ( ] νβ W, p νβ W, p 2m−1 2m+1 ( ) ∈ λ , λ ≤ min C P ( )∈ λ P ,λ . (3.20) m∈Z νβ W, q νβ W, q |m|≤c log(β) [ ] ν (W,p) Since the terms P νββ (W,q) ∈ (λ2m−1 , λ2m+1 ] sum to 1 over m ∈ Z, the above must be smaller 1

than C(1/⌊c log β⌋) 2 , thus implying (3.19). ( ) 2 b,s ( Next I will) show (3.20). Define h ∈ L (D , ν) as h = αχ ∪p(k)̸=q(k) p(k) − αχ ∪p(k)̸=q(k) q(k) for a parameter α ∈ [−1, 1]. Then for any p ∈ p and q ∈ p ⟨h, Y p ⟩ =

αn αns n sn

and

αns n ⟨h, Yq ⟩ = −

αn , sn

(3.21)

where 1 ≤ n < s n is the number of edges not shared by the paths p, q ∈ Γnb,s . Notice that ( ) ( ) ( ) ( ) Mβ W + h, p νβ W + h, p β⟨h,Y p ⟩−β⟨h,Yq ⟩ Mβ W, p 2βαn νβ W, p ( ) = ( ) = e ( ) = e ( ). νβ W + h, q Mβ W + h, q Mβ W, q νβ W, q (3.22) Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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= exp{⟨W, h⟩ − 12 ∥h∥2H }. Applying the Cauchy–Schwarz Define ˆ P to have derivative dP dP inequality, ) ) [ ( [ ( ( ] )] [ −1 ] [ ] νβ W, p νβ W, p 1 ∥h∥2 −⟨W,h⟩+ Hχ 2 ( ) ∈ λ ,λ =ˆ ( ) ∈ λ−1 , λ P E e νβ W, q νβ W, q [( ]1 )2 ] 12 [ νβ (W, p) [ ] 2 1 ∥h∥2 −⟨W,h⟩+ ˆ H 2 ( ) ∈ λ−1 , λ ≤ˆ E e P . νβ W, q ˆ

Since the law ˆ P is a shift of P by h, the above is equal to ) [ ( ]1 [ ] 2 νβ W + h, p 1 2 ( ) ∈ λ−1 , λ = e 2 ∥h∥H P νβ W + h, q ( ) ]1 [ n [ −1 ] 2 α 2 (bs) 2βαn νβ W, p n ( ) ∈ λ ,λ =e , P e νβ W, q where the second inequality is by (3.22). With α ranging over [−1, 1], the above implies (3.20) with C := exp{1/bn } and c := log(2n)/ log(λ). □ 3.6. Chaos expansion The proof of the proposition below is in Appendix A. Proposition 3.3. Let S ⊂ E b,s be finite and Γ Sb,s be nonempty. Define the measure µ(n) S such that ( ) µ A ∩ G (n) (n) S µ S (A) = (3.23) ( ) , µ G (n) S { } n b,s where G (n) such that S ⊂ ∪sk=1 [ p]n (k). Then the sequence µ(n) S is the set of p ∈ Γ S n∈N converges vaguely to a limiting probability measure µ S . The following defines a generalization of the measure ρk (x1 , . . . , xk ; d p) on Γ b,s ; see Definition 1.23. Definition 3.4. Let x1 , . . . , xk be distinct elements in E b,s . If Γ{xb,s1 ,...,xk } is nonempty, define the measure ρk(n) (x1 , . . . , xk ; ds) such that for a Borel set A ⊂ Γ b,s ρk(n) (x1 , . . . , xk ; A) = bγ

(n) ({x ,...,x }) k 1

µ(n) {x1 ,...,xk } (A) ,

(3.24)

where γ (n) (S) is defined for n ∈ N and a finite set S ⊂ E b,s as follows: γ (n) (S) =

n−1 ( ∑

⏐{ ⏐ }⏐) |S| − ⏐ e ∈ E kb,s ⏐ e ∩ S ̸= ∅ ⏐ .

k=0

If

Γ{xb,s1 ,...,xk }

is empty, I define ρk(n) (x1 , . . . , xk ; d p) = 0.

{ Corollary 3.5. For fixed, distinct x1 , . . . , xk ∈ E b,s , the sequence of measures ρk(n) (x1 , } . . . , xk ; d p) n∈N converges vaguely to ρk (x1 , . . . , xk ; d p). Moreover, if A ⊂ Γ Nb,s , then ρk(n) (x1 , . . . , xk ; A) = ρk (x1 , . . . , xk ; A) for all n > N . Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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Remark 3.6. I drop superscripts in the following cases: γ ≡ γ (∞) , ρk ≡ ρk(∞) , and identify ρk (x1 , . . . , xk ) ≡ ρk (x1 , . . . , xk ; Γ b,s ). Remark 3.7. The measure ρk(n) (x1 , . . . , xk ; d p) is equal to rk(n) (x1 , . . . , xk ; p)µ(d p), where the density can be written as rk(n) (x1 , . . . , xk ; =

k ∏

p) = b

γ (n)

( ) χG (n) ( p) sn ⋃ {x1 ,...,xk } nk [ p]n (ℓ) ( ) = b χ {x1 , . . . , xk } ⊂ µ G (n) {x1 ,...,xk } ℓ=1

Y p(n) (xℓ ) .

ℓ=1

Proposition 3.8. Let Mβ(n) be the GMC measure over the field (W, βY (n) ). For a Borel set A ⊂ Γ b,s , the random variable Mβ(n) (A) has the chaos expansion ∫ ∞ ∑ βk µ(A) + ρk(n) (x1 , . . . , xk ; A)W(x1 ) · · · W(xk )ν(d x1 ) · · · ν(d xk ) . (3.25) b,s )k k! (D k=1 The hierarchy of functions {ρk(n) (x1 , . . . , xk ; A)}k∈N satisfies ∫ ( ) (n) (I). Db,s ρk(n) x1 , . . . , xk ; A ν(d xk ) = ρk−1 (x1 , . . . , xk−1 ; A) and ∫ ) (n) ( (II). (Db,s )k ρk x1 , . . . , xk ; A ν(d x1 ) · · · ν(d xk ) = µ(A). Proof. Recall that ∫ (n) (n) 2 β2 Mβ(n) (A) = eβ⟨W,Y p ⟩− 2 ∥Y p ∥H µ(d p) ,

(3.26)

A

(⋃ n ) where Y p(n) ∈ H is equal to Y p(n) = bn χ sk=1 [ p]n (k) . The integrand above has the chaos expansion ∫ ∞ k ∑ ∏ (n) 2 (n) β2 βk Y p(n) (xℓ )W(x1 ) · · · W(xk )ν(d x1 ) · · · ν(d xk ). eβ⟨W,Y p ⟩− 2 ∥Y p ∥H = 1 + b,s )k k! (D k=1 ℓ=1 ∏ By Remark 3.7, kℓ=1 Y p(n) (xℓ ) is equal to the density, rk(n) (x1 , . . . , xk ; p), of the measure ρk(n) (x1 , . . . , xk ; d p). Thus, the expressions (3.25) and (3.26) are equal. The equalities (I) and (II) also follow from Remark 3.7 since, for instance, ∫ ∫ ∫ ρk(n) (x1 , . . . , xk ; A)ν(d xk ) = rk(n) (x1 , . . . , xk ; p)µ(d p)ν(d xk ) D b,s

D b,s

A

∫ ∏ k

∫ = D b,s

Y p(n) (xℓ )µ(d p)ν(d xk ) ,

A ℓ=1

and switching the order of integration and using that =

∫ ∏ k−1

∫

D b,s

Y p(n) (x)ν(d x) = 1 yields

Y p(n) (xℓ )µ(d p) = ρk(n) (x1 , . . . , xk−1 ; A) . □

A ℓ=1

Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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Proof of Theorem 1.25. By Proposition 3.2, the sequence {Mβ(n) (A)}n∈N converges in L 2 (Γ b,s , BΓ , µ) to Mβ (A). Thus, ⟨ (n) ⟩ ( ) ρk (·, A) ≡ µ(A), ρ1(n) (x1 ; A), ρ2(n) (x1 , x2 ; A), . . . , viewed as an element of (⋃ ) ∞ ∞ ∞ ⨁ ⨁ β 2k k ⊗k 2 b,s k ν , L (D ) , BD , k! k=0 k=0 k=0 converges as n → ∞ to a limit ⟨˜ ρk (·, A)⟩. However, if A ∈ BΓ(n) := P(Γnb,s ), Corollary 3.5 implies that ⟨˜ ρk (·, A)⟩ = ⟨ρk (·, A)⟩ □ Appendix A. Further discussion of the diamond hierarchical lattice The discussion in this section is applicable to any b, s ∈ {2, 3, . . .}. A.1. The vertex set I will show how the set of vertices, Vnb,s , on the graph Dnb,s are embedded in D b,s . For b,s n ∈ N, I can label elements in Vnb,s \Vn−1 by ( )n−1 ( ) b,s Vnb,s \Vn−1 ≡ {1, . . . , b} × {1, . . . , s} × {1, . . . , b} × {1, . . . , s − 1} .    b,s ≡ E n−1 b,s Given an element v = (b1 , s1 ) × · · · × (bn , sn ) ∈ Vnb,s \Vn−1 , define Uv = L v ∪ Rv ⊂ Db,s

for {

L v := (b1 , s1 ) × · · · × (bn , sn ) ×

∞ ∏ j=1

⏐ } ⏐ ⏐ ˆ ˆ (b j , s) ⏐ b j ∈ {1, . . . , b} ⊂ Db,s

and { Rv :=

(b1 , s1 ) × · · · × (bn , sn + 1) ×

∞ ∏ j=1

⏐ } ⏐ ⏐ ˆ ˆ (b j , 1) ⏐ b j ∈ {1, . . . , b} ⊂ Db,s .

Pairs x, y ∈ Uv satisfy d D (x, y) = 0, and Uv is the maximal equivalence class with that property. Thus, v is canonically identified with an element in D b,s . The root vertices V0b,s = {A, B} of the graph are identified with the subsets of Db,s given by {∏ } {∏ } ∞ ∞ ⏐ ⏐ ⏐ ⏐ A := (ˆ b j , 1) ⏐ ˆ b j ∈ {1, . . . , b} and B := (ˆ b j , s) ⏐ ˆ b j ∈ {1, . . . , b} . j=1

j=1

A.2. The metric space D b,s Next I prove the points in Proposition 2.4. Note that each element of E b,s is equivalent to a nested sequence en ∈ E nb,s . Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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Completeness: Let {xk }k∈N be a Cauchy sequence in D b,s . The sequence ˜ π (xk ) ∈ [0, 1] must be Cauchy and thus convergent to a limit λ ∈ [0, 1]. If λ is a multiple of b−N for some nonnegative integer N ∈ {0, 1, 2, . . .}, let N be the smallest such value. For large k, the terms xk must become arbitrarily close to generation N vertices. Since the generation N vertices are at least a distance 1/s N apart, the terms must be close to the same generation N vertex and thus convergent. If λ is not a multiple of b−N for any N ∈ N, then there must exist a nested sequence of b,s edge sets en ∈ ∪∞ k=1 E k such that each closure en contains a tail of the sequence {x k }k∈N . The sequence {xk }k∈N converges to the unique element in the set ∩∞ n=1 en . Compactness: Let {xk }k∈N be a sequence in D b,s . Since D b,s is covered by sets e for e ∈ E nb,s , the pigeonhole principle implies that there must exist a nested sequence of edge sets en ∈ E nb,s such that each en contains xk for infinitely many k ∈ N. Thus, there is a subsequence of {xk }k∈N converging to the unique element in ∩∞ n=1 en . Hausdorff dimension: The contractive maps Si, j ’s defined in Remark 2.3 are the similitudes of the fractal D b,s . The open set O := D b,s \{A, B} is a separating set since ⋃ Si, j (O) ⊂ O and Si, j (O) ∩ Sk,l (O) = ∅ i, j

for (i, j) ̸= (k, l). Since the Si, j ’s have contraction constant 1/s and there are bs maps, the Hausdorff dimension of D b,s is log(bs)/ log s; see [11, Section 11.2] for a discussion of Hausdorff dimension and self-similarity. A.3. The measures Finally, I prove Propositions 2.5, 2.9 and 3.3. Uniform measure on the diamond lattice: Let B E := {A ∩ E b,s =| A ∈ B D } be the restriction of B D to E b,s = D b,s \V b,s . Every element A ∈ B D can be decomposed as A = A1 ∪ A2 for A1 ⊂ V b,s and A2 ∈ B E . I define ν as zero on V b,s , and it remains to define ν on B E . The b,s Borel σ -algebra B E is generated by the semi-algebra, C E = ∪∞ n=0 E n , i.e., the collection of cylinder sets C(b1 ,s1 )×···×(bn ,sn ) . This follows since for any open set O ⊂ D b,s we can write ⋃ O\V b,s = ex , (A.1) x∈O\V b,s

where ex ∈ C E is the biggest set in C E satisfying x ∈ ex ⊂ O. A premeasure ˆ ν can be placed on −1 C E by assigning ˆ ν(e) = |E nb,s | = (bs)−n to each e ∈ E nb,s . The finite premeasure (E b,s , C E ,ˆ ν) extends uniquely to a measure (E b,s , B E , ν) through the Carath´eodory procedure. b,s Uniform measure on paths: Consider the semi-algebra CΓ := ∪∞ of subsets of Γ b,s . n=0 Γn b,s An arbitrary open set O ⊂ Γ can be written as a disjoint union of elements in CΓ in analogy to (A.1). Indeed, each element in CΓ is an open ball with respect to the metric dΓ . A finite −1 premeasure ˆ µ is defined on CΓ by assigning each q ∈ Γnb,s the value ˆ µ(q) = |Γnb,s | . Again, b,s by Carath´eodory’s technique, the measure ˆ µ extends to a measure (Γ , BΓ , µ). Next I will argue that µ and ν are related ∫ ∫1 ( ) through the identity (2.1). To construct a measure ˜ ν on D b,s satisfying ˜ ν(R) = µ 0 1 R p(r ) µ(d p), I can follow the approach used above for constructing ν by defining how ˜ ν acts on V b,s and the cylinder sets. In particular, if ˜ ν agrees b,s with ν on V and the cylinder sets, then they must be equal. The vertex set V b,s has ˜ ν-measure Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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zero because a path p : [0, 1] → D b,s only passes through vertex points at the countable collection are integer multiples of s −n for some n ∈ N, and thus ∫1 ( of )times r ∈ [0, 1] that b,s . A cylinder set R = C(b1 ,s1 )×···×(bn ,sn ) has ˜ ν-measure (bs)−n 0 1V b,s p(r ) = 0 for all p ∈ Γ since a path chosen uniformly at random will pass through R with probability b−n and, in that event, it will spend a duration of s −n in R. Therefore ˜ ν = ν. Uniform measure on paths through a finite subset of Eb,s : Let S ⊂ E b,s be finite and Γ Sb,s be nonempty. There exists an N ∈ N such that no two elements in S fall into the same equivalence class e ∈ E Nb,s . For n > N , dµ(n) S = JS(n) dµ

where

( ) JS(n) ( p) := bn|S|−γ (S) χ [ p]n ∩ Γ Sb,s ̸= ∅ .

Moreover, JS(n) forms a nonnegative, mean-one martingale with respect to the filtration Fn = Γnb,s . If g∫ : Γ b,s → R is measurable with respect to Fm for some m ∈ N, then the sequence Γ b,s g( p)µ(n) (d p) is constant for n ≥ m and thus convergent. A continuous function h : Γ b,s → R must be uniformly continuous since Γ b,s is a compact metric space. Thus, given ϵ > 0, there exists an n ∈ N such that |g( p) − g(q)| < ϵ when [ p]n = [q]n . It follows that ∫ Yn g = Γ b,s g( p)µ(n) (d p) converges to a limit Y∞ g as n → ∞. Therefore, µ(n) S converges vaguely to a limiting probability measure µ S on Γ b,s . Appendix B. Random shift The following argument from [26] shows that βY defines a random shift. Proof of Theorem 1.12. Let] ˜ Eβ refer[ to the expectation ]with respect to ˜ Pβ . The calculation [ ⟨W,ψ⟩ ˜ below shows that Eβ e = Eβ Mβ (W, Γ b,s )e⟨W,ψ⟩ for any ψ ∈ H, and thus that Mβ (W, Γ b,s ) is the Radon–Nikodym derivative of ˜ Pβ with respect to P. ∫ [ ] [ ] ˜ E e⟨W+βY p ,ψ⟩ µ(d p) Eβ e⟨W,ψ⟩ := Γ b,s ∫ [ ] = E e⟨W,ψ⟩ eβ⟨Y p ,ψ⟩ µ(d p) Γ b,s

Since Y

(n)

converges strongly to Y by part (i) of Proposition 1.21, the above is equal to ∫ (n) 1 2 = e 2 ∥ψ∥H lim eβ⟨Y p ,ψ⟩ µ(d p) n→∞ Γ b,s ∫ (n) 2 (n) 2 β2 1 = lim e 2 ∥ψ+βY p ∥H − 2 ∥Y p ∥H µ(d p) . n→∞ Γ b,s

[ ] { } With E exp{⟨W, φ⟩} = exp 12 ∥φ∥22 , I have the equality ∫ [ (n) ] (n) 2 β2 = lim E e⟨W,ψ+βY p ⟩ e− 2 ∥Y p ∥H µ(d p) n→∞ Γ b,s [(∫ ) ] (n) (n) 2 β2 = lim E eβ⟨W,Y p ⟩− 2 ∥Y p ∥H µ(d p) e⟨W,ψ⟩ n→∞ b,s [ Γ( ] ) (n) = lim E Mβ W, Γ b,s e⟨W,ψ⟩ . n→∞

Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.

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Finally, the limit can be brought inside the expectation as a consequence of part (ii) of Proposition 3.2 [ ( ] ) = E Mβ W, Γ b,s e⟨W,ψ⟩ . □ References [1] T. Alberts, J. Clark, S. Kocic, The intermediate disorder regime for a directed polymer model on a hierarchical lattice, Stoch. Process. Appl. 127 (2017) 3291–3330. [2] T. Alberts, K. Khanin, J. Quastel, The continuum directed random polymer, J. Stat. Phys. 154 (1–2) (2014) 305–326. [3] T. Alberts, K. Khanin, J. Quastel, The intermediate disorder regime for directed polymers in dimension 1 + 1, Ann. Probab. 42 (3) (2014) 1212–1256. [4] J. Barral, A. Kupiainen, M. Nikula, E. Saksman, C. Webb, Basic properties of critical lognormal multiplicative chaos, Ann. Probab. 43 (2015) 2205–2249. [5] N. Berestycki, An elementary approach to Gaussian multiplicative chaos, Electron. J. Probab. 22 (27) (2017) 22. [6] F. Caravenna, R. Sun, N. Zygouras, Polynomial chaos and scaling limits of disordered systems, J. Eur. Math. Soc. (JEMS) 19 (2017) 1–65. [7] F. Caravenna, R. Sun, N. Zygouras, Universality in marginally relevant disordered systems, Ann. Appl. Probab. 27 (5) (2017) 3050–3112. [8] J.T. Clark, High-temperature scaling limit for directed polymers on a hierarchical lattice with bond disorder, J. Stat. Phys. 174 (6) (2019) 1372–1403. [9] B. Duplantier, R. Rhodes, S. Sheffield, V. Vargas, Renormalization of critical Gaussian multiplicative chaos and KPZ relation, Comm. Math. Phys. 330 (1) (2014) 283–3330. [10] B. Duplantier, S. Sheffield, Liouville quantum gravity and KPZ, Invent. Math. 185 (2) (2011) 222–393. [11] G. Folland, Real Analysis: Modern Techniques and their Applications, second ed., Wiley, 1999. [12] G. Giacomin, H. Lacoin, F.L. Toninelli, Hierarchical pinning models, quadratic maps, and quenched disorder, Probab. Theory Related Fields 145 (2009). [13] L. Goldstein, Normal approximation for hierarchical structures, Ann. Appl. Probab. 14 (2004) 1950–1969. [14] R.B. Griffith, M. Kaufman, Spin systems on hierarchical lattices. Introduction and thermodynamical limit, Phys. Rev. B 26 (9) (1982) 5022–5032. [15] B.M. Hambly, J.H. Jordan, A random hierarchical lattice: the series-parallel graph and its properties, Adv. Appl. Probab. 36 (2004) 824–838. [16] B.M. Hambly, T. Kumagai, Diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice, Adv. Appl. Probab. 36 (2004) 824–838. [17] S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997. [18] J. Junnila, E. Saksman, The uniqueness of the Gaussian multiplicative chaos revisted, Electron. J. Probab. 22 (2017) 1–31. [19] J.P. Kahane, Sur le chaos multiplicative, Ann. Sci. Math. Québec 9 (2) (1985) 105–150. [20] H. Lacoin, Hierarchical pinning model with site disorder: disorder is marginally relevant, Probab. Theory Related Fields 2 (1–2) (2010) 159–175. [21] H. Lacoin, G. Moreno, Directed Polymers on hierarchical lattices with site disorder, Stochastic Process. Appl. 120 (4) (2010) 467–493. [22] H. Lacoin, R. Rhodes, V. Vargas, Complex Gaussian multiplicative chaos, Comm. Math. Phys. 337 (2) (2015) 569–632. [23] R. Rhodes, V. Vargas, Gaussian multiplicative chaos and applications: a review, Probab. Surv. 11 (2014) 315–392. [24] P.A. Ruiz, Explicit formulas for heat kernels on diamond fractals, Comm. Math. Phys. 364 (2018) 1305–1326. [25] E. Saksman, C. Webb, The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line, (preprint) arXiv:1609.00026. [26] A. Shamov, On Gaussian multiplicative chaos, J. Funct. Anal. 270 (2016) 3224–3261. [27] C. Webb, The characteristic polynomial of random unitary matrix and Gaussian multiplicative chaos, Electron. J. Probab. 20 (104) (2015) 21.

Please cite this article as: J.T. Clark, Continuum directed random polymers on disordered hierarchical diamond lattices, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.05.008.