On the Liouville heat kernel for k-coarse MBRW and nonuniversality

We study the Liouville heat kernel (in the $L^2$ phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each $\varepsilon>0$ there exists such a field, whose covariance is a bounded perturbation of that of the two dimensional Gaussian free field, and such that the associated Liouville heat kernel satisfies the short time estimates, $$ \exp \left( - t^{ - \frac 1 { 1 + \frac 1 2 \gamma^2 } - \varepsilon } \right) \le p_t^\gamma (x, y) \le \exp \left( - t^{- \frac 1 { 1 + \frac 1 2 \gamma^2 } + \varepsilon } \right) , $$ for $\gamma<1/2$. In particular, these are different from predictions, due to Watabiki, concerning the Liouville heat kernel for the two dimensional Gaussian free field.


Introduction
In recent years, there has been much interest and progress in the understanding of two dimensional Liouville quantum gravity, and associated processes. We do not provide an extensive bibliography and refer instead to the original articles and surveys [9,10,5] for background. The starting point for this study is the construction of Liouville measure, which is the exponential of the Gaussian free field and is constructed rigorously using Kahane's theory of Gaussian multiplicative chaos [17].
One aspect that has received attention is the construction of Liouville Brownian motion using the Liouville measure and the theory of Dirichlet forms. Mathematically, this has been achieved in [11] (see also [4]), and properties of the associated Liouville heat kernel have been discussed in [12,15,2]. One important motivation behind the study of the Liouville heat kernel is that it can be used to study the geometry (and critical exponents) of Liouville quantum gravity. Indeed, a particularly nice application of the construction of the Liouville heat kernel is that it allows for a clean derivation of the so-called KPZ relations [3]. Another important motivation, discussed in [15], are the predictions of Watabiki [18] concerning the short time behavior of the Liouville heat kernel. See the discussion in [15,2] for existing (weak) estimates on the diffusivity exponents of the Liouville heat kernel.
An important aspect of the class of logarithmically correlated Gaussian fields (of which the 2D Gaussian free field is arguably the prominent example) is the universality of many quantitites, e.g. Hausdorff dimensions, statistics of the maximum, etc., see [17,7]. One could naively expect that for Gaussian fields in this class, the predicted exponents of the Liouville heat kernel would be universal.
Our goal in this paper is to show that this is not the case, in the sense that the explicit predictions on Liouville heat-kernel exponents (appearing in [18] and discussed in [15,2]) do not hold for some two dimensional logarithmically correlated Gaussian fields which are bounded perturbations of the Gaussian free field. Namely, we study in this paper the heat kernel for Liouville Brownian motion constructed with respect to a particular logarithmically correlated field, introduced in [6] under the name k-coarse modified branching random walk (MBRW for short). Given k > 0 integer, this is the centered Gaussian field on the torus T = R 2 /(4Z) 2  We will show in Section 2.1 that for all k, where λ is continuous in (0, 2] and |λ| ≤ 6k. Fixing γ ∈ (0, 2), we introduce in Section 2.3, following [11], the Liouville measure µ γ , Liouville Brownian motion (LBM) {Y t }, and Liouville heat kernel (LHK) p γ t (x, y), associated with (γ, h). Formally, the Liouville measure on T is defined as µ γ (dx) := e γh(x)− 1 2 γ 2 Eh 2 (x) dx; one then introduces the positive continuous additive functional (PCAF) with respect to µ γ as where {X t } denotes a standard Brownian motion (SBM) on T. The LBM is then defined formally as Y t := X F −1 (t) , and the LHK p γ t (x, y) is then the density of the Liouville semigroup with respect to µ γ , i.e.
where the superscript x is to recall that Y 0 = X 0 = x. Let P denote the Gaussian law of h. The main result of this paper is as follows.
Remark 1.2. Our result shows that the exponent of the LHK with respect to the k-coarse MBRW is for large k and small γ, roughly (1 + o k (1))/(1 + γ 2 /2). In particular, it does not match values one could guess from Watabiki's formula, see [18,15], based on which one would predict that for γ small, the exponent is (1+o(γ))/(1+7γ 2 /4). This is yet another manifestation of the expected nonuniversality of exponents related to Liouville quantum gravity, across the class of logarithmically correlated Gaussian fields. See [6,8] for other examples.
Heuristic. We describe the strategy behind the proof of the lower bound, and the upper bound is similar. First, represent hierarchically the MBRW as follows. Let h j be independent centered Gaussian fields on T with covariance Formally, h = ∞ j=0 h j . For given t, choose r such that t = 2 −kr(1+ 1 2 γ 2 −o(1)) , and decompose the field h into a coarse field ϕ r and a fine field ψ r , with with respective covariances Note that much like the MBRW, the fine field is not defined pointwise but only in the sense of distributions.
With k, r fixed, we partition T into 2 2(kr+2) boxes of side length s = 2 −kr , elements of We call the elements of BD r s-boxes. Similarly to [6], we will find a sequence of neighboring s-boxes B i , 1 ≤ i ≤ I (with I ≤ 2 kr(1+δ) , δ chosen below) connecting x to y, so that the following properties (of the B i 's) hold. The coarse field ϕ r throughout each B i is bounded above by δkr log 2, where δ > 0 is small and will be chosen according to ε in Theorem 1.1. With probability at least s δ , the LBM associated with the fine field ψ r crosses each B i within time s 2−δ . Forcing the original LBM to pass through this sequence of boxes, we will then conclude that it spends time at most crossing from x to the s-box containing y. This happens with probability at least , and, modulu a localization argument, completes the proof of the lower bound.
Structure of the paper. The preliminaries Section 2 is devoted to the study of the covariance of the k-coarse MBRW h, and in particular to verifying that its covariance is a bounded perturbation of that of the Gaussian free field. We also discuss the power law spectrum of h and the construction of the LBM with its corresponding PCAF. In addition, Section 2.2 is devoted to a study of the coarse field ϕ r , and results in estimates on its fluctuations and maximum in a box. Section 3 is devoted to a study of the fine field; we introduce the notions of slow and fast points/boxes and estimate related probabilities. (The property of being fast is used in the proof of the lower bound, and that of being slow is used in the upper bound.) Finally, the proof of lower bound is contained in Section 4, and that of upper bound is contained in Section 5. Both these sections borrow crucial arguments from [6].
Notation convention. Throughout the paper, we restrict attention to 0 ≤ γ < 1/2. T is equipped with the natural metric inherited from the Euclidean distance. We choose δ > 0 small and k large integer (as functions of ε) and keep them fixed throughout. We let C i , i = 0, 1, . . . be universal positive constants, independent of all other parameters. With r as described above, we let BD r (x) denote the unique element of BD r containing x. For ℓ > 0, an ℓ-box means a box of side length ℓ. Let B ℓ (x) denote the ℓ-box centered at x, and let B(x, ℓ) denote the ball centered at x with radius ℓ. For any box B, let c B denote the center of B. If B is an ℓ-box, denote by B * the (5ℓ)-box centered at c B . We use P and E to denote the probability and expectation related to the Gaussian field h. Let P x and E x be the probability and expectation related to the SBM starting at x. We let F x and F x r be the PCAFs for the LBM and ψ r -LBM started at x, respectively. When the starting point x needs not be emphasized, we drop the superscript x.

Preliminaries
Subsection 2.1 is devoted to the proof of (1). In Subsection 2.2, we study the coarse field ϕ r and bound its maximum on small boxes as well as the fluctuation across such boxes. Subsection 2.3 is devoted to a quick review of the construction and existence of the LBM and the LHK.

The coarse field
Note that g j (x, y) is a positive definite kernel on L 2 (T), since, with R = R j = 2 −kj , and therefore, for any f ∈ L 2 (T), Since g j (x, y) is Lipshitz continuous, Kolmogorov's criterion implies that the associated Gaussian field x → h j (x) is continuous almost surely (more precisely, there exists a version of the field which is continuous almost surely). Consequently, the coarse field ϕ r is also smooth. In this subsection, we estimate the maximum value as well as the fluctuations of ϕ r in a box. We begin by recalling an easy consequence of Dudley's criterion.
Lemma 2.2. Let {η z : z ∈ B} be a Gaussian field on a finite index set B. Set σ 2 = max z∈B Var(η z ). Then for all λ, a > 0, Proof. Use the notation in Subsection 2.1. Let d = |x − y|, r 0 = r 0 (d). By (6) and (8), Corollary 2.4. Suppose k is large. Let B denote a box of side length ℓ, and set M := max z∈B ϕ r (z).
Proof. We discretize B by dividing B into 2 2n identical boxesB's and identifying the lower left cornerc of eachB as a point in Z 2 . Denote by M n the maximum value of ϕ r over thesec's. By the continuity of the coarse field, M n increases to M as n → ∞. By Proposition 2.3, we can apply Lemma 2.1 to ϕ r / √ 2 kr 2ℓ and conclude that EM n ≤ √ 2C 0 √ 2 kr ℓ. The monotone convergence theorem yields the result.
Corollary 2.5. There exist r 0 = r 0 (k, δ) such that the following holds for k large and r ≥ r 0 . Enumerate the boxes in BD r arbitrarily as Proof. Note that, for all x, Eϕ r (x) 2 = kr log 2. By Corollary 2.4,

by a union bound and symmetry,
where in the last inequality we use r ≥ r 0 (k, δ).

Construction of the LBM and LHK
There are several ways to construct the Liouville measure µ γ with respect to h, say, via the method of Gaussian multiplicative chaos [13]. In our case, since we deal with γ < 1/2, it is particulaly simple since L 2 methods apply. So, in the rest of this section we concentrate on the construction of the LBM and LHK. Suppose ε = 2 −kr . Then, since A(εx, εy; 2 −k(r+j) ) = A(x, y; 2 −kj ). By (6), Let Ω ε be a Gaussian field independent of h, with EΩ ε = 0 and EΩ ε (x)Ω ε (y) = G (1) r (εx, εy). Actually, Ω ε is a copy of the coarse field ϕ r if we regard x as εx. Then is a constant depending on q (as well as γ). Thus For any 2 −k(r+1) < ε ≤ 2 −kr , we take C(q) =Ĉ(q)2 −kξ(q) and conclude that Recall that the coarse field ϕ r is smooth, so With (10) and (11), one can follow the arguments in [11, Section 2] and obtain the following conclusions. Let F denote the PCAF associated with µ γ . Then, P-a.s., the limit of H r in P xprobability exists and it is the PCAF F ; that is, P x (sup 0≤t≤T |F (u) − H r (u)| > a) → r→∞ 0, for all a > 0 and T > 0. Further, the process Y t := X F −1 (t) is a strong Markov process, which is called the LBM with respect to µ γ . The LHK p γ t (x, y) exists and satisfies E x f (Y t ) = f (y)p t (x, y)µ γ (dy). Furthermore, by [12, Theorem 0.1] and parallel arguments in [15], p γ t (x, y) is continuous in (t, x, y).

Fast/slow points/boxes of the fine field
This section is devoted to the study of properties of the fine field. For the lower bound on the LHK, we need to construct regions which are fast to cross for the LBM, while for the upper bound we will need to create obstacles, i.e. regions which force the LBM to be slow. Toward this end, we introduce in Definitions 3.1 and 3.2 the notions of fast/slow points and boxes, and estimate, in Lemma 3.3 and 3.4, the probability that a point/box is fast/slow. Throughout, we fix s = 2 −kr for an appropriate integer r ≥ 1 (as explained in the introduction, r, and hence s, are chosen so that t = s 1+ 1 2 γ 2 +o (1) ). This choice determines the fine field ψ r , see (4). With this choice, one can construct the PCAF F r based on ψ r in the same way as F was constructed, by replacing the measure µ γ with the truncated measure µ γ r written formally as µ γ r (dx) = e γψr(x)− γ 2 2 Eψ 2 r (x) dx (as before, the actual construction involves the smooth cutoff ψ r,w := w j=r h j and taking the limit as w → ∞). Formally, we write We note also that the sequence of approximating PCAF converges as w → ∞, in the sense described at the end of Section 2, to F r . Fix δ 1 , δ 2 , δ 3 , ε 1 , ε 2 , ε 3 > 0 small, possibly depending on k, γ and s. Fix z ∈ T and recall that B ℓ (z) denotes the ℓ-box centered at z. Let σ z,ℓ denote the time that the SBM (starting from z) hits ∂B ℓ (z).

Definition 3.1 (Fast points and boxes).
The set of fast points is denoted by F. An s-box B is said to be fast if |B ∩ F| ≥ δ 3 s 2 .

Definition 3.2 (Slow points and boxes).
A point z is said to be slow if The set of slow points is denoted by S. An s-box B is said to be slow if |B ∩ S| ≥ ε 3 s 2 .
We emphasize that the notions of fast/slow points and boxes depend on the fine field ψ r only. Further, a point (or box) may be fast and slow simultaneously.
Our fundamental estimate concerning fast/slow points is contained in the next lemma.
Lemma 3.3. There exist universal positive constants C 1 , C 2 , C 3 such that the following hold.

It follows that
Note thatÎ is a random variable depending only on the SBM {X}. By [16,Theorem 4.33], EÎ < ∞.
The next lemma estimates the probability that an s-box B is fast/slow.
Proof. (i) By Lemma 3.3(i) and the translation invariance of the fine field ψ r , E|B ∩ F| ≥ (1 − C 1 δ 1 δ 2 )s 2 . Since |B ∩ F| ≤ |B| ≤ s 2 , |B ∩ F| ≤ |B ∩ F|1 |B∩F |<δ 3 s 2 + |B ∩ F|1 |B∩F |≥δ 3 s 2 ≤ δ 3 s 2 + s 2 1 |B∩F |≥δ 3 s 2 . Hence, E|B ∩ F| − δ 3 s 2 ≤ s 2 P(|B ∩ F| ≥ δ 3 s 2 ) = s 2 P(B is fast). Therefore, P(B is fast) ≥ 1 (ii) Our strategy is as follows. We will divide B into n 2 identical boxesB of side length s = s/n, where n is to be chosen properly to support the following arguments. In each boxB, one can find O(s 2 /n 2 ) slow points in average, by Lemma 3.3(ii). Then, we would like to use large deviations to show that, with high probability, there are at least δ 3 s 2 slow points in B, i.e. B is slow. Unfortunately, the random variables |B ∩ S|'s, measuring the size of the cluster of slow points in the smaller boxesB, are heavily dependent. To obtain the appropriate large deviation estimates by independence, we will replace σ z,s in (14) by σ z,s , and use a new parametersε 1 to define the property of a point to be slow. LetS consist of slow points. Then, the random variables |B i ∩S|'s are almost independent, and good large deviation estimates for their sums can be obtained. Finally, we will show that by choosingε 1 properly, B ∩S ⊂ B ∩ S with high probability, completing the proof.
The actual proof is in four steps. In the first step, we set the parameters n andε 1 , and give the definition of being slow. In the second step, we will show |B ∩S| ≥ δ 3 s 2 with high probability. In the third step, we will show B ∩S ⊂ B ∩ S with high probability. In the last step, we collect the results obtained and show (ii).
Denote byS the set of slow points.
Without loss of generality, we suppose B = [0, s) 2 . We next partition B into n 2 identicalsboxes, from which we pick those of the form [4as, (4a + 1)s) × [4bs, (4b + 1)s), a, b ∈ Z ∩ [0, n/4), and enumerate them arbitrarily asB i , i = 1, · · · , (n/4) 2 . Note thatB i ∩S depends on the restriction of the fine field ψr to the (2s)-box centered at cB i , and ψr(w) is independent of ψr(w ′ ) if |w − w ′ | ≥ 2s. It follows that the random variables |B i ∩S|'s are mutually independent. Let Now we estimate the right hand side of (23) via large deviations. Note that the χ i 's are Bernoulli random variables, with P (χ i = 1) ≥ a, see (22), and therefore Using independence and Chebyshev's inequality we get Recall thats = s/n, a = 60C 3 e −6kγ 2 , see (21), and ε 3 ≤ C 2 3 e −12kγ 2 = ( a 60 ) 2 by assumption. Thus, Together with (24) and (23), we conclude that Step We are going to compare F z r (σ) with F z r (σ), and show below that which we will use in the next step. Before doing that, we first complete the proof of (27). Let φ = ψ r − ψr, which has covariance G r,r (w 1 , w 2 ) = k log 2r SetB = 2 krB , which has side length 2. Note that A(w 1 , w 2 , 2 −kj ) = A(ŵ 1 ,ŵ 2 , 2 −k(j−r) ), wherê w i = 2 kr w i . Therefore, {φ(w), w ∈B} is a copy of the coarse field {ϕ r 0 (ŵ), w ∈B}, with w being identified asŵ = 2 kr w, where we recall that r 0 =r − r and is defined in (18). By Corollary 2.4, Since Eφ(w) 2 = kr 0 log 2 = log n for all w, we have where we use Lemma 2.2, and the last inequality holds by (20). Noting for all z ∈ B, thes-box centered at z is contained inB, we have X u ∈B for u ≤σ, where we drop the superscript z in X u . Therefore, on the event {M < 2n log n}, it holds that for all z ∈ B, where in the first equality we use the independence of ψr and φ. By the definition ofε 1 in (19), P z (F r (σ) ≥ ε 1 s 2 ) ≥ P z (Fr(σ) ≥ e γ2n log n+ γ 2 2 log n ε 1 s 2 ) = P z (Fr(σ) ≥ε 1s 2 ).
Step 4. If |B ∩S| ≥ ε 3 s 2 and B ∩S ⊂ B ∩ S, we have |B ∩ S| ≥ ε 3 s 2 , i.e. B is slow. Hence, By (25) and (28), it follows that where in the second inequality we use (20) and in the last two inequalities we use (18). This implies (ii) and completes the proof of the lemma.
The next lemma bounds below F z r (σ z,3s ) uniformly in z in slow boxes.
Lemma 3.5. There exists a universal positive constant C 4 such that the following holds. Suppose B is slow. Then, P z (F r (σ z,3s ) ≥ ε 1 s 2 ) ≥ C 4 ε 2 ε 3 for all z in the closure of B.

Lower Bound
We continue to take s := 2 −kr = t 1 1+ 1 2 γ 2 +o (1) . To obtain the lower bound on the LHK, we will force the LBM {Y x u }, started at x ∈ T, to hit y ∈ T according to the following three steps. First, we will force the LBM to hit inside BD r (y) a point which is very fast (a notion to be defined below), then hit inside B(y, s 1+β ′ ) (where β ′ > 0 is a parameter to be chosen), and finally we force the LBM to hit y. We will allow time about t/3 for each step, and show that these steps respectively bring factors e −s −(1+o(1)) , s 2+2β ′ +o(1) and O(1) for the lower bound of the heat kernel. This will give the lower bound e −s −(1+o (1) The argument is naturally split according to these steps. In Subsection 4.1, we compute the probabilities of the first step in Lemma 4.1 and of the second one in Lemma 4.3, after introducing the notion of very fast points; in that section, r will be arbitrary, i.e. not tied to the value of t. We pick the value of r according to t in Subsection 4.2, where we will deal with the third step and show the lower bound.
Lemma 4.1. There exist positive constants c, k 0 = k 0 (δ), c 0 = c 0 (k, δ) and r 0 = r 0 (x, y, γ, δ, k), not depending on r but possibly depending on k, γ, such that the following holds for k ≥ k 0 and r ≥ r 0 . Suppose D is a random (with respect to h) set and D ⊂ BD r (y). Let ς 1 be the hitting time of D by the LBM started from x. Then, with P-probability at least 1 − e −c 0 r − P(|D| < δ 3 s 2 ), Proof. We construct a sequence of neighboring s-boxes connecting x and y, as follows. Discretize T by regarding each B ∈ BD r (equivalently, its center c B ) as a point in Z 2 . We investigate the discrete Gaussian field Φ := {ϕ r (c B ), B ∈ BD r }, together with the Bernoulli process Ξ := {ξ B , B ∈ BD r } defined by ξ B := 1 if B is fast. Next we will apply [6, Theorem 1.7] to (Φ, Ξ). Set N = 2 kr , and correspond B, ϕ r (c B ), ξ B respectively to w ∈ Z 2 , ϕ N,w , ξ N,w in [6]. Then, • Ξ is independent of Φ, since Ξ depends on the fine field while Φ depends on the coarse field.
• The collection of random variables {ξ B } B∈BDr has finite range dependence, in particular ξ B is independent of ξ B ′ if |c B − c B ′ | ∞ > 9s. (In the language of [6], Ξ is q-dependent for q = 9.) • P (ξ B = 1) is equal to a same value p for all B.

Remark 4.2. (i)
The space is the torus T here, while it is a box in [6]. One can identify the torus as [0, 4) 2 , and consider the box [1, 3] 2 where we locate x and y, noting that h(z) is independent of h(w) if |z − w| ≥ 2. (ii) To achieve (31), it is not crucial whether one uses balls B(x, R) (as in our situation) or boxes B 2R (x) (as in [6]) to define A(x, y; R). That is, the proof of (31) is similar to that of [6,Theorem 1.7].
Let E 2 be the event that the following properties hold.
(b ′ ) x is fast.
By Corollary 2.5, P(a ′ ) ≥ 1 − e −r . By Lemma 3.3, P(b ′ ) ≥ 1 − C 1 δ 1 /δ 2 = C 1 2 −kδr . Take c 0 such that (C 1 + 1) 1 400 2 − kδ 400 r + e −c 0 r + e −r + C 1 2 −kδr ≤ e −c 0 r . Then, we have Next, we are going to show that (30) holds on E, completing the proof. Suppose E holds. We will force the SBM to follow this sequence of boxes; to control the LBM time, we will force also passage through fast points, and some additional properties, as follows. Recall that {X x u } is the SBM starting from x. Construct a sequence of hitting times σ i as follows. Let σ 1 = 0. Then . Suppose that σ i has been defined, such that Informaly, τ i is the time it takes for the SBM to cross B i into the next box B i+1 and hit a fast point. Note that (a) together with (a ′ ) implies that (a ′′ ) For all z ∈ ∪ i B * i , ϕ r (z) ≤ cδkr log 2.
In order to take advantage of (a ′′ ), we need to also control the path of the SBM when traveling from x i to B i+1 ∩ F. Toward this end, definẽ Thus,τ i is the time it takes the SBM to exit B * i when starting at x i . We will force the events τ i ≤ s 2 and τ i ≤τ i to ensure that the LBM stays inside B * i and spends a short enough time to hit B i+1 ∩ F.
Let β ′ > 0 be fixed. Abbreviate B = BD r (y), and set A = B ∩ B(y, s 1+β ′ ). Denote by τ A (respectively, τ * ) the times that the SBM hits A (respectively, ∂B * ). A point z ∈ B is called very fast if P z (F r (s 2 ) ≤ s 2−δ |τ A ≤ s 2 ≤ τ * ) ≥ 1/2. Let VF denote the set of very fast points. Note that VF ⊂ B. We would like to mention that the very fast property does not imply the fast property. (ii) Let ς 2 denote the time that the LBM hits A. Then, there exists r 1 = r 1 (δ, γ, k) such that the following holds for r ≥ r 1 . With P-probability at least 1 − 2e − 1 8 δ 2 kr log 2 , Proof. The proof of (i) is parallel to Lemma 3.4(i) combined with Lemma 3.3(i), while that of (ii) is parallel to (33).

Proof of the lower bound in (2)
We take and set s = 2 −krt so that The following lemma is a straight forward adaptation of [15,Corollary 5.20]. We omit the details.
Furthermore, by Corollary 2.5, with probability at least 1 − e −c 0 r − e −r , we have (a), (b) and the following property (c) all hold.
Remark 5.3. When a discrete path is identified as a sequence of points v 0 , v 1 , · · · on Z 2 , v i+1 may not be a neighbour of v i . However, we have |v i+1 − v i | ∞ ≤ 2 for all i. Then, the proof in [6, Theorem 1.5] automatically extends to the current setup.