# Multi-modal Learnable Queries for Image Aesthetics Assessment
††thanks: This research/project is supported, in part, by the National
Research Foundation, Singapore and DSO National Laboratories under the AI
Singapore Programme (AISG Award No: AISG2-RP-2020-019); Alibaba Group through
Alibaba Innovative Research (AIR) Program and Alibaba-NTU Singapore Joint
Research Institute (JRI) (Alibaba-NTU-AIR2019B1), Nanyang Technological
University, Singapore; the RIE 2020 Advanced Manufacturing and Engineering
(AME) Programmatic Fund (No. A20G8b0102), Singapore.

Zhiwei Xiong1,2, Yunfan Zhang1,2, Zhiqi Shen1, Peiran Ren3 and Han Yu1
1College of Computing and Data Science, Nanyang Technological University,
Singapore
2Alibaba-NTU Singapore Joint Research Institute, Nanyang Technological
University, Singapore
3Alibaba Group, Hangzhou, China
{zhiwei002, yunfan.zhang, zqshen<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Image aesthetics assessment (IAA) is attracting wide interest with the
prevalence of social media. The problem is challenging due to its subjective
and ambiguous nature. Instead of directly extracting aesthetic features solely
from the image, user comments associated with an image could potentially
provide complementary knowledge that is useful for IAA. With existing large-
scale pre-trained models demonstrating strong capabilities in extracting high-
quality transferable visual and textual features, learnable queries are shown
to be effective in extracting useful features from the pre-trained visual
features. Therefore, in this paper, we propose MMLQ, which utilizes multi-
modal learnable queries to extract aesthetics-related features from multi-
modal pre-trained features. Extensive experimental results demonstrate that
MMLQ achieves new state-of-the-art performance on multi-modal IAA, beating
previous methods by 7.7% and 8.3% in terms of SRCC and PLCC, respectively.

###### Index Terms:

image aesthetics assessment, multi-modal analysis

##  I Introduction

Image aesthetics assessment (IAA) refers to evaluating the quality of images
based on human perception. With the prevalence of social media, IAA has
recently attracted much attention, especially in downstream applications such
as image aesthetic enhancement [1], and aesthetic-aware text-to-image
generation [2]. Due to the subjective and ambiguous nature of the definition
of aesthetics, the “standards” of image aesthetics are usually determined by
opinion scores from different reviewers. Based on these opinion scores, IAA
can be treated as a data-driven problem where the ground truth is either the
mean opinion score (MOS) or the distribution of opinion scores (DOS).

With direct supervision solely from MOS or DOS, various IAA methods were
proposed to extract local to global [3], low to high-level features [4], and
address layout [5] to theme-aware [6] image aesthetics. However, direct
supervision from MOS or DOS may suffer from a lack of details regarding how
these decisions are made. User comments associated with the images could be
utilized to provide additional aesthetic guidance. Fig. 1 shows two example
images and some of their corresponding user comments from the AVA [7] and AVA-
Comments [8] datasets, respectively. Comments keywords such as “phenomenal”,
“magical”, and “love” for the left image and “blurry”, “out of focus”, and
“messy” for the right image express strong inherent sentiments that could be
potentially beneficial for IAA.

Figure 1: Example images with their corresponding image ID, ground truth MOS,
and selected user comments from the AVA [7] and AVA-Comments [8] datasets.
Figure 2: The overall structure of the proposed multi-modal learnable queries
(MMLQ) method for IAA. The visual and textual LQs learn visual and textual
aesthetic features through a replaceable self-attention layer, a separate
multi-modal cross-attention layer with corresponding visual and textual pre-
trained features, and a replaceable feed-forward layer. Finally, the averaged
visual LQs and the averaged textual LQs are concatenated and fed into linear
layers followed by Softmax to output the estimated aesthetic DOS of the input
image.

Recently, large-scale pre-trained models [9, 10] have demonstrated a strong
capability in providing rich transferable knowledge for downstream tasks.
Learnable queries and prompts [10, 11, 12] are shown to be effective ways to
extract useful task-specific features from such pre-trained backbones for
different modalities. Therefore, in this paper, we propose to utilize multi-
modal learnable queries (MMLQ) with large-scale frozen visual and textual
encoders to extract multi-modal aesthetic features from both input images and
their associated user comments. With a proper design for the multi-modal
interaction blocks (MMIB) selected with comprehensive experiments, our
proposed MMLQ efficiently extracts and processes multi-modal aesthetic
features and reaches new state-of-the-art performance on multi-modal IAA.

##  II Related Works

Early works [13, 14] of IAA attempted to develop hand-crafted features
corresponding to human-understandable photographic rules summarized by
professional photographers. However, such photographic rules may not cover
implicit aesthetic patterns that are difficult to summarize using human
language. Consequently, later works [3, 4, 5, 6] proposed various deep
learning models to capture such implicit aesthetic features showing promising
results. Recently, as the prevalence of large-scale pretraining, frozen pre-
trained image encoders [15, 16, 17] have demonstrated an outstanding
capability in providing rich transferable knowledge for IAA. Learnable queries
[17] is one efficient solution to extract relevant aesthetic features from
such frozen image encoders.

With additional aesthetic guidance from user comments, a few recent works [18,
19] attempted to extract multi-modal aesthetic features. Zhu et al. [18]
extracted multi-modal aesthetic features with attribute-based cross-modal
attention and a gate unit. Li et al.[19] extracted attribute-grounded multi-
modal aesthetic features with multi-modal memory network and cross-modal
memory attention. However, their methods require additional pre-training on
aesthetic attributes and suffer from performance degradation for images
without dominant attributes. Unlike these attribute-based methods, we propose
to utilize multi-modal learnable queries without attribute guidance to extract
more diverse multi-modal aesthetic features from patch-level image features
and token-level comment features with frozen visual and textual encoders.

##  III The Proposed Approach

In this section, we present more details about our proposed MMLQ method with
an overall structure shown in Fig. 2.

###  III-A Multi-modal Pre-trained Features

For a given image $I$, a frozen pre-trained vision transformer (ViT) is
employed to split the processed image into $N_{p}$ patch tokens and a prepend
[CLS] token. The extracted pre-trained image features are a collection of
visual token embeddings:

$\boldsymbol{F}_{v}=\\{\boldsymbol{v}_{i}\\}_{i=1}^{N_{p}+1},$ (1)

where $\boldsymbol{v}_{i}\in\mathbb{R}^{H_{v}}$ with $H_{v}$ denoting the size
of the visual features. Meanwhile, we accumulate all corresponding user
comments on the given image $I$ into a long user comment $C$. A frozen pre-
trained text encoder is employed to encode the comment into a collection of
token-level embeddings:

$\boldsymbol{F}_{t}=\\{\boldsymbol{t}_{j}\\}_{j=1}^{N_{w}},$ (2)

where $\boldsymbol{t}_{j}\in\mathbb{R}^{H_{t}}$ with $H_{t}$ denoting the size
of the textual features. $N_{w}$ denotes the max comment length allowed
including both word-level and special tokens at the beginning and end of the
sentence (i.e., [CLS] and [SEP]). Comment lengths longer than $N_{w}$ are
truncated to $N_{w}$ and comment lengths shorter than $N_{w}$ are filled with
[PAD] tokens till $N_{w}$.

###  III-B Multi-modal Interaction Block

Before the first multi-modal interaction block (MMIB), a set of learnable
queries (LQs) are created to extract visual and textual aesthetic features,
respectively. Specifically, they are a set of learnable embeddings that are
shared among samples:

$\displaystyle\boldsymbol{E}_{v}=\\{\boldsymbol{e}_{k}\\}_{k=1}^{N_{v}},\boldsymbol{E}_{t}=\\{\boldsymbol{e}_{l}\\}_{l=1}^{N_{t}},$
(3)

where $\boldsymbol{e}_{k},\boldsymbol{e}_{l}\in\mathbb{R}^{H_{q}}$ are visual
and textual LQs with $H_{q}$ denoting the query embedding size. $N_{v}$ and
$N_{t}$ denote the number of visual and textual LQs, respectively. The LQs are
then normalized with separate layer normalization.

In each MMIB, both visual and textual LQs first go through a replaceable
multi-head self-attention (SA) layer. The added SA layer could be shared among
multi-modal LQs or separated based on modality, which can be expressed as:

$\displaystyle\boldsymbol{Q}_{mh},\boldsymbol{K}_{mh},\boldsymbol{V}_{mh}=\boldsymbol{E}_{m}\boldsymbol{W}_{mh}^{Q},\boldsymbol{E}_{m}\boldsymbol{W}_{mh}^{K},\boldsymbol{E}_{m}\boldsymbol{W}_{mh}^{V},$
(4)
$\displaystyle\boldsymbol{A}_{mh}=\text{Softmax}(\frac{\boldsymbol{Q}_{mh}\boldsymbol{K}_{mh}^{T}}{\sqrt{d}})\boldsymbol{V}_{mh},$
(5)
$\displaystyle\boldsymbol{E}_{m}=\text{LN}(\text{Concat}(\\{\boldsymbol{A}_{mh}\\}_{h}^{N_{h}})\boldsymbol{W}_{m}^{O}+\boldsymbol{E}_{m}),$
(6)

where $\boldsymbol{E}_{m}$ denotes either $\boldsymbol{E}_{v}$ or
$\boldsymbol{E}_{t}$ in a separate SA layer while
$\boldsymbol{E}_{m}=\text{Concat}(\boldsymbol{E}_{v},\boldsymbol{E}_{t})\in\mathbb{R}^{(N_{v}+N_{t})\times
H_{q}}$ in a shared SA layer. Weight matrices
$\boldsymbol{W}_{mh}^{Q},\boldsymbol{W}_{mh}^{K},\boldsymbol{W}_{mh}^{V}\in\mathbb{R}^{H_{q}\times
d}$ are employed to get down-sampled embeddings
$\boldsymbol{Q}_{mh},\boldsymbol{K}_{mh},\boldsymbol{V}_{mh}$. $d$ is the
hidden size of each head, and the scaling of $\frac{1}{\sqrt{d}}$ is to
improve training stability. Finally, weight matrice
$\boldsymbol{W}_{m}^{O}\in\mathbb{R}^{(N_{h}\times d)\times H_{q}}$ is
employed to transform the concatenated multi-head attention results into the
original dimension $H_{q}$ of $\boldsymbol{E}_{m}$, which is added into the
original $\boldsymbol{E}_{m}$ followed by a layer normalization (LN) to get
the new $\boldsymbol{E}_{m}$.

After the replaceable SA layer, the visual and textual LQs would extract
aesthetic features from pre-trained visual and textual features respectively
through a separate multi-head cross-attention (CA) layer. Different from the
SA layer, keys and values are constructed from the pre-trained visual or
textual features, which can be expressed as:

$\boldsymbol{Q}_{mh},\boldsymbol{K}_{mh},\boldsymbol{V}_{mh}=\boldsymbol{E}_{m}\boldsymbol{W}_{mh}^{Q},\boldsymbol{F}_{m}\boldsymbol{W}_{mh}^{K},\boldsymbol{F}_{m}\boldsymbol{W}_{mh}^{V},$
(7)

where $m$ denotes $v$ or $t$. Weight metrices
$\boldsymbol{W}_{mh}^{Q}\in\mathbb{R}^{H_{q}\times d}$,
$\boldsymbol{W}_{vh}^{K},\boldsymbol{W}_{vh}^{V}\in\mathbb{R}^{H_{v}\times
d}$,
$\boldsymbol{W}_{th}^{K},\boldsymbol{W}_{th}^{V}\in\mathbb{R}^{H_{t}\times d}$
are employed to transform LQs $\boldsymbol{E}_{m}$, visual features
$\boldsymbol{F}_{v}$, textual features $\boldsymbol{F}_{t}$, respectively.
Then, $\boldsymbol{Q}_{mh},\boldsymbol{K}_{mh},\boldsymbol{V}_{mh}$ go through
a similar multi-head attention and an add & norm procedure as illustrated in
equations 5, 6 to obtain the new $\boldsymbol{E}_{m}$.

Followed by the replaceable SA layer and the separate CA layer, a replaceable
feed-forward (FF) layer is added to further process the LQs. It comprises two
linear layers, a GELU activation [20], and an add & norm procedure as in the
SA and CA layers, which can be expressed as:

$\boldsymbol{E}_{m}=\text{LN}(\text{FC}(\text{GELU}(\text{FC}(\boldsymbol{E}_{m})))+\boldsymbol{E}_{m}),$
(8)

where the LQs are projected to a larger intermediate dimension $H_{i}=4\times
H_{q}$, and then back to $H_{q}$. Similar to the SA layer, the added FF layer
could be shared among multi-modal LQs or separated based on modality.

###  III-C Prediction Head for IAA

After $N$ MMIBs, both visual and textual LQs accumulate adequate aesthetic
patterns for the final prediction. We take the average of LQs within each
modality and concatenate the two averaged LQs. One linear layer (output
dimension of $H_{q}$) with GELU activation and another linear layer (output
dimension of 10) with Softmax activation are employed on the concatenated LQs
to output the final prediction:

$\displaystyle\boldsymbol{E}_{a}=\text{Concat}(\bar{\boldsymbol{E}}_{v},\bar{\boldsymbol{E}}_{t}),$
(9)
$\displaystyle\hat{\boldsymbol{D}}_{a}=\text{Softmax}(\text{FC}(\text{GELU}(\text{FC}(\boldsymbol{E}_{a})))),$
(10)

where $\bar{\boldsymbol{E}}_{v},\bar{\boldsymbol{E}}_{t}\in\mathbb{R}^{H_{q}}$
denote the averaged visual and textual LQs, respectively.
$\boldsymbol{E}_{a}\in\mathbb{R}^{2H_{q}}$ denotes the concatenated multi-
modal aesthetic features. $\hat{\boldsymbol{D}_{a}}\in\mathbb{R}^{K}$ denotes
the predicted K-scale aesthetic DOS. Following previous works [5, 18, 19], we
adopt Earth Mover’s Distance (EMD) loss to optimize the prediction:

$\mathcal{L}(\boldsymbol{D}_{a},\hat{\boldsymbol{D}}_{a})=\sqrt{\frac{1}{K}\sum_{k=1}^{K}\lvert
CDF_{\boldsymbol{D}_{a}}(k)-CDF_{\hat{\boldsymbol{D}}_{a}}(k)\rvert^{2}},$
(11)

where $CDF_{\boldsymbol{D}_{a}}$ and $CDF_{\hat{\boldsymbol{D}}_{a}}$ are
cumulative density functions for the ground truth and predicted aesthetic DOS,
respectively.

##  IV Experimental Evaluation

Figure 3: The histogram of the accumulated comment lengths for each image in
AVA-Comments [8]. The vertical dotted line in red indicates a comment length
of 512, which is the maximum comment length ($N_{w}$) allowed in our
experiments.

###  IV-A Dataset

The dataset we use for experiments is the benchmark IAA dataset, the AVA
dataset [7]. It contains more than 250,000 images downloaded from the
DPChallenge website, where each image received 78 to 549 aesthetic scores
(average of 210) on a scale of 1 to 10, where 1 and 10 denote the lowest and
highest aesthetics, respectively. The AVA-Comments [8] dataset provides the
corresponding user comments on the images in AVA, which were crawled from the
original links of the images with a histogram of accumulated comment length
for each image as in Fig 3. We use the same split as in [5], where 235,510
images are used for training, and 19,998 images are used for testing.

TABLE I: Comparison with the state-of-the-art methods on the train-test split from [5]. Methods |  Modality |  Input size | SRCC$\uparrow$ | PLCC$\uparrow$ | Acc(%)$\uparrow$ | MSE$\downarrow$ | EMD$\downarrow$
---|---|---|---|---|---|---|---
NIMA [21] |  Image |  224$\times$224 | 0.612 | 0.636 | 81.51 | - | 0.050
AFDC [22] |  Image |  320$\times$320 | 0.649 | 0.671 | 83.24 | 0.271 | 0.045
HLA-GCN [5] |  Image |  300$\times$300 | 0.665 | 0.687 | 84.60 | 0.255 | 0.043
KD-IAA [15] |  Image |  Full resolution | 0.732 | 0.751 | 85.30 | - | -
TCD-IAA [23] |  Image |  Full resolution | 0.767 | 0.783 | 86.20 | - | -
AAM-Net [18] |  Multi-modal |  - | 0.780 | 0.803 | 85.69 | 0.176 | 0.037
AMM-Net [19] |  Multi-modal |  - | 0.816 | 0.830 | 87.58 | 0.160 | 0.035
MMLQ (CLIP) |  Multi-modal |  224$\times$224 | 0.876 | 0.896 | 89.03 | 0.094 | 0.029
MMLQ (EVA-CLIP) |  Multi-modal |  224$\times$224 | 0.879 | 0.899 | 88.99 | 0.092 | 0.028

###  IV-B Experiment Settings

Following [10, 17], we set the query hidden size $H_{q}$ to 768, and the
default number of visual and textual queries $N_{v},N_{t}$ to 2. We use
$N_{h}=12$ heads in multi-head attention layers, yielding a hidden size $d=64$
for each head. For input images, we only apply horizontal flipping with
$p=0.5$ during training and directly resize the image to $224\times 224$.
Frozen pre-trained ViTs with $14\times 14$ patch sizes are applied as the
visual encoder, resulting in $N_{p}=16\times 16=256$ patches for each input
image. For input user comments, the frozen text encoder we adopt is
BERT${}_{\text{base}}$ [24] which outputs token-level textual features with
$H_{t}=768$. The max comment length $N_{w}$ is set to the maximum length
allowed in BERT${}_{\text{base}}$ (i.e., 512), approximately the 90th
percentile of the histogram in Fig 3. We train MMLQ with a batch size of 128
for 10 epochs with the Adam optimizer. The learning rate is initially set to
$3\times 10^{-5}$, and multiplied by 0.1 every two epochs.

###  IV-C Comparison with State-of-the-art Methods

TABLE II: Performance with different MMIB design. ✗, ✓, ✓✓denote no, shared, and separate layers, respectively.  SA | FF | SRCC$\uparrow$ | PLCC$\uparrow$ | Acc(%)$\uparrow$
---|---|---|---|---
✗ | ✗ | 0.874 | 0.894 | 88.92
✗ | ✓ | 0.876 | 0.896 | 89.03
✗ |  ✓✓ | 0.876 | 0.896 | 88.96
✓ | ✗ | 0.875 | 0.894 | 88.88
✓ | ✓ | 0.874 | 0.894 | 88.72
✓ |  ✓✓ | 0.873 | 0.893 | 88.73
✓✓ | ✗ | 0.873 | 0.892 | 88.78
✓✓ | ✓ | 0.873 | 0.893 | 88.81
✓✓ |  ✓✓ | 0.875 | 0.894 | 88.77

We compare the performance of MMLQ with previous image-based and multi-modal
methods in Table I. Following previous works [5, 18, 19], we evaluate MMLQ
with binary accuracy (Acc) for binary aesthetic classification (where a MOS of
5 is the boundary); with Spearman’s rank correlation coefficient (SRCC),
Pearson linear correlation coefficient (PLCC), Mean Squared Error (MSE) for
MOS regression; with EMD loss for DOS prediction. In general, multi-modal
methods have higher performance compared to image-based methods, due to the
additional aesthetic knowledge in the user comments. We report the results of
MMLQ with ViT-L/14 from CLIP [9] and ViT-G/14 from EVA-CLIP [25], where the
former is the default vision encoder in our experiments with a $H_{v}$ of
1024, and the latter has a $H_{v}$ of 1408. With multi-modal learnable queries
to extract multi-modal aesthetic features, our method reaches state-of-the-art
performance with both vision encoders on all the metrics with a low input
image resolution (i.e., $224\times 224$). The performance of MMLQ (EVA-CLIP)
beats previous methods by 7.7% and 8.3% in terms of SRCC and PLCC,
respectively.

###  IV-D Ablation Studies

Effects of different MMIB design: We first explore how MMLQ’s performance
varies with different MMIB designs. Specifically, we set $N$ of MMIBs to 6,
and focus on the designing of the self-attention (SA) and feed-forward (FF)
layers. Table II shows the performance with no / shared / separate SA layer
paired with no / shared / separate FF layer. Overall, MMIB with different
designs results in similar performance. It shows the main contribution of MMIB
comes from the cross-attention (CA) between multi-modal LQs and their
corresponding multi-modal pre-trained features. The best performance occurs
when shared FF layers are added after the CA layer of each modality. The
inferior performance with shared / separate SA layers could be caused by the
feature space gap among LQs of different modalities and over-emphasis on the
difference among LQs of the same modality, respectively.

TABLE III: Performance with different modalities.  Visual | Textual | SRCC$\uparrow$ | PLCC$\uparrow$ | Acc(%)$\uparrow$
---|---|---|---|---
✓ | ✗ | 0.725 | 0.743 | 85.25
✗ | ✓ | 0.822 | 0.852 | 85.90
✓ | ✓ | 0.876 | 0.896 | 89.03

Effects of different modality: To explore the contributions from each
modality, we keep the number of LQs (i.e., $N_{v}$ or $N_{t}$) for one
modality unchanged, and remove the LQs, encoder, and cross-attention layer
corresponding to the other modality. As shown in Table III, the performance
with multi-modality is superior to single visual or textual modality, which
reflects the complementarity of the two modalities in extracting aesthetic
features. Interestingly, the performance of using textual modality only is
superior to using visual modality only. It could be attributed to the rich
sentiment clues available in user comments that are beneficial for IAA.

Figure 4: Performance with different numbers of MMIBs.

Effects of different model complexity: Since model performance is often
directly affected by model complexity, we compare model performance with
different model complexity. Specifically, we explore the effect of different
numbers of MMIBs in Fig. 4, and different numbers of LQs in Fig. 5.
Surprisingly, involving more LQs tends to degrade the performance of MMLQ. It
could be due to the additional knowledge captured by additional LQs being
redundant for image aesthetics. On the contrary, the performance of MMLQ is
fairly stable with varying $N$. These observations demonstrate the
effectiveness of a limited number of MMIBs and LQs in capturing adequate
visual and textual aesthetic features.

Figure 5: Performance with different numbers of LQs.

###  IV-E Example Results

Several example results of MMLQ (EVA-CLIP) on images from the test set of AVA
are shown in Fig 6. The examples cover a wide range of aesthetic levels (i.e.,
relatively high, moderate, and low ground truth MOSs from top row to bottom
row) and visual content (e.g., landscape, humans, animals, plants, objects,
etc.). The small errors between their predicted and ground truth MOSs
demonstrate the strong capability of MMLQ in IAA for images with various
aesthetic levels and visual content.

TABLE IV: Performance when different portions of user comments are available during inference.  Comment(%) | SRCC$\uparrow$ | PLCC$\uparrow$ | Acc(%)$\uparrow$
---|---|---|---
100 | 0.879 | 0.899 | 88.99
60 | 0.866 | 0.888 | 87.71
20 | 0.818 | 0.846 | 83.86

###  IV-F Limitation

Although multi-modal IAA methods [18, 19] like MMLQ demonstrate extraordinary
performance in IAA, they share a common practical limitation of unpredictable
performance when only a limited portion of user comments are available. In
Table IV, we compare the performance of MMLQ (EVA-CLIP) when different
portions of user comments are available during inference. It can be observed
that the performance of MMLQ does degrade as fewer user comments are
available. It could be due to the mismatch in sentiment between the randomly
selected comments and all comments which is critical in IAA because everyone’s
opinion matters. Nevertheless, with only 20% of the user comments during
inference, the performance of MMLQ still outperforms previous multi-modal IAA
methods in terms of SRCC and PLCC, indicating the effectiveness of MMLQ in
capturing multi-modal aesthetic details for IAA.

Figure 6: Example results on images from the test set of the AVA dataset with
MMLQ. The blue and (green) numbers below each image indicate their
corresponding predicted and (ground truth) aesthetic MOSs, respectively.

##  V Conlusion and Future Work

In the paper, we propose the multi-modal learnable queries (MMLQ) method,
which adopts learnable queries with frozen image and text encoders to
efficiently extract multi-modal aesthetic features from input images and their
associated user comments. Extensive experiments have demonstrated that our
proposed MMLQ method largely outperforms existing state-of-the-art IAA methods
on the benchmark AVA dataset with user comments provided by the AVA-Comments
dataset.

In future research, we plan to tackle the limitations as mentioned in Section
IV-F by incorporating additional submodules based on aesthetic-aware image-to-
text retrieval or image captioning, both of which no longer rely on explicit
corresponding user comments during inference.

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where $c_{2,\delta}:=e^{\frac{\gamma^{2}}{2}(q^{2}-q)M_{g}}\Expe{\para{\int_{0}^{1}e^{ \overline{\omega}^{\delta}(x)}\dx}^{q}}$ where $\omega^{\delta}$ the exact-scaling field of height $\delta$.

Furthermore, for the positive moments in $p\in (0,1)$ of lower truncated GMC $\eta_{n}(0,t):=\int_{0}^{t}e^{U_{n}(s)}\ds$, we have the  bound

\begin{equation}\label{lowertrunp01}
\Expe{\para{\eta_{n}(0,t)}^{p}}\lessapprox~t^{\zeta(p)}, \forall t\geq 0.
\end{equation}

When $p\in(-\infty,0)\cup (1,\frac{2}{\gamma^{2}})$, we have

\begin{equation}
\Expe{\para{\eta_{n}(0,t)}^{p}}\leq \Expe{\para{\eta(0,t)}^{p}} , \forall t\geq 0.
\end{equation}

§.§ Comparison of unit circle and real line

Here we study the moments for $Q_{H}(a)$ because we need them in the ratio computations <ref>. We only study the convex case.

Fix $q\in (-\infty,0]\cup [1,\frac{2}{\gamma^{2}})$. For $t\in [0,1)$ we have the bound

\begin{equation}
\Expe{\para{\eta_{H}(0,t)}^{q}}\leq c t^{\zeta(q)}
\end{equation}

and for $t\in [1,\infty)$ we have the bound

\begin{equation}
\Expe{\para{\eta_{H}(0,t)}^{q}}\lessapprox \branchmat{\ceil{t}^{q}  &q>0\\\floor{t}^{q}& q<0  }.
\end{equation}

As explained in <ref>, we have the covariance: for points $x,\xi\in [0,1)$, with 0 and 1 identified,  and $\varepsilon\leq r$ we find for $y:=|x-\xi|$:

[H_ε(x)H_r(ξ)]:=   2log(2)+log1/2sin(πy)     y> 2/π arctan(π/2)
log(1/)+(1/2)log(π^2ε^2+4)+2/πarctan(π/2)/ε   y≤2/π arctan(π/2)

-log(π)- y ε^-1-log(cos(π/2y))   .

For this covariance it is shown in <cit.>, that it is comparable to the exact-logarithmic field $\omega$ on any compact metric space  i.e.

\begin{equation}\label{eq:covarianceunitcirclerealline}
\sup_{n\geq 1}\sup_{x,\xi\in [0,1),0\sim 1}\abs{\Exp[H_{\e_{n}}(x)H_{\e_{n}}(\xi)]-\Exp[\omega_{\e_{n}}(x)\omega_{\e_{n}}(\xi)]}<K
\end{equation}

for $\e_{n}\to 0$.
Case $t\in [0,1$ and $q\in (-\infty,0]\cup [1,\frac{2}{\gamma^{2}})$
For $q\in (-\infty,0]\cup [1,\frac{2}{\gamma^{2}})$ we have that $F(x)=x^{q}$ is convex, and so we use the <ref> and Kahane inequality, to dominate $H$ by the field $\omega+N(0,K)$ where $N(0,K)$ is an independent Gaussian. So from here we proceed as before to get the bound $t^{\zeta(q)}$.

Case $t\in [1,\infty)$ and $q\in (-\infty,0]\cup [1,\frac{2}{\gamma^{2}})$
Here we have to be careful of the inherent periodicity of $H$. If we have $t=n+r$ for $n\in \mathbb{N},r\in [0,1]$, we write

\begin{equation}
\eta_{H}(0,t)=n\eta_{H}(0,1)+\eta_{H}(0,r).
\end{equation}

We again apply Kahane inequality, to dominate by

\begin{equation}
\end{equation}

So from here we get the bound $n^{q}$ for $q<0$ and the bound $(n+1)^{q}$ for $q>0$.

§ SMALL BALL ESTIMATES FOR $\ETA$

For small ball we can use the following theorem from <cit.> and its application to GMC in <cit.> for the truncated field with $\delta=1$. See also the work in <cit.>, <cit.> for more small deviation estimates. We will use the notion of stochastic domination:

X≽Y X≥x≥Y≥x.

Let $W$ and $Y$ be positive random variables satisfying stochastic dominating relations

Y ≽W_0 Y_0 + W_1 Y_1,

where $(Y_0,Y_1)$ is an independent pair of copies of $Y$, independent of $(W_0,W_1)$, and $W_0 \eqdis W_1 \eqdis W$. Suppose further that there exist $\gamma > 1$ and $x' \in ]0,1[$ such that

(W ≤x) ≤exp( - c (-logx)^γ)   for all  x ≤x'.

Then for any $\alpha \in [1,\gamma[$ there exists a constant $t_\alpha > 0$ such that for all $t \geq t_\alpha$ we have

e^-t Y ≤exp(- c_α(logt)^α)   for all  t ≥t_α.

Therefore, we have the following small deviation estimate for $t\geq t_{\alpha}$ and $u=\frac{M}{t}$ for some $M>0$

u≥Y≤e^M exp(- c_α(logM/u)^α).

Then in <cit.>, it is shown that the above theorem applies to the total mass of the exact-scaling case $Y=\eta_{\omega}^{1}(0,1)$ by checking the two assumptions <Ref>,<Ref> for $\alpha\in [1,2)$. But here we need it for $Y=\eta_{\omega}^{1}(0,t)$ for $t\geq 0$. Combining this with Kahane's inequality we have the following bound for the Laplace transform.

Fix $\alpha\in [1,\beta^{-1})$. We have the following estimate for the Laplace transform of $\eta^{\delta}$ with field $U^{\delta,g}$. Let $r=r(t)>0$ satisfying $rt\geq t_{\alpha}\geq 1$ from above <Ref>. When $t\in  [0,\delta]$ then

and when $t\in [\delta,\infty)$ then

where $\e\in (0,1)$ can be optimized,

σ_t^2:=γ^2(lnδ/t+M_g)   σ^2:=γ^2M_g,

m_2(r,t):=lnrt-γ^2/2(lnδ/t+M_g)>0   m_2(r,t):=lnrt-γ^2/2M_g>0.

Case $t\in [0,\delta]$
We start with $\eta(  t)=  t\int_{0}^{1}e^{ \overline{U}^{\delta,g}(  tx)}\dx$. The field's $U^{\delta,g}(  tx)$ covariance is upper bounded by:

lnδ/tx-y-1+    tx-y/δ+g_δtx-y_x-y≤1,δ/t≤lnδ/t+ ln1/x-y+M_g

for all $\abs{x-y}\leq 1$. When $\delta, t\leq 1$, then we can simply ignore the nonpositive term $\ln\delta\leq 0$. (For the field $\omega^{\delta}$ we have the same covariance bound). Therefore, we apply Kahane's inequality for the fields $U^{\delta,g}(tx)$ and $N(0,\ln\frac{\delta}{t}+M_{g} )+\omega^{1}(x)$ to upper bound by

-rt∫_0^1e^ U^δ(tx)≤    -rte^γN(0,lnδ/t+M_g) ∫_0^1e^ ω^1(x).

Here one can possibly try to use the results on the Laplace transform of lognormal, in particular the asymptotic estimate <cit.> to get better estimates. For now we just use the above estimate <cit.> for $Y:=\int_{0}^{1}e^{ \overline{\omega}^{1}(x)}\dx$ and some $t_{\alpha}\geq 1$ to upper bound by

- c_αlogrte^γN(0,lnδ/t+M_g)  ^αrte^γN(0,lnδ/t+M_g) ≥t_α≥1

+   γN(0,lnδ/t+M_g)≤-lnrt/t_α.

We study the first term. Using layercake and change of variables, the first expectation is

where  $\sigma_{t}^{2}:=\gamma^{2}(\ln\frac{\delta}{t}+M_{g})$, $m_{1}:=\frac{1}{\gamma}\ln\frac{t_{\alpha}}{rt}+\frac{\gamma}{2}(\ln\frac{\delta}{t}+M_{g})<0$, $\tand     m_{2}:=\ln{rt}-\frac{\gamma^{2}}{2}(\ln\frac{\delta}{t}+M_{g})=\abs{m_{1}}+\ln t_{\alpha}>0$.  We split into ranges $[0,\infty)=[0,m_{2}(1-\e)]\cup [m_{2}(1-\e),\infty),$ for some $\e>0$. For the first range we estimate by

-m_2/γσ_t≥N(0,1)∫_0^m_2(1-) e^-c_αx^αx^α-1≤c-1/2m_2/γσ_t^2,

and for the second range we estimate by

∫_m_2(1-)^∞ e^-c_αx^αx^α-1=1/α    ∫_m_2(1-)^α^∞ e^-c_αx⪅ e^-c_αm_2(1-)^α.

For the probability term, we use the usual Gaussian tail estimate

where $m_{3}:=\frac{1}{\gamma \sigma_{t}}\para{\ln\frac{rt}{t_{\alpha}}-\frac{\gamma^{2}}{2}(\ln\frac{\delta}{t}+M_{g})}=\frac{1}{\gamma \sigma_{t}}\para{m_{2}-\ln t_{\alpha}}$. Due to the constraint $rt\geq t_{\alpha}\geq 1$, we see that as $t\to 0$, we have $r\to +\infty$. In this asymptotic, we see that $m_{3}^{2}$ and $\para{\frac{m_{2}}{\sigma_{t}}}^{2}$ have similar growths whereas $m_{2}^{\alpha}$ for $\alpha$ close to 2 has much faster growth due to the absence of $\frac{1}{\sigma_{t}}$. Therefore, for small $t\approx 0$ we can upper bound by the slowest bound, $3\cdot c\expo{-\frac{1}{2}\para{\frac{\e m_{2}}{\gamma \sigma_{t}}}^{2}}$. But for general $t\in [0,\delta]$, we simply take maximum.

Case $t\in [\delta,\infty)$
Here for the field $U^{\delta,g}( tx)$ we upper bound its covariance by:

lnδ/tx-y-1+    tx-y/δ+g_δtx-y_x-y≤1,δ/t≤ln1/x-y+M_g

for all $\abs{x-y}\leq 1$.  (For the field $\omega^{\delta}$ we have the same covariance bound). So we will apply Kahane's inequality for the fields $U^{ \delta}(tx)$ and $N(0,M_{g})+\omega^{1}(x)$ to upper bound by:

-rt∫_0^1e^ U^δ(tx)≤    -rte^γN(0,M_g) ∫_0^1e^ ω^1(x).

This expression is similar to <Ref> with the only difference of replacing $\ln\frac{\delta}{t}+M_{g}$ by $M_{g}$. So we have

- c_αlogrte^γN(0,M_g)  ^αrte^γN(0,M_g) ≥t_α≥1+γN(0,M_g)≤-lnrt/t_α.

Here we are not dividing $\frac{1}{\sigma_{t}}$ and so the slowest bound will be the $\alpha<2$ and so we bound by

where $\wt{m}_{2}:=\ln{rt}-\frac{\gamma^{2}}{2}M_{g}>0$.

We need corollaries here that simplifies that above bounds for the case of exponential-choices for the parameters and $g=0$. First, we study a simple case for $\delta=\rho_{*}$-scale.

We fix $\rho\in (0,1)$. We also fix integer $\ell\geq 1$. We set

r:=ρ_1^-ℓ,t:=ρ_2^ℓ    δ:=ρ,

for $ \rho\geq \rho_{2}>\rho_{1}$ defined as

\begin{equation}
\rho_{1}=\rho^{a_{1} }\tand \rho_{2}=\rho^{a_{2}}.
\end{equation}

The exponents $a_{i}$ need to satisfy $\frac{1}{\ell}<a_{2}<a_{1}$. For small enough $\rho$ we have the bound

c_1:= 1/32(a_1-a_2) /√(βa_2-1/ℓ  )   -√(βa_2-1/ℓ  )   ^2.

The condition $rt\geq t_{\alpha}$ translates to

which is true for small enough $\rho$ since $\ell\geq 1$. Here we have the case $\delta\geq t$ and so we study the exponents $\frac{ m_{2}(r,t)}{\sigma_{t}}$ and $m_{2}(r,t)$. We write explicitly the first exponent

m_2(r,t)/σ_t=lnrt/√(βlnδ/t)   -√(βlnδ/t)   =   √(ℓ) lnρ_2/ρ_1/√(βlnρ^1/ℓ/ρ_2 )   -√(βlnρ^1/ℓ/ρ_2 )

=   √(ℓ) √(ln1/ρ)(a_1-a_2)/√(βa_2-1/ℓ  )   -√(βa_2-1/ℓ  )

≥   √(ln1/ρ)√(ℓ)b_1,1,

b_1,1:=(a_1-a_2) /√(βa_2-1/ℓ  )   -√(βa_2-1/ℓ  ) .

The second exponent is

m_2(r,t)=lnrt-βlnδ/t    =   ℓlnρ_2/ρ_1-βlnρ^1/ℓ/ρ_2

=   ℓln1/ρa_1-a_2-βa_2-1/ℓ

≥   ℓln1/ρ b_1,2,

Therefore, we are comparing

To simplify the bounds we take $\alpha>1$ and small enough $\rho$ so that

1/32b_1,1^2<c_αb_1,20.5^α ln1/ρ^α-1.

That way we only keep the first exponential term in <Ref>.

Next we study the more complicated situation of varying scale $\delta=\rho_{*}^{n}$.

We fix $\rho\in (0,1),\e\geq 0$. We also fix integers $n\geq k\geq 1$ and $z,\ell\geq 0$. We set

r:=ρ_2^-n-zρ_1^-ℓ,t:=ρ_2^k+z+ℓ+ρ_3^n-k    δ:=ρ_2^n,

for $ \rho_{3}>\rho_{2}>\rho_{1}$ defined as

\begin{equation}
\rho_{3}=\rho^{a_{3}},\rho_{2}=\rho^{a_{2}},\rho_{1}=\rho^{a_{1} }.
\end{equation}

The exponents $a_{i}$ need to satisfy

Case $n=k$ and $\ell\geq 1$
For small enough $\rho$ we have the bound

c_1:= 1/32(a_1-a_2(1+) /√(βa_2(1+)+z  )   -√(βa_2(1+)+z  )   ^2.

Case $n>k$ and $\ell\geq 0$ and $z=0$
We have

-rη^δ(t)≤cρ^ℓc_2      ℓ> a_2-a_3/a_2(n-k)

ρ^ℓc_3,1+(n-k) c_3,2      0≤ℓ≤a_2-a_3/a_2(n-k),

c_2:=   1/32a_1-a_2(1+)(1+β)/√(βa_21+  )      ^2

c_3,1:=   (a_1-a_2(1+))ln1/ρ^α-1c_3,2:=(a_2-a_3)ln1/ρ^α-1.

For ease of notation we let

ρ_2,:=ρ_2^1+/ℓ a_2,:=a_2(1+/ℓ),

ρ_2,,z:=ρ_2^1+/ℓ+z/a_2 a_2,:=a_2(1+/ℓ+z/a_2).

Case $n=k$ and $\ell\geq 1$
The condition $rt\geq t_{\alpha}$ translates to

which is true for small enough $\rho$. Here we have the case $\delta\geq t$

and so we study the exponents $\frac{ m_{2}(r,t)}{\sigma_{t}}$ and $m_{2}(r,t)$. We write explicitly the first exponent

m_2(r,t)/σ_t=lnrt/√(βlnδ/t)   -√(βlnδ/t)   =   √(ℓ) lnρ_2,/ρ_1/√(βln1/ρ_2,,z )   -√(βln1/ρ_2,,z )

=   √(ℓ) √(ln1/ρ)(a_1-a_2,)/√(βa_2,,z  )   -√(βa_2,,z  )

≥   √(ln1/ρ)√(ℓ)b_1,1,

b_1,1:=a_1-a_2(1+) /√(βa_2(1++z/a_2)  )   -√(βa_2(1++z/a_2)  ) .

The second exponent is

m_2(r,t)=lnrt-βlnδ/t    =   ℓlnρ_2,/ρ_1-βln1/ρ_2,,z

=   ℓln1/ρa_1-a_2,-βa_2,,z

≥   ℓln1/ρ b_1,2,

Therefore, we are comparing

To simplify the bounds we take $\alpha>1$ and small enough $\rho$ so that

1/32b_1,1^2<c_αb_1,20.5^α ln1/ρ^α-1.

That way we only keep the first exponential term in <Ref> because it has smaller decay.

Case $n>k$ and $\ell\geq 0$ and $z=0$
The condition $rt\geq t_{\alpha}$ translates to

which is true for small enough $\rho$.
Subcase $t< \delta$
This subcase happens when

δ/t> 1ℓ+>  ln1/ρ_2 ^-1lnρ_3/ρ_2  (n-k)=a_2-a_3/a_2(n-k)
and so we have $\ell>n- k\geq 1$. The first exponent now is more complicated because it contains $n-k$ too

m_2(r,t)/σ_t=   lnrt/√(βlnδ/t)   -√(βlnδ/t)

=   √(ℓ) lnρ_2,/ρ_1/√(βlnρ_2/ρ_3^(n-k)/ℓ1/ρ_2, )   -√(βlnρ_2/ρ_3^(n-k)/ℓ1/ρ_2, )

+n-k/√(ℓ)lnρ_3/ρ_2/√(βlnρ_2/ρ_3^(n-k)/ℓ1/ρ_2, )

=   √(ℓ) √(ln1/ρ)(a_1-a_2,)/√(βa_2,-n-k/ℓ(a_2-a_3)  )   -√(βa_2,-n-k/ℓ(a_2-a_3)  )

+n-k/√(ℓ)√(ln1/ρ)(a_2-a_3)/√(βa_2,-n-k/ℓ(a_2-a_3)  )

≥   √(ln1/ρ)√(ℓ)b_2,1+n-k/√(ℓ)b_2,2,

b_2,1:=   a_1-a_2(1+)(1+β)/√(βa_21+  )

b_2,2:=   (a_2-a_3)/√(βa_21+  ) .

The second exponent is

m_2(r,t)=   lnrt-βlnδ/t

=   ℓlnρ_2,/ρ_1-βlnρ_2/ρ_3^(n-k)/ℓ1/ρ_2, +(n-k)lnρ_3/ρ_2

=   ℓln1/ρa_1-a_2,-βa_2,-n-k/ℓ(a_2-a_3)+(n-k)ln1/ρ(a_2-a_3)

≥   ln1/ρℓb_3,1+(n-k)b_3,2,

b_3,1:=   a_1-a_2(1+)(1+β)b_3,2:=a_2-a_3.

As before to simplify the bounds we take $\alpha>1$ and $\rho$ small enough so that we keep the first bound. And we only keep the first part in terms of $\ell$ and let $c_{2}:=b_{2,1}.$

Subcase $t\geq \delta$
In this case the exponent is simply

m_2(r,t)^α=lnrt^α=    ℓ(a_1-a_2,)+(n-k)(a_2-a_3)^αln1/ρ^α.

To assist with having large coefficients we take $\alpha>1$ and lower bound by

(<ref>)≥   ℓc_3,1+(n-k)c_3,2ln1/ρ,

c_3,1:=   (a_1-a_2(1+))ln1/ρ^α-1c_3,2:=(a_2-a_3)ln1/ρ^α-1.

§.§  Rate of convergence to Lebesgue measure

This section is not used in this article but our above Laplace-transform estimate gives a small ball estimate for 1d-GMC which might be of independent interest. In particular, we use ideas from <cit.> to estimate the rate of convergence

where the height of the field $U^{n}$ is $\delta_{n}$, as $\Delta\to 0$ or $n\to+\infty$, and $A=[M_{1},M_{2}]$ (or by translation invariance just $[0,M]$). Then we transfer that to the convergence of the inverse $Q^{n}(x)\to x$.
Fix $A:=\spara{0,M}~\tforsome M>0$ and $\alpha\in [1,\beta^{-1})$. For $m>n$ and $\Delta>0$, we have the following estimates

η_m^n(A)<A-Δ   ≤c_1 e^rA-Δ -A/4δ_nlnr δ_n^α=:B_n,m^1(Δ,A),

η_m^n(A)>A+Δ   ⪅1/Δ^p Aδ_n^p-1=:B_n^2(Δ,A),

where $p\in (0,2]\cap [0,\frac{2}{\gamma^{2}})$, $c_{i}>0,i=1,2,3$ and the  $r=r(\delta_{n})>0$ is required from <Ref> to satisfy

If we allow $\gamma<1$, we have

η_m^n(A)-A>Δ ⪅   1/Δ^2A^2-γ^2δ_n^γ^2    A<δ_n
1/Δ^2δ_n^2+(A-δ_n)δ_n    δ_n≤A .

The case of measure zero sets $\abs{A}=0$ can be ignored because GMC is bi-and so the estimates are true immediately.
Estimating $\Proba{ \eta^{n}(A)<\abs{A}-\Delta}$
Here we will use the Laplace transform function $e^{-rX}$, a convex function of $X$. We use <cit.> which says that $X_{m}:=\eta_{m}$ is a martingale in $m$. So its convex function is a submartingale and hence by Jensen's

In order to utilize the independence of $\eta^{n}(A)$ for distant intervals, we consider the collection

C:= [δ_nk, δ_n(k+1)]: k=0,...,M/δ_n

which will cover the set $A$ and use $C_{odd},C_{even}$ for odd and even $k$  respectively. Therefore, first we apply Chernoff inequality and then let $Z_{I}:=\eta^{n}\para{A\cap I}$ to obtain an upper bound

η^n_m(A)<A-Δ≤    e^rA-Δ-rη^n(A)

=   e^rA-Δ-r∑_I∈C_odd   Z_I-r∑_I∈C_even   Z_I

Cauchy-Schwartz≤    e^rA-Δ√(-2r∑_I∈C_odd   Z_I     )√(-2r∑_I∈C_even   Z_I     )

ind=    e^rA-Δ √(∏_I∈C_odd  -2r Z_I     )√(∏_I∈C_even  -2rZ_I     )

=   e^rA-Δ √(∏_I∈C  -2r Z_I     ).

Here we only keep the intervals $I$ that are fully contained in $I\subset [0,M]$ and there are about

# I⊂A≥1/2A/δ_n.

These $Z_{I}$ are and so using that we have at least $\frac{1}{2}\ceil{\frac{\abs{A}}{\delta_{n}}}$ intervals, we bound the above by

e^rA-Δ -2r 0,δ_nδ_n ^A/4δ_n.

We apply the exponential bound <Ref> for $t=\delta_{n}$ and $r=r(\delta_{n})$ satisfying $r\delta_{n}\geq t_{\alpha}\geq 1$ for $\alpha\in [1,\beta^{-1})$

\begin{align*}
&e^{r\para{\abs{A}-\Delta}} c   \expo{-c_{n}\frac{\abs{A}}{\delta_{n}}\frac{1}{4}\para{m_{2}}^{\alpha}}=e^{r\para{\abs{A}-\Delta}} \expo{-\frac{\abs{A}}{4\delta_{n}}\para{\ln r \delta_{n}}^{\alpha}},
\end{align*}

where in <Ref> we set $M_{g}=0$ and $t=\delta_{n}$ and $\e=\frac{1}{2}$.

Estimating $\Proba{ \eta^{n}(A)>\abs{A}+\Delta}$
We start with a Markov inequality for  $p\in (0,\frac{2}{\gamma^{2}}\wedge 2)$ (but allowing $p=2$ when $\frac{2}{\gamma^{2}}> 2$)

η^n_m(A)>A+Δ≤1/Δ^p η_m^n(A)-A^p.

Here we will use the same decomposition with $Z_{I}:=\abs{\eta_m^{n}(A\cap I)-\abs{A\cap I}}$

η_m^n(A)-A=∑_I∈C_odd   Z_I+∑_I∈C_even   Z_I

to get the upper bound

η_m^n(A)-A^p≤2^p-1∑_I∈C_odd   Z_I^p +2^p-1∑_I∈C_even   Z_I ^p.

Then we use that for independent centered random variables and $p\in[1,2]$, we have the upper bound by Bahr-Esseen   <cit.>

2^p∑_I∈C_odd   Z_I^p +2^p∑_I∈C_even   Z_I^p

and for $p\in (0,1)$ this immediately follows by subaddivity of concave functions. We use the scaling law $\Expe{(\eta^{n}(A\cap I))^{p}}\leq \Expe{(\delta_{n} \eta^{1}(0,1))^{p}}$ and bound by the sum:

When $p>1$, we can again use submartingale  <cit.> to bound by

and when $p\in (0,1)$, we use <Ref> in <Ref>

So all together since there are $\frac{\abs{A}}{\delta_{n}}$ elements in the sum, we have for $p>1$

η_m^n(A)>A+Δ   ≤c_p/Δ^p Aδ_n^p-1η^1(0,1)^p+1

and for $p\in (0,1)$

η_m^n(A)>A+Δ   ≤c_p/Δ^p Aδ_n^p-12.

Case $\gamma<1$
Here we simply take the $L^{2}$-moment

η^n(M)-M^2=   η^n(M)^2-M^2

=   ∬_[0,M]^2e^U̅^n(s)+U̅^n(t)-M^2

≤   ∬_[0,M]^2e^U̅^n(s)+U̅^n(t)s-t≤δ_n

≤   ∬_[0,M]^2δ_n/s-t^γ^2s-t≤δ_n

⪅   M^2-γ^2δ_n^γ^2    M<δ_n
δ_n^2+(M-δ_n)δ_n    δ_n≤M .

In the second inequality, we only kept the logarithmic part of the covariance by bounding the other part $\frac{\abs{s-t}}{\delta_{n}}-1\leq 0$.

The second estimate goes to zero as $n\to +\infty$ for all $\Delta>0$ and $p>1$. In the first estimate, we have to be careful on the choice of $\Delta>0$. To get a decaying bound in the first estimate $B_{n,m}^{1}$, we would need to make a careful choice of $\Delta$.  Because $\frac{1}{\delta_{n}}$ beats the $(\ln\delta_{n})^{2}$ growth, we can make choices of $\Delta$ and $r_{n}$ so that we have the assumption $r_{n}\delta_{n}\geq t_{a}\alpha\geq 1$ and the above bound goes to zero as $n\to +\infty$. For example, we take $r_{n}:=b\frac{t_{\alpha}}{\delta_{n}}$ for some fixed $b>1$, then the bound becomes

-1/δ_n1/4Alnb t_α ^2- b t_α A-Δ   =:-1/δ_n c_Δ   .

For fixed $\abs{A}$, we can make a choice of $\Delta$ close enough to $\abs{A}$ so that

1/4Alnb t_α ^2- b t_α A-Δ>0

and so the overall exponent decays in $n$ for all $\Delta\in [\abs{A}-\lambda,\abs{A})$ for some small enough $\lambda>0$.

By converting to the analogous estimate for $\eta^{n}$, we similarly have the rate of convergence for the inverse.

For the inverse $Q^{n}$ we have the rates

where in this second estimate $x>\Delta$ is forced otherwise the first probability is zero.

The extension of <ref> to intervals

is unclear. This is because the current proof using decoupling doesn't work as evaluating GMCs over two distant subintervals $I,J\subset \spara{Q_{a},Q_{a}+x} $ are still correlated.

§ MOMENTS OF THE MAXIMUM AND MINIMUM OF MODULUS OF GMC

In this section we study tail estimates and small ball estimates of the maximum/minimum of shifted GMC.

We first recall the positive/negative moments of the supremum of GMC from <cit.>.

Moments $p\in [1,\frac{2}{\gamma^{2}})$ using union bound
For $L,\delta,x\geq 0$  and $\delta\leq 1$ we have

\begin{equation}\label{eq:maxmodulusetapone}
\Expe{\para{\supl{T\in[0,L] }\etamu{T,T+x}{\delta}}^{p}}\leq c\ceil{\frac{L}{x}}\Expe{\para{\etamu{0,x}{\delta}}^{p} }.
\end{equation}

Moments $p\in [1,\frac{2}{\gamma^{2}})$ using decoupling
We have the bound

∫_0^xsup_T∈[0,L]e^U(T+s)∩U(T) _+(s)^p  ≤c x^p(1-α)     L/x≥e
c x^p(1-α̃)     L/x≤e,

for $\alpha:=12\sqrt{2}\gamma+\beta (p-1)+\epsilon<1$ and $\tilde{\alpha}:=\beta (p-1)+\epsilon<1$.
Moments $p\in (0,1)$
We have the bound

∫_0^xsup_T∈[0,L]e^U(T+s)∩U(T) _+(s)^p  ≤    c x^p(1-12√(2)γ)ln1/x^p/2,         L/x≥e
c ln1/x^p/2,         L/x≤e,

where $c$ is a constant that diverges as $12\sqrt{2}\gamma\to 1$.

We have for $p>0$

T∈[0,L] T,T+xδ^-p≤   cL/xη^δ0,x/2^-p

≤    cL/δ x^α_2δ(p),

where $\alpha_{2\delta}(p)=\zeta(-p)-1$ if $\frac{x}{2}\leq \delta$ and $\alpha_{2\delta}(p)=-p$ if $\frac{x}{2}\geq \delta$.

§.§ Maximum/Minimum of shifted GMC

Here we study the maximum/minimum of the doubly-shifted GMC. These estimates show up when studying the inverse in the context of the semigroup formula. Recall from <Ref> the semigroup notation

\begin{equation}
\set{Q_{b}\bullet T\geq t}=\set{b\geq \eta(T,T+t)}.
\end{equation}

For the moments $p\in [1,\frac{2}{\gamma^{2}})\tand \delta \leq 1$ we have for $L,x,b>0$

0≤T≤LQ_b∙T+T,Q_b∙T+T+xδ^p⪅1+(L+2x)^1/pb^q/p^p (2+L+x) x^α(p),

where the exponents satisfy the constraints:

* $\alpha(p)=\zeta(p)-1$ when $x\leq 1$ and $\alpha(p)=p$ when $x\geq 1$.

* $q$ is large enough so that  $\lambda q/p>1$ for some fixed $\lambda\in (0,1)$.

As in the proof of <Ref> for the increasing-case, we start with a decomposition $I_{k}:= [a_{k},a_{k+1}]$ ,for the diverging sequence $a_{k}:=k^{\lambda}$ for $\lambda\in (0,1)$ and $k\geq 0$,

\begin{equation}
O_{k,T}:=\set{Q_{b}\bullet T\in I_{k}}=\set{\eta^{\delta}(T,T+a_{k+1})\geq b\geq \eta^{\delta}(T,T+a_{k})}.
\end{equation}

Here we see that $a_{k}=0$ gives
We temporarily take $1/p$ power in order to apply $L_{p}$-triangle inequality. We decompose for each $T$ and upper bound

\begin{equation}
\Expe{\para{\maxl{0\leq T\leq L}\sum_{k\geq 0}\maxl{Y\in I_{k}}\etamu{Y+T,Y+T+x}{\delta}\ind{O_{k,T}}  }^{p}}^{1/p}.
\end{equation}

Then we apply triangle inequality for the $\max$ and the Lp-triangle inequality for $p\geq 1$

\begin{equation}
\sum_{k\geq 0}\para{ \Expe{\para{\maxl{Y\in I_{k}}\maxl{0\leq T\leq L}\etamu{Y+T,Y+T+x}{\delta}\maxl{0\leq T\leq L}\ind{O_{k,T}}}^{p} } }^{1/p}
\end{equation}

and we also we used that for non-negative functions $\maxl{x} f_{1}(x)f_{2}(x)\leq \maxl{x}f_{1}(x)\maxl{x}f_{2}(x)$. Since $a_{k}\to +\infty$, in the event $O_{k,T}$ we keep only the lower bound and apply the FKG inequality as in <Ref> to bound by their product

\begin{equation}
\Expe{\para{\maxl{Y\in I_{k}}\maxl{0\leq T\leq L}\etamu{Y+T,Y+T+x}{\delta} }^{p} }\Expe{\maxl{0\leq T\leq L}\ind{b\geq \eta^{\delta}(T,T+a_{k})}}.
\end{equation}

The first factor was estimated in <Ref> for $p\in [1,\frac{2}{\gamma^{2}})$

\begin{equation}
\Expe{\para{\supl{\tilde{T}\in[0,L+\abs{I_{k}}] }\etamu{\tilde{T},\tilde{T}+x}{\delta}}^{p}}\lessapprox  c(1+L+\abs{I_{k}}+x)x^{\alpha(p)},
\end{equation}

where $\alpha(p)=\zeta(p)-1$ when $x\leq 1$ and $\alpha(p)=p$ when $x\geq 1$.

Next we study the second factor and its sum in $k\geq 0$. The expression is

\begin{equation}
\sum_{k\geq 0}\para{\Expe{\maxl{0\leq T\leq L}\ind{b\geq \eta^{\delta}(T,T+a_{k})}}}^{1/p}=\sum_{k\geq 0}\para{\Proba{b\geq \minl{0\leq T\leq L}\eta^{\delta}(T,T+a_{k})}}^{1/p} .
\end{equation}

This minimum was studied in <Ref>

\begin{equation}
\Expe{\para{\infl{T\in[0,L] }\etamu{T,T+x}{\delta}}^{-q}}\lessapprox~x^{a_{\delta}(-q)}\para{\frac{L}{x}+2},
\end{equation}

where $a(q):=\zeta(-q)-1$ when $x\leq \delta$ and $a(q):=-q$ when $x\geq \delta$.

We set $a_{k}:=k^{\lambda}$ for $\lambda\in (0,1)$ and $k\geq 0$. Then by subadditivity we have $\abs{I_{k}}=(k+1)^{\lambda}-k^{\lambda}\leq 1$.  Since $\delta\leq 1$, for the $k=0$ term we just bound by $1$. Whereas for $k\geq 1$ we have $a_{k}\geq 1$ and so we can use the $a_{\delta}(-q):=-q$ exponent and in turn require

\begin{equation}
\sum_{k\geq 1} \frac{1}{k^{\lambda (q-\e_{q})/p}}<\infty,
\end{equation}

which means $\lambda (q-\e_{q})/p>1$. This is possible by taking $q>0$ large enough and small enough $\e_{q}$.

One could also possibly study the tail of the above maximum using disjoints unions:

=   _k≥0⋃_0≤T≤LQ_b∙T+T,Q_b∙T+T+xδ≥t∩O_k,T

=   ∑_k≥0⋃_0≤T≤LQ_b∙T+T,Q_b∙T+T+xδ≥t∩O_k,T .

Next we study the negative moments of the minimum of shifted GMC.

For $p,b>0$ and $\delta\leq 1$ we have

min_0≤T≤LQ_b∙T+T,Q_b∙T+T+xδ    ^-p    ⪅2+(L+2)^1/pp_2b^q/pp_2^p(4+L+x)x^(α(p))

= c(L,x,b) x^(α(-p)),

where $c(L,x,b)<\infty$ when uniformly bounded $L,x,b\leq B$ and $\alpha(-p)=\zeta(-p)-1$ when $x\leq \delta$ and $\alpha(-p)=-p$ when $x\geq \delta$.

The proof will be the analogue of the infimum case in <Ref>. We start with a decomposition

\begin{equation}
O_{k,T}:=\set{Q_{b}\bullet T\in [a_{k},a_{k+1}]=:I_{k}}=\set{\eta^{\delta}(T,T+a_{k+1})\geq b\geq \eta^{\delta}(T,T+a_{k})},
\end{equation}

for some diverging sequence $a_{k}$ to be chosen latter. We decompose for each $T$ and upper bound

min_0≤T≤LQ_b∙T+T,Q_b∙T+T+xδ    ^-p

≤Y∈I_k0≤T≤L∑_k≥0Y+T,Y+T+xδ^-1 O_k,T  ^p.

We temporarily take power $1/p$ and we apply triangle inequality for the $\max$ and the $L_p$-triangle inequality for $p\geq 1$

Y∈I_k0≤T≤L∑_k≥0Y+T,Y+T+xδ^-1 O_k,T  ^p^1/p

≤∑_k≥0 Y∈I_k0≤T≤LY+T,Y+T+xδ^-1 0≤T≤LO_k,T^p  ^1/p.

We set $a_{k}:=k^{\lambda}$ for $\lambda\in (0,1)$ and $k\geq 0$. Because the two factors are highly correlated we decouple them via
Y∈I_k0≤T≤LY+T,Y+T+xδ^-1 0≤T≤LO_k,T^p ^1/p

≤   Y∈I_k0≤T≤LY+T,Y+T+xδ ^-pp_1 ^1/pp_10≤T≤Lb≥η^δ(T,T+a_k)^1/pp_2,

for $\frac{1}{p_{1}}+\frac{1}{p_{2}}=1$. The first factor was estimated in <Ref>

\begin{equation}
\Expe{\para{\infl{T\in[0,L+\abs{I_{k}}] }\etamu{T,T+x}{\delta}}^{-pp_{1}}}\lessapprox~x^{a_{\delta}(-pp_{1})}\para{\frac{L+\abs{I_{k}}}{x}+2},
\end{equation}

where $a(q):=\zeta(-q)$ when $x\leq \delta$ and $a(q):=-q$ when $x\geq \delta$.

For the second factor, when $k\leq 2^{1/\lambda}$, we bound  by $1$. When $k\geq 2^{1/\lambda}$, we obtain sum as in the proof of the maximum-case in <Ref>. Namely we apply <Ref> for $q>0$

\begin{equation}
\para{\Expe{\maxl{0\leq T\leq L}\ind{b\geq \eta^{\delta}(T,T+a_{k})}}}^{1/pp_{2}}\leq \sum_{k\geq 1} \frac{1}{k^{\lambda (q-1)/pp_{2}}},
\end{equation}

and for finiteness we require $\lambda (q-1)/pp_{2}>1$. This is possible by taking $q>0$ large enough. Finally, we are free to take $p_{1}=1+\e$ for arbitrarily small $\e>0$ since $\eta$ has all its negative moments. That will force $p_{2}=\frac{1+\e}{\e}$ to be arbitrarily large and so again we need to ensure that we take $q$ large enough.

§.§ Version with perturbation

Next we go over the versions we need for <Ref>. We have the maximum version.

Fix $L\geq 0$, $P:=[e^{-r},e^{r}]$ for $r<2$ and $c<1$. For $p>0$ we have

max_T∈[0,L]u∈Pe^-uηQe^u-e^-rc∙T+T,Qe^u-e^-rc∙T+T+t   ^-p ⪅c(L,t) t^α(p),

for the same exponents as in <Ref> and $c(L,t)$ is uniformly bounded for bounded $t,L<B$.

And the following is the minimum version.

Fix $L\geq 0$, $P:=[e^{-r},e^{r}]$ for $r<2$ and $c<1$. For $p>0$ we have

min_T∈[0,L]u∈Pe^-uηQe^u-e^-rc∙T+T,Qe^u-e^-rc∙T+T+t   ^-p ⪅c(L,t) t^(α(-p)-1),

for the same exponents as in <Ref> and $c(L,t)$ is uniformly bounded for bounded $t,L<B$.

Their proofs are modifications of <Ref> and <Ref>, so we only provide the proof for the minimum case.

The proof is the same as of <Ref> with the difference that we have to keep track of $u$. We let $b_{u}:=\para{e^{u}-e^{-r}}c$ and start with a decomposition

\begin{equation}
O_{k,T,u}:=\set{Q_{b_{u}}\bullet T\in [a_{k},a_{k+1}]=:I_{k}}=\set{\eta(T,T+a_{k+1})\geq b_{u}\geq \eta(T,T+a_{k})},
\end{equation}

for some diverging sequence $a_{k}$ to be chosen latter. We decompose for each $T$ and upper bound

min_0≤T≤Lu∈Pe^-uQ_b_u∙T+T,Q_b_u∙T+T+t    ^-p

≤Y∈I_ku∈P0≤T≤L∑_k≥0e^-uY+T,Y+T+t^-1 O_k,T,u  ^p.

We temporarily take power $1/p$ and we apply triangle inequality for the $\max$ and the $L_p$-triangle inequality for $p\geq 1$

Y∈I_ku∈P0≤T≤L∑_k≥0e^-uY+T,Y+T+tδ^-1 O_k,T,u  ^p^1/p

≤e^r ∑_k≥0 Y∈I_k0≤T≤LY+T,Y+T+tδ^-1 u∈P0≤T≤LO_k,T,u^p  ^1/p.

We set $a_{k}:=k^{\lambda}$ for $\lambda\in (0,1)$ and $k\geq 0$. Because the two factors are highly correlated we decouple them via
Y∈I_k0≤T≤LY+T,Y+T+tδ^-1 u∈P0≤T≤LO_k,T,u^p ^1/p

≤   Y∈I_k0≤T≤LY+T,Y+T+tδ ^-pp_1 ^1/pp_1

for $\frac{1}{p_{1}}+\frac{1}{p_{2}}=1$. For the $u$-maximum we just upper bound

\begin{equation}
b_{u}=\para{e^{u}-e^{-r}}c\leq \para{e^{r}-e^{-r}}c \leq e^{2}.
\end{equation}

From here the proof proceeds as in the proof of <Ref>.

§ INVERSE AND SHIFTED GMC MOMENTS

Here we state the moments for the inverse and shifted GMC from <cit.>.

Let $p\in \left(-\dfrac{\left(1+\frac{\gamma^{2}}{2}\right)^{2}}{2\gamma^{2}}, \infty \right)$, the height $\delta\leq 1$ and $a,x>0$.

* (Positive moments) for all $p>0$

(Q^δ(a,a+x))^p≤    c_1x^p_1C_pos(a,δ)+c_2x^p_2,

for $p_{1},p_{2}$ constrained as

\begin{equation}
\zeta(-p_{1})+p>1\tand p_{2}>p.
\end{equation}

* (Negative moments)For all  $p\in \left(0,\dfrac{\left(1+\frac{\gamma^{2}}{2}\right)^{2}}{2\gamma^{2}} \right)$

(Q^δ(a,a+x))^-p≤    c̃_1x^-p̃_1+c̃_2x^-p̃_2,

for $\tilde{p}_{1},\tilde{p}_{2}$ constrained as

\begin{equation}
\zeta(\tilde{p}_{1})>p+1    \tand     p>\tilde{p}_{2}.
\end{equation}

The constants dependence is

C_pos(a,δ)   :=δ^ζ(-p_1)+p-2,C_inf,1(α,δ,ρ),C_inf,2(α,δ,ρ)

C_neg(a,δ)   :=C_sup(a,δ,ρ)

with the constants $C_{inf,1}$ , $C_{inf,2}$,$C_{sup}$ coming from <Ref>.

Let $t>0$.

* For positive moments $p\in [1,\beta^{-1})$ we have

Q^δ_a,Q^δ_a+tδ_0^p    ≤t^α(p)C_sup(a,δ,ρ),

where $\alpha(p):=\zeta(p)-1$ when $t\leq \delta_{0}$ and $\alpha(p):=p$ when $t\geq \delta_{0}$.

* For positive moments $p\in (0,1)$  we have

Q^δ_a,Q^δ_a+tδ_0^p    ≤C_sup(a,δ,ρ)t^p(1-12√(2)γ)ln1/t^p/2     t≤ρδ/e
ln1/t^p/2     t≥ρδ/e.

For simplicity we further bound by $t^{p\alpha}$ for $\alpha=1-12\sqrt{2}\gamma-\epsilon$ for small $\e>0$ to avoid carrying the log-term around.

* For negative moments we have

Q^δ_a,Q^δ_a+tδ_0^-p  ≤   t^α_(p)C_inf(α,δ,δ_0,ρ),

where $\alpha_{\e}(p):=\frac{\zeta(p(1+\e))-1}{1+\e}$ for $t\leq 2\delta_{0}$ and $\alpha_{\e}(p):=p$ for $t\geq 2\delta_{0}$.

§.§ Precomposed term $\eta^{k}(Q^{k+m}(x))$

Here we use the change of variables formula <cit.>: Consider any increasing, possibly infinite, right-continuous function $A:[0,\infty)\to [0,\infty]$ with inverse $C_{s}$, then if $f$ is a nonnegative Borel function on $[0,\infty)$ we have

\begin{equation}
\int_{[0,\infty)}f(u)dA_{u}=    \int_{[0,\infty)}f(C_{s})\ind{C_{s}<\infty}\ds.
\end{equation}

So this change of variables gives

\begin{equation}
\eta^{k}(Q^{k+m}(x))=\int_{0}^{x}e^{U^{k}_{k+m}(Q^{k+m}(s))}\ds
\end{equation}

and so we can now condition by $\mathcal{U}^{k+m}_{\infty}$. Here as $m\to +\infty$, we simply get $\eta^{k}(x)$.

Fix $\gamma<\sqrt{73} - 6 \sqrt{2}$, $x,\delta_{k+m}$, then we have

\begin{equation}\label{Bl2uppertr}
\Expe{\para{\eta^{k}(Q^{k+m}(x))-x}^{2}}\lessapprox \delta_{k}^{p} x^{2-p},
\end{equation}

for $p\in (0,1)$.

We start by scaling the $\delta_{k}$

In what follows we replace $\frac{x}{\delta_{k}}$ by $x$ for ease of notation and change it back for the final result. By first conditioning on $\mathcal{U}^{m}_{\infty}$ we have:

≤∬_[0,x]^2∩Δ_1 δ_1/Q^m(s)-Q^m(t)∨δ_m ^γ^2- AreaΔ_1,

where $\Delta_{1}:=\set{(s,t)\in [0,x]^{2}: \abs{Q^{m}(s)-Q^{m}(t)}\leq \delta_{1} }$.

First we estimate the expected size of the complement of the diagonal set

\begin{equation}\label{eq:areaofdiagonalset}
\Expe{\text{Area}\spara{\Delta_{1}}}=  2\iint\limits_{[0,x]^{2}, s<t } \Proba{Q^{m}(s,t)\leq  \delta_{1}}\ds\dt=2\iint\limits_{[0,x]^{2}, s<t } \Proba{t-s\leq \eta^{m}(Q^{m}(s),Q^{m}(s)+\delta_{1})}\ds\dt
\end{equation}

and then bound by Markov for $p_{1}\in (0,1)$ and apply <ref>:

(<ref>)≤   ∬_[0,x]^2, s<t (t-s)^-p_1η^m(Q^m(s),Q^m(s)+δ_1)^p_1

⪅   δ_1^p_1α∬_[0,x]^2, s<t (t-s)^-p_1≈δ_1^p_1αx^2-p_1.

For the double integral, we first study the case $\abs{Q^{m}(s)-Q^{m}(t)}\leq  \delta_{m}$. Here we get an upper bound for

δ_1/δ_m ^γ^2 AreaΔ_m,

where $\Delta_{m}:=\set{(s,t)\in [0,x]^{2}: \abs{Q^{m}(s)-Q^{m}(t)}\leq \delta_{m} }$. We further scale

(<ref>)≤   δ_m^-γ^2∬_[0,x]^2, s<t (t-s)^-p_1η^m(Q^m(s),Q^m(s)+δ_m)^p_1

=   δ_m^2-γ^2∬_0,x/δ_m^2, s<t (t-s)^-p_1η^1(Q^1(s),Q^1(s)+δ_1)^p_1

⪅   δ_m^2-γ^2 x/δ_m^ 2-p_1≈δ_m^p_1-γ^2 x^2-p_1.

This is non singular in $\delta_{m}$ by taking $p_{1}>\gamma^{2}$. Next we study the case $\abs{Q^{m}(s)-Q^{m}(t)}\geq  \delta_{m}$ to get:

2∬_[0,x]^2∩Δ_1,m δ_1/Q^m(t)-Q^m(s) ^γ^2,

for $\Delta_{1,m}:=\set{(s,t)\in [0,x]^{2}:t>s\tand  \delta_{m}\leq Q^{m}(t)-Q^{m}(s)\leq \delta_{1} }$. Using the layercake, we shift to GMC

A_1,m1/Q^m(t)-Q^m(s)  ^γ^2
=   ∫_c_1^c_mη^mQ^m(s),Q^m(s)+1/r^1/γ^2≥t-s   ,

for $ A_{1,m}:=\set{\delta_{m}\leq Q^{m}(t)-Q^{m}(s)\leq \delta_{1}}$ and $c_{m}:=\para{\frac{1}{\delta_{m}}}^{\gamma^{2}}$. We again apply Markov for $p_{2}\in (0,1)$ and <ref>

(<ref>)⪅    ∬_[0,x]^2, s<t (t-s)^-p_2·∫_c_1^c_m 1/r^p_2α/γ^2 .

Here to avoid getting a divergent factor in $\delta_{m}$, we require $\frac{p_{2}\alpha}{\gamma^{2}}>1$ and so

α>  γ^2γ<√(73) - 6 √(2)≈0.0587224

This is the most singular factor and so we just bound by $x^{2-p_{2}}$.