#  IReNe: Instant Recoloring of Neural Radiance Fields

Alessio Mazzucchelli1,2 Adrian Garcia-Garcia1 Elena Garces3,5 Fernando Rivas-
Manzaneque4,8 Francesc Moreno-Noguer6 Adrian Penate-Sanchez7
1 Arquimea Research Center 2 Universitat Politècnica de Catalunya 3
Universidad Rey Juan Carlos
4 Volinga AI 5 SEDDI 6 Institut de Robòtica i Informàtica Industrial, CSIC-UPC
7Universidad de las Palmas de Gran Canaria, IUSIANI 8Universidad Politécnica
de Madrid

###### Abstract

Advances in NERFs have allowed for 3D scene reconstructions and novel view
synthesis. Yet, efficiently editing these representations while retaining
photorealism is an emerging challenge. Recent methods face three primary
limitations: they’re slow for interactive use, lack precision at object
boundaries, and struggle to ensure multi-view consistency. We introduce IReNe
to address these limitations, enabling swift, near real-time color editing in
NeRF. Leveraging a pre-trained NeRF model and a single training image with
user-applied color edits, IReNe swiftly adjusts network parameters in seconds.
This adjustment allows the model to generate new scene views, accurately
representing the color changes from the training image while also controlling
object boundaries and view-specific effects. Object boundary control is
achieved by integrating a trainable segmentation module into the model. The
process gains efficiency by retraining only the weights of the last network
layer. We observed that neurons in this layer can be classified into those
responsible for view-dependent appearance and those contributing to diffuse
appearance. We introduce an automated classification approach to identify
these neuron types and exclusively fine-tune the weights of the diffuse
neurons. This further accelerates training and ensures consistent color edits
across different views. A thorough validation on a new dataset, with edited
object colors, shows significant quantitative and qualitative advancements
over competitors, accelerating speeds by 5$\times$ to 500$\times$.

Figure 1: IReNe enables instant 360∘ recoloring of pre-trained NeRFs using
only a single image edited by the user (top row). We introduce an optimization
scheme to avoid color bleeding at object boundaries and ensure consistency in
view-dependent effects. Furthermore, as illustrated in the bottom row, various
types of recoloring are possible, including full-object, partial-object, and
multiple-object recoloring.

##  1 Introduction

Neural Radiance Fields (NeRFs) [24] have gained traction due to their ability
to construct realistic 3D environments from 2D images and render high-
fidelity, photorealistic novel viewpoints. Such advancements unlocked many
possibilities, from immersive virtual environments [4, 3, 28] to applications
like augmented reality [27, 11]. Nonetheless, the challenge of seamlessly and
efficiently editing these neural representations while preserving photorealism
remains a critical and largely unexplored frontier. Current NeRF color editing
techniques [18, 20, 14] face several limitations that impede their practical
applications. First, total required time to perform edition is more than 1
minute for the fastest methods. This makes them ill-suited for interactive
applications that demand real-time feedback. Second, current methods often
lack accuracy, especially when managing color edits within object boundaries.
Lastly, existing editing techniques encounter difficulties in sustaining
consistent edits across various viewpoints, especially for scenes intended for
360-degree rendering.

In our paper, we present IReNe, a novel approach that facilitates the Instant
Recoloring of Neural radiance fields, effectively addressing the challenges
highlighted earlier. Leveraging an off-the-shelf, pre-trained NeRF model (in
our case, Instant-NGP [25]) and a single training image featuring user-applied
color adjustments to one or various objects, IReNe swiftly fine-tunes the
network parameters within seconds. This rapid fine-tuning enables the model to
dynamically generate novel scene views, faithfully preserving the color
modifications made in the training image.

Our approach is founded on four pivotal contributions. Firstly, we augment the
pre-trained NeRF model with a lightweight, trainable segmentation module,
providing enhanced control over color edits within object boundaries.
Secondly, speed is achieved by selectively fine-tuning the last layer of the
network, leveraging only a single training image provided by the user.
Thirdly, we show (and take advantage of) that neurons in this last layer show
specialisation. Some neurons are responsible for rendering view-dependent
appearance while others exclusively contribute to diffuse appearance. We
implement an automated classification methodology to distinguish between these
neuron types, enabling us to exclusively fine-tune the neurons associated with
diffuse appearance. This strategy not only expedites the training process but
also guarantees uniform color edits across various viewpoints, enhancing the
model’s consistency and performance.

Up until now, there has been no dataset available for the quantitative
assessment of NeRF color editing methodologies. The fourth contribution of our
work involves the development of such a dataset, where the color of certain
objects within the NeRF Synthetic [24], LLFF [23] and Mip-NeRF 360 [3] scenes
have been meticulously edited manually using Photoshop. Through a
comprehensive quantitative and qualitative evaluation on this dataset, it
becomes evident that IReNe outperforms [18] by a significant margin. Moreover,
it presents considerably better results than [20], addressing issues such as
color bleeding outside object boundaries and challenges in maintaining
viewpoint consistency. Most notably, our architecture’s retraining can be
accomplished in under 5 seconds—a stark contrast to the 1-minute requirement
of [20] and the 30-120 minutes reconstruction time of [18]. This capability
opens the door to seamlessly integrating IReNe into interactive image editing
pipelines that demand immediate response. The Dataset is available in our
project page.

Figure 2: Overview of IReNe. We use a pre-trained NeRF and a user-edited
training image $\textrm{{I}}^{\textrm{edit}}$. Our pre-trained NeRF is an
Instant NGP [25] with: a density $\textrm{MLP}_{\sigma}$, with multiresolution
hash features $f$, and a color $\textrm{MLP}_{c}$. Mapping the user’s edits
into the NeRF involves the following steps: 1) Automatic detection of the
diffuse neurons in the last layer of $\textrm{MLP}_{c}$. 2) Training an
$\textrm{MLP}_{s}$, ruled by the features $f$, to estimate a volumetric soft-
segmentation $\alpha$ of the edited region. 3) Fine-tuning the weights of the
diffuse neurons in the last layer of $\textrm{MLP}_{c}$. 4) Alpha blending
with the mask $\alpha$, to estimate the color of a 3D point x as a linear
combination of the color $c_{\textbf{x}}$ predicted by the frozen weights with
the color $c^{\prime}_{\textbf{x}}$ predicted by the retrained weights. 5)
Volumetric rendering to obtain the edited image
$\textrm{I}^{\textrm{render}}$. $\textrm{MLP}_{s}$ and the trainable last-
layer neurons are optimized through standard RGB loss computation between
$\textrm{I}^{\textrm{render}}$ and $\textrm{{I}}^{\textrm{edit}}$ in under 5
seconds.

##  2 Related Work

Novel view synthesis of 3D scenes is a well-established field. Recently,
Neural Radiance Fields (NeRF) [24] marked a breakthrough in this domain.
However, editing this new neural representation presents challenges because
the 3D information is implicitly encoded in each neuron of a multi-layer
perceptron (MLP). Consequently, several works have aimed to extend the
original NeRF formulation for different kinds of edits, such as scene
relighting [5, 6, 39, 29], scene composition [19, 21, 37], object manipulation
[16], inpainting [35], or text-based editing [33, 15, 32, 34, 36]. In this
work, we focus on the problem of editing the color of the NeRF, addressing the
challenges associated with creating predictable, globally coherent, and fast
edits.

Color Editing. Editing the color of an image has traditionally been done using
two paradigms. Edit-propagation techniques, which involve using local colored
cues, such as user scribbles or points, that are propagated to similar areas
of the image [13, 2, 10, 1]. The challenges lie in finding a similarity metric
that can match the image features of the input exemplar and identify similar
points in both local and non-local areas of the image. Deep learning-based
approaches have been successful in addressing said challenges, leveraging the
effectiveness of convolutional networks as feature extractors [22, 38]. The
other paradigm is to use pre-computed color palettes [9, 30, 12]. These
palettes contain the most relevant colors and serve as a basis for describing
the rest through linear compositing. Editing palettes is a choice among
artists who might combine it with object segmentation to perform global or
local modifications.

NeRF Color Editing. Directly applying color editing methods to a NeRF is not
possible as the color values of the scene are implicitly stored in the
neurons. A NeRF contains view-dependent effects such as specular reflections
that need to be handled appropriately to avoid artifacts. Some of the existing
strategies focus on extending the original NeRF architecture so palette-based
edits can be applied. RecolorNeRF [14] estimates a color palette in pixel
space and, through optimization, decomposes each point into a weighted
combination of color bases shared across the scene. Palette-NeRF [18] improves
it by accounting for specular (or view-dependent) effects. While producing
compelling results, this method is very costly, taking hours to train, and is
limited to global editions. Although, after said long training, multiple
editions can be performed without additional computational cost. Others, such
as ICE-NeRF [20], leverage user scribbles to propagate the edition to the full
volume encoded by NeRF. In particular, ICE-NeRF utilizes user inputs to apply
volume segmentation, which is later used for handling local color edits. As
opposed to PaletteNeRF and RecolorNeRF, which optimize the full NeRF, ICE-NeRF
fine-tunes only a specific set of neurons of a pre-trained color MLP,
achieving faster convergence. However, while fast, its edits suffer from color
bleeding at the borders. Furthermore, as acknowledged in the original work,
the reprojection method utilized to propagate user inputs is not able to
handle complex $360^{\circ}$ scenes. Concurrently to us, ProteusNeRF [32],
proposes a framework for interactive editing of NeRF, by propagating edits
among views based on image features of a pre-trained model [7]. However, their
results heavily rely on the view-consistency of these features, which are
susceptible to failure under view-dependent effects such as specularities.
Also, their edits take between 10 to 70 seconds in contrast to the 5 seconds
of our work.

Inspired by these ideas, our method is also retrained from an existing NeRF.
However, rather than a complete retraining, we selectively reuse information
in the color MLP, akin to ICE-NeRF, expediting convergence. Additionally, we
integrate a learnable segmentation branch capable of segmenting editable
regions across images, even in sequences with $360^{\circ}$ views. As
demonstrated in the results section, this segmentation, coupled with a
selection and retraining of neurons responsible for diffuse rendering,
effectively prevents color bleeding beyond the editable region and guarantees
view-consistent edits.

##  3 Background

Many current NeRF methods use two modules. The first is a set of explicit
trainable features that encode each 3D point x in space in a high-dimensional
feature vector $f_{\textbf{x}}$. The second is a neural network, implemented
as two multi-layer perceptrons, ${\textsc{MLP}}_{\sigma}$ and
${\textsc{MLP}}_{c}$. ${\textsc{MLP}}_{\sigma}$ takes as input the feature
value for each point and outputs a density value, $\sigma$, and a hidden
feature vector, $h$. To estimate the color of the given point, an encoding
$\gamma(\cdot)$ [24, 25] is applied to the view direction, $\theta$, that,
along with $h$, is fed into a second ${\textsc{MLP}}_{c}$ that outputs the
color $c$ of the point x. Each MLP is represented by a set of trainable
parameters ($\phi_{\sigma},\phi_{c}$). Formally,

$\displaystyle{\textsc{MLP}}_{\sigma}(\it{f}_{\textbf{x}},\phi_{\sigma})$
$\displaystyle=\sigma_{\textbf{x}},h_{\textbf{x}}$ (1)
$\displaystyle{\textsc{MLP}}_{c}(h_{\textbf{x}},\gamma(\theta),\phi_{c})$
$\displaystyle=c_{\textbf{x},\theta}$ (2)

Our method builds on a pre-trained NeRF using Instant-NGP [25], which
implements $\gamma$ using spherical harmonics. Another key characteristic of
this method is that the learnable feature representation $\it{f}_{\textbf{x}}$
is arranged into $L$ levels of a multiresolution hash grid where the number of
features per level is determined as a geometric progression. At each level $l$
a hash grid function $\it{f}_{\textbf{x},l}$ produces the value of the
features for the given resolution. The final feature is a concatenation of the
features from each level:

$\it{f}_{\textbf{x}}=\\{\it{f}_{\textbf{x},0}\oplus\it{f}_{\textbf{x},1}\oplus...\oplus\it{f}_{\textbf{x},L}\\}.$
(3)

The feature encodings $\\{\it{f}_{\textbf{x}}\\}$ and the MLP weights for
density and color ($\phi_{\sigma}$ and $\phi_{c}$) are learned at the same
time in an end-to-end approach by minimizing a standard RGB loss between the
volume rendered RGB value for each pixel ${C}_{ij}$ and the original RGB value
from the training image ${C}^{\prime}_{ij}$,

$\mathcal{L}=\sum{\|{C}_{ij}-{C}^{\prime}_{ij}\|}_{2}^{2}$ (4)

Figure 3: Volumetric rendering for the activations of 3 neurons in the last
layer. Points with similar color in 3d space will share a similar activation
pattern.

##  4 Method

Given as input a single re-colored view of the NeRF
$\textrm{{I}}^{\textrm{edit}}\in\mathbb{R}^{W\times H\times 3}$, our goal is
to instantly propagate this edition to the entire NeRF taking into account the
local or global nature of the edit and view-dependent effects. Addressing this
requires overcoming several challenges: First, achieving near real-time NeRF
editing is crucial. We accomplish this by selectively retraining only the last
layer of the color MLP (refer to Sec. 4.1). This approach significantly
reduces the number of parameters requiring modification compared to
alternative methods [20, 14, 18]. Secondly, since the edition is being
performed only on a single view, being able to reproduce coherent view-
dependent effects is inherently complex. To solve it, we propose to use the
view-dependent information already present in the pre-trained NeRF (Sec. 4.2).
Finally, the NeRF edits need to be restricted to the modified regions. We
prevent the edits from propagating to undesired areas by introducing a
lightweight soft-segmentation network trained on the existing features (Sec.
4.3). Fig. 2 presents an overview of the method.

###  4.1 Last Layer Re-training

Given a pre-trained NeRF model, the color ${\textsc{MLP}}_{c}$ can be seen as
a mapping ruled by the output hidden vector $h_{\textbf{x}}$ of the density
${\textsc{MLP}}_{\sigma}$, by the view direction encoding $\gamma(\theta)$,
and by the parameters $\phi_{c}$ of the ${\textsc{MLP}}_{c}$, as seen in Eq.
2. A naïve approach to perform the color edition would be to completely
retrain the color MLP, obtaining a new set of parameters $\phi^{\prime}_{c}$
for the new set of colors $c^{\prime}_{x,\theta}$, as follows,

$\displaystyle{\textsc{MLP}}_{c}(h_{\textbf{x}},\gamma(\theta),\phi^{\prime}_{c})$
$\displaystyle=c^{\prime}_{\textbf{x},\theta}.$ (5)

However, besides being a costly process due to the full retraining, this also
leads to changes of important and useful information already available in the
NeRF. Thus, instead of re-training the full MLP, we propose to retrain only
the _last_ _layer_ of the color ${\textsc{MLP}_{c}}$, formally,

$\displaystyle{\textsc{MLP}}_{last}(\it{h}_{\textbf{x},\theta},\phi^{\prime}_{last})$
$\displaystyle=c^{\prime}_{\textbf{x},\theta},$ (6)

where $\it{h}_{\textbf{x},\theta}$ is the output of the penultimate layer,
that depends on the position x and the view direction $\theta$, and
$\phi^{\prime}_{last}$ is the new set of parameters for the last layer of the
color ${\textsc{MLP}_{c}}$. Although simple, this strategy enables us to
achieve not only rapid edits but also, as demonstrated in Sec. 5.3, to
generate higher-quality edits than full re-training. The updated color
${\textsc{MLP}_{c}}$ is obtained by exclusively retraining its last layer to
match the single image user edit, using Eq. 4. In Fig. 3 we show insights on
why our approach performs so well without retraining the whole model. It can
be seen that the neuron activations of the last layer show strong evidence of
what seems to be color specialisation.

Figure 4: Rendered activations of the same pose while varying the view directional encoding. Top, diffuse neuron. Bottom, view-dependent neuron.  |  | NeRF Synthetic [24] | LLFF [23] | Mip NeRF 360 [3]
---|---|---|---|---
Method |  |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  LPIPS$\downarrow$ |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  LPIPS$\downarrow$ |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  LPIPS$\downarrow$
INGP (ref) [25] |  32.88 |  0.968 |  0.025 |  25.95 |  0.786 |  0.161 |  27.32 |  0.720 |  0.278
PaletteNeRF [18] |  |  29.92 |  0.960 |  0.038 |  24.02 |  0.763 |  0.200 |  22.24 |  0.666 |  0.351
RecolorNeRF [14] |  |  29.00 |  0.943 |  0.048 |  24.07 |  0.714 |  0.340 |  - |  - |  -
IReNe |  |  30.33 |  0.959 |  0.035 |  25.63 |  0.783 |  0.165 |  26.80 |  0.714 |  0.292
Table 1: Quantitative results and comparison with the state-of-the-art.

###  4.2 View-dependent Effects Preservation

The color ${\textsc{MLP}_{c}}$ contains all the information necessary to
render the color of the scene, including view-dependent effects. Thus, by
directly modifying $\phi_{last}$ using a single image to supervise the
training, there is a high risk of modifying this information in an
inconsistent manner, generating artifacts. To solve this, we propose a method
to reuse the existing view-dependent information in the pre-trained NeRF.

Within the provided $\it{h}$, the 64-dimensional vector coming from the output
of the penultimate layer, we identified two distinct behaviors: certain
entries in this vector exhibit significant changes as the viewing direction
changes, while others maintain constancy, encapsulating what we refer as
geometry information (as shown in Fig. 4). We preserve much of the view-
dependent information by freezing part of the weights of this last layer. By
doing so, pre-existing information is retained after the edition process.

To achieve this, we propose a method to automatically identify entries in
$\it{h}_{k}$ exhibiting such behavior. Specifically, for the pixels in the
input edited view $\textrm{{I}}^{\textrm{edit}}$, we compute the neuron
activations of the last layer, yielding $\it{h}_{\theta}\in\mathbb{R}^{W\times
H\times 64}$, for a given view direction $\theta$. We then uniformly sweep
$\theta$ along the azimuth every $\psi$ degrees, obtaining a set of view-
dependent feature vectors
$\\{\it{h}_{\theta_{0}},\it{h}_{\theta_{1}},...,\it{h}_{\theta_{Q}}\\}$ where
the value of $Q$ depends on the angular resolution of the scene as we discuss
in the implementation details. We aggregate the values of this set along the
spatial dimensions obtaining $A\in\mathbb{R}^{64\times Q}$, where each
$A_{k}\in\mathbb{R}^{Q}$ is a profile of the angular behavior of each output
color neuron $k$ across the whole scene (see Fig. 4).

Subsequently, we label the neuron $k$ as a view-dependent neuron if either of
the two conditions is met: 1) if
$\frac{\textrm{std}(A_{k})}{\mu(A_{k})}<\tau_{1}$, where $\textrm{std}(\cdot)$
and $\mu(\cdot)$ are the mean and standard deviation, respectively; or 2) if
$\min(A_{k})\leq\tau_{2}$. During training, we preserve the weights of neurons
identified as view-dependent, thereby retaining their original color. Our
strategy of preserving the color of the view-dependent effects is consistent
with related methods that explicitly separate view-dependent effects from
diffuse reflections using optimization techniques [18].

###  4.3 Soft 3D Segmentation from 2D User Edits

Since the information encoded on the color ${\textsc{MLP}_{c}}$ is global,
performing modifications such as the one depicted in Eq. 6 may lead to
undesired edits in other regions of the scene. To address this issue, we
introduce a 3D segmentation step to selectively apply edits only in the target
regions. Since our goal is to achieve near-real-time interactive editing, we
avoid relying on 2D segmentation models [8] or user scribbles [20] and the
subsequent propagation of segmentation using depth information. Instead, we
propose to reuse the already existing information in the feature grid to
implement a soft 3D segmentation model for our NeRF. We introduce a small
${\textsc{MLP}_{s}}$ that learns a soft-segmentation field
$\alpha_{\textbf{x}}={\textsc{MLP}_{s}}(\it{f}_{\textbf{x}})$ for the edited
area using as input the features $\it{f}$ of the multiresolution hash grid.
The output of this network is a single value that represents the probability
$\alpha$ of a feature at a specific point being part of the edited area.

We use this probability to perform alpha blending between the outputs of the
network using the original set of parameters for the color MLP ($\phi_{c}$)
and the new ones ($\phi^{\prime}_{last}$):

$c_{\textbf{x}}=\alpha_{\textbf{x}}c^{\prime}_{\textbf{x}}+(1-\alpha_{\textbf{x}})c_{\textbf{x}},$
(7)

where $c^{\prime}_{\textbf{x}}$ is the new color and $c_{\textbf{x}}$ is the
original one.

The new ${\textsc{MLP}_{s}}$ introduced to perform the soft segmentation is
trained alongside the last layer of the color ${\textsc{MLP}_{c}}$ using the
loss presented in Eq. 4, without the need for any ground truth segmentation
mask. Convergence is typically achieved in less than 5 seconds.

Figure 5: Qualitative comparison with state of the art methods. For each
approach we show as a small overlayed image the input that the method requires
from the user. For PaletteNeRF we show the original and the edited color
palettes of the image. For IReNe, we show the region the user selects using
Photoshop (or any similar editing tool). On that region we can interactively
perform several color edits by modifying the HSV color within the region.
Figure 6:  Additional qualitative results of IReNe. First column: Original
scene, Other columns: Edited scene. Rows 1-2: Two partial editions to the
flower scene. Rows 3-4: Multicolor edition on the same scene. The third row
has 3 edits: lego body (yellow to blue), lego emergency light (red to green),
oven mitten (red to green). The fourth row has 2 edits. Rows 5-6: Single edit
to recolor the vegetation.

Implementation details. The soft segmentation ${\textsc{MLP}_{s}}$ is
implemented as a two-layer MLP. The input layer takes the 32-dimensional
multiresolution hash grid feature vector $\it{f}$. The hidden layer has 64
neurons, the output is the one-dimensional probability ($\alpha$). Selected
values of $\tau_{1}$ and $\tau_{2}$ are 0.5 and 100 respectively. Q is equal
to 30 for all the scenes. For Blender and MIP360 we sweep the whole $\psi$.
Due to its forward-facing nature, only half of the $\psi$ is considered for
LLFF. We used the Pytorch [26] implementation of Instant-NGP [25], our models
were trained on a single NVIDIA RTX 3090 GPU. Every scene was trained for 200
iterations over the same edited image using the Adam Optimizer [17] with a
constant learning rate set to 0.01.

##  5 Evaluation

In this section we describe how we achieved the first quantitative results of
NeRF recoloring methods by creating ground truth edits that allow meaningful
and detailed comparisons. We also show qualitative results to visually compare
our method with the state of the art.

Dataset. To the best of our knowledge, existing methods lack dedicated metrics
to quantitatively evaluate the quality of the edition with respect to ground
truth edits. We address this issue by creating a new benchmark building on
existing datasets. From NeRF Synthetic [24], we used Lego, Mic, Drums, Ficus,
and Hotdog (some texture files for Ship, Materials, and Chair are missing),
and edited the color map before re-rendering the scene. We then used 1 view
for training and 200 for evaluation. From LLFF [23] and MIP360 [3], we used
all scenes, for which we apply manually the same edits using Photoshop to 11
views (1 for training, 10 for evaluation).

###  5.1 Quantitative Evaluation

We quantitatively compare our method with PaletteNeRF [18] and RecolorNeRF
[14]. Unfortunately, ICE-NeRF’s code is not publicly available, so we just
provide qualitative comparisons in the following subsection. Both PaletteNeRF
and RecolorNeRF are methods that require a color palette, while our dataset
contains full re-colored images as input. Thus, for a fair comparison with
these methods, we re-trained the models using our edited image in the RGB loss
and froze all the parameters except the palette. In this way, we simulate the
process of finding the most appropriate palette in an automated manner
(applying stochastic gradient descent). As a result, we obtained the best
palette for our edits and were able to emulate our full-image edits.

Quantitative evaluations are reported in Table 1. As both IReNe and
PaletteNeRF are trained from Instant-NGP [31], we include metrics from this
model on the ten evaluation images (without color edition) as a reference for
the maximum achievable metric. Notably, IReNe outperforms state-of-the-art
methods across the three metrics—PSNR, SSIM, and LPIPS. Furthermore, on real
datasets (LLFF and Mip NeRF 360), IReNe achieves metrics comparable to
Instant-NGP, suggesting proximity to the performance limit imposed by the pre-
trained model. Results for RecolorNeRF on the Mip NeRF dataset are not
reported due to this method not being able to converge on those scenes. In
terms of computation time, our model achieves remarkable speed-ups compared to
other approaches. Specifically, the average training time for IReNe is 5
seconds, while RecolorNeRF requires 15-20 minutes, and PaletteNeRF takes
between 30 minutes and 2 hours. ICE-NeRF reports training times of 25 seconds.
In summary, IReNe achieves between 5$\times$ and 500$\times$ speed-ups
compared to existing approaches.

Figure 7: Qualitative ablation study. Rendered images from the results detailed in Table 2. |  |  | NeRF Synthetic [24] | LLFF [23] | Mip NeRF 360 [3]
---|---|---|---|---|---
Method |  |  |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  LPIPS$\downarrow$ |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  LPIPS$\downarrow$ |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  LPIPS$\downarrow$
Full MLP |  |  28.25 |  0.950 |  0.047 |  24.86 |  0.777 |  0.172 |  24.31 |  0.685 |  0.325
Last layer |  |  29.09 |  0.951 |  0.044 |  24.99 |  0.779 |  0.176 |  25.35 |  0.702 |  0.310
Soft Segmentation |  |  30.20 |  0.956 |  0.037 |  24.77 |  0.784 |  0.163 |  25.68 |  0.707 |  0.295
IReNe |  |  30.33 |  0.959 |  0.035 |  25.63 |  0.783 |  0.165 |  26.80 |  0.714 |  0.292
Table 2:  Ablation study. In this ablation comparison we show the results of
learning the full color MLP instead of just the last layer (Full MLP), the
result of only training the last layer but not applying the soft segmentation
MLP or the neuron selection (Last layer), the result of using the last layer
and the soft segmentation (Soft Segmentation), and the full method IReNe.

###  5.2 Qualitative Evaluation

We offer qualitative comparisons of our method with PaletteNeRF and ICE-NeRF.
For PaletteNeRF, we manually selected the palette to achieve the most visually
appealing image. Unfortunately, no code was provided for ICE-NeRF, so we had
to use the result images from their paper and supplemental material. However,
due to the low resolution of ICE-NeRF renders, we chose not to provide zoomed
regions in the images to ensure a fair comparison.

Fig. 5 shows the comparison between different methods. Additionally, we
present the original image and the supplementary information generated by each
method (user scribbles for ICE-NeRF, color palette for PaletteNeRF, and the
region selection that the user creates in Photoshop for IReNe before applying
the HSV color change in said region). The user edit that IReNe uses is a
region selection that the user does on one image, on which we then perform the
desired color change in HSV space. For the first 2 results the camera pose of
the render was not available for ICE-NeRF so we had to find by hand a similar
pose and thus the slight misalignment between the pose used by PaletteNeRf and
IReNe and the pose in ICE-NeRFs results.

When analyzing the results in Fig. 5 it can be seen that ICE-NeRF performs
poorly in the real scenes from the Mip NeRF 360 dataset, where the color
bleeds out into the rest of the scene. Their results are better in Synthetic
NeRF dataset but it can be seen that ICE-NeRF destroys part of the contained
information when optimizing color. As a way to check if that degradation
happens due to the method, the green chair images for the ground truth are
obtained from the same ICE-NeRF source and serve as a quality baseline. In the
LLFF dataset ICE-NeRF results are more robust. As PaletteNeRF is only able to
support global recoloring, the edition is applied outside of the interest
region (_e.g._ Horns scene). It is also spilling some of the color edition
into the view-dependent effects of other regions (_e.g._ the pink color for
the Lego is propagated to the yellow table reflections). We show additional
results of our method in Fig. 6, with multicolor edits and other complex color
editions.

###  5.3 Ablation Study

Tab. 2 provides quantitative results on the quality of different variants of
our method. Training uniquely the last layer of ${\textsc{MLP}_{c}}$ results
in better quality metrics than training the full MLP. Also, adding the soft
segmentation further increases the quality of IReNe, while being able to
retain existing view-dependent information in the pre-trained model leads to
the highest quality. The effect of each of the contributions can be
qualitative observed in Fig. 7.

##  6 Conclusions

We introduced IReNe, a new method that enables interactive NeRF editing with a
single user edit. The proposed method not only outperforms current state-of-
the-art in terms of speed and interactivity but also overcomes the main
limitations of the existing methods, namely multi-view consistency and
precision at object boundaries. Besides the high-quality results obtained by
IReNe, we also provide insights on how color is encoded in NeRFs. We hope that
this work may serve as inspiration to further lines of research, as well as
enable use cases and edition workflows that were possible using traditional
2D/3D tools and, now, can be achieved in NeRFs thanks to IReNe.

Limitations and Future Work. One of the main limitations of our work is the
need to rely on external edition tools, such as Photoshop, to achieve the
complete edition. Also, while the last layer retraining and the method for
discriminating which weights to freeze are very robust to the chosen image,
the soft segmentation model results may be poor on some occasions. Besides
trying to solve these limitations, future work should include the capacity to
not only recolor regions but also being able to affect the indirect
illumination produced by edited objects onto other objects.

##  7 Ackowledgements

This work is supported by Arquimea Research Center and Horizon Europe, Teaming
for Excellence, under grant agreement No 101059999, project QCircle, and by
project MoHuCo (PID2020-120049RB-I00). Elena Garces was partially supported by
a Juan de la Cierva - Incorporacion Fellowship (IJC2020-044192-I).

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#  On the large interaction asymptotics of the free energy density of the
Ising chain with disordered centered external field

Orphée Collin

###### Abstract

This article completes [6] by identifying explicitly the leading coefficient
in the asymptotic development of the free energy density of the centered
Random Field Ising Chain as the spin-spin interaction of the chain goes to
infinity, under general assumptions on the disorder law.

_AMS subject classification (2010 MSC)_ : 60K37, 82B44, 60K35,

_Keywords_ : random field Ising chain, disordered systems, random matrix
products

##  1 Introduction

We denote by $\mathbb{N}:=\\{1,2,\dots,\\}$ the set of all positive integers.
Let $\mu$ be a law on $\mathbb{R}$. We consider, under a probability measure
denoted $\mathbb{P}_{\mu}$, an i.i.d. sequence $h=(h_{n})_{n\in\mathbb{N}}$ of
real random variables following law $\mu$.

For every realization of the sequence $h=(h_{n})_{n\in\mathbb{N}}$, for every
$N\in\mathbb{N}$, every $(a,b)\in\\{-1,+1\\}^{2}$ and every
$J\in\mathbb{R}^{+}$, we define the partition function of the Ising chain on
$\\{1,\dots,N\\}$ with external field $h$, boundary conditions $(a,b)$ and
spin-spin interaction $J$ as:

$Z_{N,h}^{a,b}(J):=\sum_{\sigma\in\\{-1,1\\}^{\\{0,1,\dots,N\\}}:\sigma_{0}=a,\sigma_{N}=b}\exp\Big{(}-2J\sum_{n=1}^{N}{\mathbf{1}}_{\\{\sigma_{n}\neq\sigma_{n-1}\\}}+\sum_{n=1}^{N}\sigma_{n}h_{n}\Big{)}\,.$
(1)

We say that _configuration_ $\sigma$ exhibits a _spin flip_ at position $n$ if
$\sigma_{n}\neq\sigma_{n-1}$.

Using Kingman’s sub-additive ergodic theorem, it is straightforward to show
the existence, for every $J\in\mathbb{R}$, of a real number $F_{\mu}(J)$ such
that for almost every realization of $h$ (i.e. $\mathbb{P}_{\mu}$-almost
surely), for every $(a,b)\in\\{-1,+1\\}^{2}$, we have

$\frac{1}{N}\log
Z_{N,h}^{a,b}(J)\underset{N\to\infty}{\longrightarrow}F_{\mu}(J)\,.$ (2)

The number $F_{\mu}(J)$ is called the free energy density of the Random Field
Ising Chain (RFIC) with disorder law $\mu$ and interaction $J$. We are
interested in the asymptotics of $F_{\mu}(J)$, as $J\to\infty$.

From now on, we assume that $\mu$ has a finite second moment, is centered and
is different from $\delta_{0}$. We denote its variance by
$\vartheta^{2}:=\mathbb{E}_{\mu}[h_{1}^{2}]$, with $\vartheta\in(0,\infty)$.
We will consider in particular the case of the centered Gaussian law with
variance $\vartheta^{2}$, which we denote by ${\mathcal{N}}_{\vartheta^{2}}$.
The aim of this article is to show that:

$F_{{\mu}}(J)\sim\frac{\vartheta^{2}}{2J}\,,\quad\text{ as }J\to\infty.$ (3)

To prove (3) we will proceed in two steps. To begin with, we will cover the
special case of disorder law ${\mathcal{N}}_{\vartheta^{2}}$ through the use
of a continuum analogue of the RFIC, introduced as a weak disorder limit in
[8], and for which the free energy density is known exactly. Then we will show
(3) under general integrability assumptions on $\mu$ using a coarse-graining
argument, together with the central limit theorem. More precisely, we are
going to show the following three theorems.

######  Theorem 1 (Gaussian disorder).

As $J\to\infty$,

$F_{{\mathcal{N}}_{\vartheta^{2}}}(J)=\frac{\vartheta^{2}}{2J}+O\left(\frac{(\log
J)^{\frac{1}{2}}}{J^{2}}\right)\,.$ (4)

######  Theorem 2 (Disorder with finite exponential moments).

If $\mu$ admits finite exponential moments, i.e.
$\mathbb{E}_{\mu}[\exp(th_{1})]<\infty$ for all $t$ in a neighbourhood of 0,
then, as $J\to\infty$,

$F_{\mu}(J)=\frac{\vartheta^{2}}{2J}+O\left(\frac{(\log
J)^{\frac{1}{3}}}{J^{\frac{4}{3}}}\right)\,.$ (5)

######  Theorem 3 (Disorder with finite polynomial moments).

Let $p\in[2,\infty)$. If $\mu$ admits a finite $p$-th moment, i.e.
$\mathbb{E}_{\mu}[|h_{1}|^{p}]<\infty$, then, for $p\in[3,\infty)$,

$\frac{\vartheta^{2}}{2J}+O\left(\frac{(\log
J)^{\frac{1}{3}}}{J^{\frac{4}{3}}}\right)\leq
F_{{\mu}}(J)\leq\frac{\vartheta^{2}}{2J}+O\left(\frac{1}{J^{\frac{4p}{3p+2}}}\right)\,,\quad\text{as
}J\to\infty\,,$ (6)

while for $p\in[2,3)$,

$\frac{\vartheta^{2}}{2J}+O\left(\frac{(\log
J)^{\frac{p-1}{2p}}}{J^{\frac{2(p-1)}{p}}}\right)\leq
F_{{\mu}}(J)\leq\frac{\vartheta^{2}}{2J}+O\left(\frac{1}{J^{\frac{2p(p-1)}{p^{2}+p-1}}}\right)\,,\quad\text{as
}J\to\infty\,.$ (7)

In particular, if $p>\frac{3+\sqrt{5}}{2}$, then (3) holds.

Often, in statistical physics, properties concerning the typical
configurations of the system can be extracted from the free energy density. We
are now going to discuss some general properties of the free energy density of
the RFIC, and use our results to derive bounds on the mean density of spin
changes in the RFIC in the thermodynamic limit.

Let us denote by $P^{a,b}_{N,h,J}$ the RFIC measure, i.e., the Gibbs measure
on $\\{-1,+1\\}^{N}$ associated to the partition function $Z^{a,b}_{N,h}(J)$.
A classical and simple computation yields expressions for the expectation and
the variance of the density of spin changes under the RFIC measure:

$E^{a,b}_{N,h,J}\left[\frac{1}{N}\sum_{n=1}^{N}{\mathbf{1}}_{\\{\sigma_{n}\neq\sigma_{n-1}\\}}\right]=-\frac{1}{2}\partial_{J}\left(\frac{1}{N}\log
Z^{a,b}_{N,h}(J)\right)\,.$ (8)

and

$\mathrm{Var}^{a,b}_{N,h,J}\left[\frac{1}{N}\sum_{n=1}^{N}{\mathbf{1}}_{\\{\sigma_{n}\neq\sigma_{n-1}\\}}\right]=\frac{1}{4N}\partial_{J}^{2}\left(\frac{1}{N}\log
Z^{a,b}_{N,h}(J)\right)\,.$ (9)

Since the left-hand sides are non-negative, we see that
$J\mapsto\frac{1}{N}\log Z^{a,b}_{N,h}(J)$ is non-increasing and convex on
$\mathbb{R}$. Taking the $\mathbb{P}_{\mu}$-almost sure limit, as
$N\to\infty$, we obtain also that $F_{\mu}$ is non-increasing and convex on
$\mathbb{R}$. We mention also that $F_{\mu}$ is non-negative: this can be seen
by lower bounding the partition function by the contribution given by
$\sigma=(a,b,b,\dots,b,b)$ and using the law of large numbers.

Furthermore, the two-by-two matrix
$(Z_{N,h}^{a,b}(J))_{(a,b)\in\\{-1,+1\\}^{2}}$ can be expressed (see [1]) as a
product of i.i.d. random matrices, distributed under a law determined by
$\mu$. Consequently, Furstenberg’s theory can be summoned: the free energy
density $F_{\mu}(J)$ is the top Lyapunov exponent of this random matrix
product and has an integral expression involving the invariant measure of a
certain Markov chain on $\mathbb{R}^{+}$. Then, using results by D. Ruelle
(see [11], or also [4]), it is standard to show that $F_{\mu}$ is analytic on
$(0,\infty)$.

As a consequence of convexity and twice differentiability, we can take the
limit $N\to\infty$ in (8) and (9). We see that the variance goes to 0 as
$N\to\infty$, so we obtain, for all $J>0$ and all $(a,b)\in\\{-1,+1\\}^{2}$,
$\mathbb{P}_{\mu}$-almost surely, the convergence

$\frac{1}{N}\sum_{n=1}^{N}{\mathbf{1}}_{\\{\sigma_{n}\neq\sigma_{n-1}\\}}\underset{N\to\infty}{\longrightarrow}-\frac{1}{2}F^{\prime}_{{\mu}}(J)\,$
(10)

in $P^{a,b}_{N,h,J}$-probability.

From our main theorems, we derive the following asymptotic results on the
limiting quantity appearing in (10).

######  Corollary 4.

As $J\to\infty$,

$-\frac{1}{2}F^{\prime}_{\mu}(J)=\frac{\vartheta^{2}}{4J^{2}}+O(\varepsilon_{J})\,.$
(11)

where $\varepsilon_{J}$ is equal to $\frac{(\log
J)^{\frac{1}{4}}}{J^{\frac{5}{2}}}$ in the case of Gaussian disorder, to
$\frac{(\log J)^{\frac{1}{6}}}{J^{\frac{13}{6}}}$ in the case of exponential
integrability, and, in the case of polynomial integrability, to
$\frac{1}{J^{\frac{3+\eta_{p}}{2}}}$ for $p\in(\frac{3+\sqrt{5}}{2},\infty)$
and to $\frac{1}{J^{\eta_{p}+1}}$ for $p\in[2,\frac{3+\sqrt{5}}{2}]$, where
$\eta_{p}$ denotes the exponent in the upper bound in Theorem 3.

###### Proof.

By convexity, for every $J>0$ and every $a\in(0,J]$, we have

$\frac{F_{{\mu}}(J)-F_{{\mu}}(J-a)}{a}\leq
F^{\prime}_{{\mu}}(J)\leq\frac{F_{{\mu}}(J+a)-F_{{\mu}}(J)}{a}\,.$ (12)

In the Gaussian case, apply this with $a=J^{\frac{1}{2}}(\log
J)^{\frac{1}{4}}$ to derive (11) from Theorem 1. In the case of exponential
integrability, apply it instead with $a=J^{\frac{5}{6}}(\log J)^{\frac{1}{6}}$
to derive (11) from Theorem 2. Finally, in the case of polynomial
integrability, when $\eta_{p}>1$, take $a=J^{\frac{3-\eta_{p}}{2}}$ to derive
(11) from Theorem 3, and when $\eta_{p}\leq 1$, recall that $F_{\mu}^{\prime}$
is non-positive, and use the lower bound in (12) with $a=J/2$ and the non-
negativity of $F_{\mu}$. ∎

Let us now discuss the links with the existing literature. In [9], the free
energy density $F_{\mu}(J)$ is computed exactly for two specific choices of
the law ${\mu}$. In [6], under the assumptions that ${\mu}$ admits finite
exponential moments, that ${\mu}$ has a density with respect to the Lebesgue
measure and that this density is uniformly Hölder continuous, a much more
detailed asymptotic development for $F_{{\mu}}(J)$ is obtained, precisely:

$F_{{\mu}}(J)=\frac{\kappa_{1}}{2J+\kappa_{2}}+O\left(\exp(-cJ)\right),\quad\text{
as }J\to\infty,$ (13)

with constants $\kappa_{1}\in(0,\infty)$, $\kappa_{2}\in\mathbb{R}$ and
$c\in(0,\infty)$ depending on $\mathcal{L}$. A formula appears in [6] for the
value of $\kappa_{1}$, but it is not very explicit: it is in terms of an
integral involving the invariant measure of a certain Markov chain on
$(0,\infty)$, see equation (3.12) in [6]. In this perspective, the
contribution of the present work is to show that $\kappa_{1}$ is simply the
variance of ${\mu}$, but it is also much more general than [6] concerning the
assumptions required on ${\mu}$. We also mention that in [9] the leading
coefficient is not recognized as being the variance of ${\mu}$.

A description of the typical configurations of the RFIC was proposed in the
physical literature, by D. Fisher, P. Le Doussal and C. Montus, in [5]. The
typical configurations are conjectured to be close to a given configuration
which we will call _the Fisher configuration_ , obtained through a
renormalization procedure. This configuration is determined by the disorder
sequence $h$ and has a density of spin changes which is equivalent to
$\frac{\vartheta^{2}}{4J^{2}}$ as $J\to\infty$. For more details on this, see
[3], where an estimation of the discrepancy density between typical
configurations and the Fisher configuration is obtained. Our present results
concerning the free energy density and the mean density of spin changes are in
accordance with the predictions of Fisher, and are beyond the reach of [3]. In
the present article, we do not make use of Fisher’s approach, except for
heuristic discussions. Let us however mention that it is possible to use the
Fisher configuration to derive the lower bound
$F_{\mu}(J)\geq\frac{\vartheta^{2}+o(1)}{2J}$, as $J\to\infty$: simply by
lower bounding the partition function by the contribution given only by the
Fisher configuration, whose spin domains are well-understood, at least to
first order, see Theorem 3 in [3].

##  2 Comparison of the discrete partition function with Gaussian disorder
with the continuum partition function

In this section, we are going to prove Theorem 1. In order to do so, we turn
to the continuous analogue of the model. The continuum RFIC was introduced in
[8] at the level of the free energy as a weak disorder limit of the discrete
RFIC. The partition function and the continuum model themselves have been
introduced in [2]. We now present the definitions of the partition function
and of the free energy density.

Consider, under a probability $\mathbb{P}$, a standard Brownian motion
$B=(B_{t})_{t\in[0,\infty)}$. For every $J\in\mathbb{R}$ and
$a\in\\{-1,+1\\}$, denote by $\mathbf{P}_{J}^{a}$ the law of a Poisson process
$(s_{t})_{t\geq 0}$ taking values in $\\{-1,1\\}$, with $s_{0}=a$ and jump
rate equal to $\exp(-2J)$, and denote by $\mathbf{E}_{J}^{a}$ the associated
expectation. Then, for every realization of $B$, every $\mathcalb{l}\geq 0$,
every $(a,b)\in\\{-1,+1\\}^{2}$ and every $J\geq 0$, we define the partition
function of the continumm Ising chain on $[0,\mathcalb{l}]$ with external
field $B$, boundary conditions $(a,b)$ and interaction $J$ as:

${\mathcal{Z}}_{\mathcalb{l},B}^{a,b}(J):=\exp(\mathcalb{l}e^{-2J})\;\mathbf{E}_{J}^{a}\left[\exp\left(\int_{0}{l}s_{t}\mathrm{d}B_{t}\right){\mathbf{1}}_{\\{s{l}=b\\}}\right]\,.$
(14)

For almost every realization of $B$, i.e. $\mathbb{P}$-almost surely, for
every $(a,b)\in\\{-1,+1\\}^{2}$, we have (see [1] together with Lemma 5.1 of
[2]):

$\frac{1}{\mathcalb{l}}\log{\mathcal{Z}}_{\mathcalb{l},J,B}^{a,b}(J)\underset{\mathcalb{l}\to\infty}{\longrightarrow}{\mathcal{F}}(J)\,,$
(15)

where ${\mathcal{F}}(J)$ is called the free energy density of the continuum
RFIC with interaction $J$, and:

${\mathcal{F}}(J):=\frac{e^{-2J}K_{-1}(e^{-2J})}{K_{0}(e^{-2J})}\sim\frac{1}{2J+\log
2+\gamma_{EM}}+O(e^{-4J}),\quad\text{ as }J\to\infty.$ (16)

In this expression, $K_{-1}$ and $K_{0}$ are the modified Bessel functions of
second kind of index respectively -1 and 0, and $\gamma_{EM}$ is the Euler-
Mascheroni constant. Only the developement
${\mathcal{F}}(J)=\frac{1}{2J}+O(\frac{1}{J^{2}})$ is relevant for our
purpose.

For the sake of simplicity, we choose to prove Theorem 1 when $\vartheta=1$.
Remark 8 will be devoted to generalizing the proof to any
$\vartheta\in(0,\infty)$.

We are going to compare the partition function of the discrete Ising chain
with disorder law ${\mathcal{N}}_{1}$ with the continuum partition function we
just introduced. This comparison is based on the following heuristic.
Splitting $\mathbb{R}^{+}$ into the segments $[n-1,n],n\in\mathbb{N}$, we
observe that:

  * •

inside each segment, the Poisson parameter of the number of jumps is
$e^{-2J}$,

  * •

the disorder energies over the segments, i.e., the variables $B_{n}-B_{n-1}$
are i.i.d. and distributed under ${\mathcal{N}}_{1}$.

Thus, it seems reasonable to modify the configurations by artificially moving
the positions of the spin changes to their integer part (this means slightly
modifiying the energy $\int_{0}{l}s_{t}\mathrm{d}B_{t}$ collected): then, we
see the discrete partition function associated to the sequence
$(B_{n}-B_{n-1})_{n\in\mathbb{N}}$ appear. Provided the error concerning the
disorder energy collected is not too big, this should yield a good estimate.
This approach is backed by the following considerations. In [2] the typical
configurations of the continuum RFIC measure (the Gibbs meaure associated to
the continuum partition function introduced in (14)) have been studied, based
on the description provided by [5], and the results suggest that their spin
domains have length of order $J^{2}$. Therefore, for any fixed segment of
length 1, we heuristically assume the probability, under the continuum RFIC
measure, of having a spin change in the segment to be small - roughly,
dominated by $O(J^{-2})$ \- hence the error term in the energy should remain
of the same order.

First, we need to slightly generalize the definition of this continuum
partition function. For every real numbers
$0\leq\mathcalb{l}\leq\mathcalc{r}$, we set:

$\displaystyle{\mathcal{Z}}_{\mathcalb{l},\mathcalc{r},B}^{a,b}(J)$
$\displaystyle:=\exp(\mathcalb{l}e^{-2J})\mathbf{E}_{J}^{a}\left[\exp\left(\int{l}{r}s_{t}\mathrm{d}B_{t}\right){\mathbf{1}}_{\\{s{r}=b\\}}\Big{|}s{l}=a\right]\,.$
(17)

Observe that
${\mathcal{Z}}_{\mathcalb{l},B}^{a,b}(J)={\mathcal{Z}}_{0,\mathcalb{l},B}^{a,b}(J)$
and that
${\mathcal{Z}}_{\mathcalb{l},\mathcalc{r},B}^{a,b}(J)={\mathcal{Z}}_{\mathcalc{r}-\mathcalb{l},\Theta{l}B}^{a,b}(J)$
if we set
$\Theta{l}B:=(B_{\mathcalb{l}+t}-B_{\mathcalb{l}})_{t\in[0,\infty)}$.

Following our heuristic, we rewrite the continuum partition function defined
in (14) in the following way. Take $N$ a positive integer and to any (cadlag)
trajectory $(s_{t})_{0\leq t\leq N}\in\\{-1,1\\}^{[0,N]}$, associate the
configuration $\sigma\in\\{-1,1\\}^{\\{0,1,\dots,N\\}}$ defined by:

$\sigma_{k}=s_{k},\qquad k=0,\dots,N.$

To each $\sigma$ corresponds a set of trajectories $(s_{t})_{0\leq t\leq N}$,
we rewrite the expectation in (14) according to this partitioning to obtain:

${\mathcal{Z}}_{N,B}^{a,b}(J)=\sum_{\sigma\in\\{-1,1\\}^{\\{0,1,\dots,N\\}}:\sigma_{0}=a,\sigma_{N}=b}\prod_{n=1}^{N}{\mathcal{Z}}_{n-1,n,B}^{\sigma_{n-1},\sigma_{n}}(J)\,.$
(18)

We define the sequence $h=(h_{n})_{n\in\mathbb{N}}$ by setting
$h_{n}=B_{n}-B_{n-1}$. As noted before, $h$ is i.i.d with law
${\mathcal{N}}_{1}$. Our aim is to compare the above sum with:

$Z_{N,h}^{a,b}(J)=\sum_{\sigma\in\\{-1,1\\}^{\\{0,1,\dots,N\\}}:\sigma_{0}=a,\sigma_{N}=b}\prod_{n=1}^{N}\exp(-2J{\mathbf{1}}_{\\{\sigma_{n-1}\neq\sigma_{n}\\}}+\sigma_{n}h_{n})\,.$
(19)

so naturally we are going to compare the continuum partition function on the
_block_ $[n-1,n]$, i.e ${\mathcal{Z}}_{n-1,n,B}^{\sigma_{n-1},\sigma_{n}}(J)$,
with
$\exp(-2J{\mathbf{1}}_{\\{\sigma_{n-1}\neq\sigma_{n}\\}}+\sigma_{n}h_{n})$.
This comparison will be provided by the next lemma, that without loss of
generality we formulate only for the first block $[0,1]$. We introduce the
random variable

$H:=\max_{0\leq s\leq t\leq 1}|B_{s}-B_{t}|\,.$ (20)

######  Lemma 5.

For every realization of $B$, for every $(a,b)\in\\{-1,+1\\}^{2}$, for every
$J\geq 0$ and $M\geq 0$, we have

${\mathcal{Z}}_{1,B}^{a,b}(J)\geq\exp\left(-2(J+M){\mathbf{1}}_{\\{a\neq
b\\}}+bB_{1}-2(H-M)_{+}\right)\,,$ (21)

and

${\mathcal{Z}}_{1,B}^{a,b}(J)\leq\exp\left(-2(J-M){\mathbf{1}}_{\\{a\neq
b\\}}+bB_{1}+2(H-M)_{+}+e^{2(H-J)}\right)\,.$ (22)

###### Proof.

The proof is based on the following observation: for any (cadlag) trajectory
$(s_{t})_{0\leq t\leq 1}\in\\{-1,1\\}^{[0,1]}$ satisfying $s_{0}=a$ and
$s_{1}=b$, denoting by $j$ its number of jumps on $[0,1]$, we have

$\left|\int_{0}^{1}s_{t}\mathrm{d}B_{t}-bB_{1}\right|\leq 2jH\,.$ (23)

We first handle the lower bound. If $a=b$ (respectively, if $a\neq b$), we
lower bound the partition function ${\mathcal{Z}}_{1,B}^{a,b}(J)$ by
restricting the expectation $\mathbf{E}_{J}^{a}$ to the set of trajectories
with no jump in $[0,1]$ (respectively with exactly one jump in $[0,1]$).
Recall the “unnormalizing” factor in front of the expectation in (16). We
obtain:

${\mathcal{Z}}_{1,B}^{a,b}(J)\geq\exp(-2(J+H){\mathbf{1}}_{\\{a\neq
b\\}}+bB_{1})\,.$ (24)

Using $H\leq M+(H-M)_{+}$ we derive the desired lower bound.

We now turn to the proof of the upper bound. Using the observation (23), we
have:

${\mathcal{Z}}_{1,B}^{a,b}(J)\leq\sum_{j}\frac{(e^{-2J})^{j}}{j!}\exp({bB_{1}+2jH})\,,$
(25)

where the sum is over even integers if $a=b$ and over odd integers if $a\neq
b$. In the first case, let us bound the sum by the sum over all non-negative
integers, in the second case let us bound by the sum over all positive
integers, and use $j!\geq(j-1)!$. We obtain :

${\mathcal{Z}}_{1,B}^{a,b}(J)\leq\exp\left(bB_{1}-2(J-H){\mathbf{1}}_{\\{a\neq
b\\}}+e^{2(H-J)}\right)\,.$ (26)

Using $H\leq M+(H-M)_{+}$ we see that the upper bound in the lemma follows. ∎

From Lemma 5, we derive the following lemma and corollary concerning the free
energy densities.

######  Lemma 6.

For every $M\geq 0$ and $J\geq 0$, we have:

${\mathcal{F}}(J)\geq F_{{\mathcal{N}}_{1}}(J+M)-\mathbb{E}[2(H-M)_{+}].$ (27)

and

${\mathcal{F}}(J)\leq
F_{{\mathcal{N}}_{1}}(J-M)+e^{2(M-J)}+\mathbb{E}\left[2(H-M)_{+}+e^{2(H-J)}\right].$
(28)

Replacing $J$ by $J-M$ (respectively $J+M$), Lemma 5 immediately gives the
following corollary.

######  Corollary 7.

For every $M\geq 0$ and $J\geq 0$, we have:

$F_{{\mathcal{N}}_{1}}(J)\leq{\mathcal{F}}(J-M)+\mathbb{E}[2(H-M)_{+}],$ (29)

and

$F_{{\mathcal{N}}_{1}}(J)\geq{\mathcal{F}}(J+M)-e^{-2J}-\mathbb{E}\left[\left(2(H-M)+e^{2(H-M-J)}\right){\mathbf{1}}_{\\{H>M\\}}\right].$
(30)

###### Proof of Lemma 6.

Recall the observation that, under $\mathbb{P}$, sequence
$h=(B_{n}-B_{n-1})_{n\in\mathbb{N}}$ is i.i.d. with law ${\mathcal{N}}_{1}$.
The lower bound is obtained using (21) in (18), taking the $\log$, dividing by
$N$ and finally sending $N$ to infinity and using (15) and the law of large
numbers. The upper bound is obtained similarly, using (22) instead of (21). ∎

We are now ready to prove Theorem 1 when $\vartheta=1$: it is just a matter of
making a choice for $M$, depending on $J$, and controlling the error terms
appearing in Corollary 7.

###### Proof of Theorem 1 when $\vartheta=1$.

We are going to use the following fact:

$\mathbb{P}[H\geq\lambda]=O\left(\exp\left(-\frac{\lambda^{2}}{8}\right)\right)\,,\qquad\text{as
}\lambda\to\infty\,.$ (31)

This can be shown by noticing that $H\leq\lambda$ implies that
$\max_{[0,1]}B_{\cdot}\geq\frac{\lambda}{2}$ or
$\min_{[0,1]}B_{\cdot}\leq-\frac{\lambda}{2}$ and by using the fact that
$\max_{[0,1]}B_{\cdot}$ and $-\min_{[0,1]}B_{\cdot}$ are distributed as the
absolute values of standard Gaussian variables (by the reflection principle).
Equation (31) yields in particular that $\mathbb{E}\left[e^{2H}\right]$ and
$\mathbb{E}[H^{2}]$ are finite.

For every $J\geq 1$, let us set $M_{J}:=6(\log J)^{\frac{1}{2}}$. Then we have
$\mathbb{E}\left[e^{2(H-M_{J}-J)}\right]=O(e^{-2J})$, and, using Cauchy-
Schwarz inequality,

$\mathbb{E}\left[(H-M_{J})_{+}]\leq\mathbb{E}[H{\mathbf{1}}_{\\{H>M_{J}\\}}\right]\leq\mathbb{E}[H^{2}]^{\frac{1}{2}}\mathbb{P}[H>M_{J}]^{\frac{1}{2}}=O(J^{-2})\,.$
(32)

Thus, Corollary 7 yields:

${\mathcal{F}}(J+M_{J})+O(J^{-2})\leq
F_{{\mathcal{N}}_{1}}(J)\leq{\mathcal{F}}(J-M_{J})+O(J^{-2})\,.$ (33)

Recalling (16), this concludes the proof of Theorem 1 in the case where
$\vartheta=1$. ∎

######  Remark 8.

We present two possible ways to adapt the present section to prove Theorem 1
for any $\vartheta\in(0,\infty)$.

  * •

The first one consits in modifying (18) by considering the continuum partition
function on $[0,N\vartheta^{2}]$ instead of $[0,N]$ and dividing this segment
in blocks of length $\vartheta^{2}$, so
$[(n-1)\vartheta^{2},n\vartheta^{2}],n=1,\dots,N$. Lemmas 5 and 6 as well as
Corollary 7 should then be rewritten for this new block length :

    * –

The random variable $H$ is replaced by $H_{\vartheta}:=\max_{0\leq s\leq
t\leq\vartheta^{2}}|B_{s}-B_{t}|$.

    * –

In Lemma 6 and Corollary 7, ${\mathcal{F}}$ gets multiplied by
$\vartheta^{2}$.

    * –

In Lemmas 5 and 6, in the right-hand sides of the inequalities, $J$ is changed
in $J-\log\vartheta$ : this comes from the fact that the Poisson parameter of
a block of length $\vartheta^{2}$ is $\vartheta^{2}e^{-2J}$. Consequently, in
Corollary 7, $J$ is replaced by $J+\log\vartheta$ in the right-hand sides of
the inequalities.

  * •

The second one consists in keeping blocks of size one but replacing the
Browian trajectory $B$ by $\vartheta B$. The variable $H$ is thus replaced by
$\vartheta H$. In Lemma 6 and in Corollary 7, ${\mathcal{F}}$ is replaced by
${\mathcal{F}}_{\vartheta}$ the free energy density of the continuum RFIC with
external field $\vartheta B$ (instead of $B$, see (14) and (15)). Using
Brownian scaling and Poisson scaling, we see that for all $J\in\mathbb{R}$:

${\mathcal{F}}_{\vartheta}(J)=\vartheta^{2}{\mathcal{F}}(J+\log\vartheta).$
(34)

In fact, since $H_{\vartheta}$ and $\vartheta H$ have the same law, both
approaches yield the same generalizations of Lemma 6 and Corollary 7. The
proof of Theorem 1 can then be conducted _mutatis mutandis_.

##  3 Gaussian approximation of the discrete RFIC partition function

Our aim is to compare the free energy density of the RFIC with disorders $\mu$
and ${\mathcal{N}}_{\vartheta^{2}}$, in order to derive Theorems 2 and 3 from
Theorem 1. We first state a lemma which bounds the difference between the free
energy densities for different disorder laws, with respect to their
Wasserstein 1-distance.

######  Lemma 9.

Consider $\nu$ and $\nu^{\prime}$ two laws on $\mathbb{R}$. Then, for all
$J\geq 0$,

$|F_{\nu}(J)-F_{\nu^{\prime}}(J)|\leq W_{1}(\nu,\nu^{\prime}),$

where $W_{1}(\cdot,\cdot)$ is the Wasserstein 1-distance, i.e.:

$W_{1}(\nu,\nu^{\prime})=\inf_{(h,h^{\prime})}\mathbb{E}[|h-h^{\prime}|].$

with the infimum running over all couplings $(h,h^{\prime})$ of $\nu$ and
$\nu^{\prime}$.

###### Proof.

Consider a coupling $(h,h^{\prime})$ of $\nu$ and $\nu^{\prime}$ and i.i.d.
sequences $h=(h_{n})_{n\in\mathbb{N}}$ and
$h^{\prime}=(h_{n}^{\prime})_{n\in\mathbb{N}}$ such that for every
$n\in\mathbb{N}$, $h_{n}$ and $h_{n}^{\prime}$ are coupled like $h$ and
$h^{\prime}$. Then, using
$\sigma_{n}h_{n}\leq\sigma_{n}h_{n}^{\prime}+|h_{n}-h_{n}^{\prime}|$, we have:

$Z_{N,h}(J)\leq
Z_{N,h^{\prime}}(J)\exp\left(\sum_{n=1}^{N}|h_{n}-h_{n}^{\prime}|\right).$

Taking the $\log$, dividing by $N$ and sending $N$ to infinity, we get from
(2):

$F_{\nu}(J)\leq F_{\nu^{\prime}}(J)+\mathbb{E}[|h-h^{\prime}|],$

using the law of large numbers for the last term. The lemma follows via a
minimization over the couplings $(h,h^{\prime})$ of $\nu$ and $\nu^{\prime}$,
and by exchanging the roles of $\nu$ and $\nu^{\prime}$. ∎

Since this lemma provides a bound which is uniform in $J$, it seems useless
for our purpose. It will however become crucial once we have introduced the
main idea of the present section, which is to make a _coarse graining of the
Ising chain_. This coarse-graining bears some similarities with the procedure
used in the previous section to link the continuum RFIC to the Gaussian
disorder discrete RFIC. We pick $L\in\mathbb{N}$. For every $N\in\mathbb{N}$,
we ”rigidify” the RFIC on $\\{1,\dots,NL\\}$ by restricting it to
configurations which exhibit spin changes only at locations $nL,n=1,\dots,N$.
Then the collection $(\sigma_{nL})_{1\leq n\leq L}$ is itself distributed as a
RFIC on $\\{1,.\dots,N\\}$ with disorder sequence
$(h_{n}^{L})_{n\in\mathbb{N}}$ defined by
$h_{n}^{L}:=\sum_{i=(n-1)L+1}^{nL}h_{i}$. We finally observe that this
sequence is i.i.d., we denote its law by $\mu^{\ast L}$, it is the law of the
sum of $L$ independent variables with law $\mu$. Our coarse-graining approach
will yield comparison estimates on the free energy densities $F_{\mu}$ and
$F_{\mu^{\ast L}}$, see Lemma 12. The lower bound is straight-forward to
obtain, while the upper bound requires some work.

With these estimates at hand, the proofs of Theorems 2 and 3 will informally
consist in the following approximations. We will let $L$ depend on $J$ and
show, as $J\to\infty$ that:

$F_{{\mu}}(J)\approx\frac{1}{L}F_{{\mu}^{\ast L}}(J)\\\
\approx\frac{1}{L}F_{{\mathcal{N}}_{L\vartheta^{2}}}(J)\approx
F_{{\mathcal{N}}_{\vartheta^{2}}}(J)\approx\frac{\vartheta^{2}}{2J}\,.$ (35)

The first and third approximations will be given by our coarse-graining
estimates. The fourth one comes from Theorem 1. The second one will be
provided by 9 together with the following lemma, which is a version of the
central limit theorem with respect to the Wasserstein-1 distance.

######  Lemma 10.

If ${\mu}$ has a finite third moment, then the sequence
$\left(W_{1}\left(\mathcal{L}^{\ast
L},\mathcal{N}_{L\vartheta^{2}}\right)\right)_{L\in\mathbb{N}^{*}}$ is
bounded. If ${\mu}$ has a finite $p$-th moment, with $p\in[2,3)$, then, as
$L\to\infty$,

$W_{1}\left(\mathcal{L}^{\ast
L},\mathcal{N}_{L\vartheta^{2}}\right)=O\left(L^{\frac{3-p}{2}}\right)\,.$
(36)

###### Proof.

This is equivalent to showing that the Wasserstein 1-distance between the law
of $(h_{1}+\dots+h_{n})/\sqrt{L}$ and ${\mathcal{N}}_{\vartheta^{2}}$ is
$O(L^{-\frac{1}{2}})$, respectively $O(L^{1-\frac{p}{2}})$. This is a known
result, see [10] for example. ∎

Lemma 10 yields that the error in the second approximation in (35) is
$O(\frac{1}{L})$, respectively $O(\frac{1}{L^{\frac{p-1}{2}}})$. How large an
$L$ we will take will depend on the error terms coming from the coarse-
graining estimates, which we are going to present now.

###  3.1 Coarse-graining of the Ising chain

As annouced, in this section, we pick $L\in\mathbb{N}$ and, denoting by
${\mu}^{\ast L}$ the law of $h_{1}+\dots+h_{L}$ (recall that the $h_{n}$’s are
i.i.d with law ${\mu}$), we will compare the free energy density
$F_{\mu}(\cdot)$ with the free energy density $F_{{\mu}^{\ast L}}(\cdot)$ by
performing a “coarse-graining” of the Ising chain.

We need to slightly generalize the notation in (1). We introduce for positive
integers $\mathcalb{l}$ and $\mathcalc{r}$ satisfying
$\mathcalb{l}\leq\mathcalc{r}$ the partition function on
$\\{\mathcalb{l},\mathcalb{l}+1,\dots,\mathcalc{r}\\}$:

$Z_{\mathcalb{l},\mathcalc{r},h}^{a,b}(J):=\sum_{\sigma\in\\{-1,1\\}^{\\{\mathcalb{l}-1,\mathcalb{l},\dots,\mathcalc{r}\\}}:\sigma_{\mathcalb{l}-1}=a,\sigma{r}=b}\exp\Big{(}-2J\sum_{n=\mathcalb{l}}^{\mathcalc{r}}{\mathbf{1}}_{\\{\sigma_{n}\neq\sigma_{n-1}\\}}+\sum_{n=\mathcalb{l}}^{\mathcalc{r}}\sigma_{n}h_{n}\Big{)}\,.$
(37)

Note that $Z_{N,h}^{a,b}(J)=Z_{1,N,h}^{a,b}(J)$ and that
$Z_{\mathcalb{l},\mathcalc{r},h}^{a,b}(J)=Z_{\mathcalc{r}-\mathcalb{l}+1,\Theta^{\mathcalb{l}-1}h}^{a,b}(J)$
if we define $\Theta^{\mathcalb{l}-1}h$ as the sequence
$(h_{\mathcalb{l}-1+n}-h_{\mathcalb{l}-1})_{n\in\mathbb{N}}$.

The coarse-graining idea goes as follows: we define a sequence
$h^{L}=(h^{L}_{n})_{n\in\mathbb{N}}$ by setting
$h^{L}_{n}:=\sum_{i=(n-1)L+1}^{nL}h_{i}$. Observe that the sequence $h^{L}$ is
i.i.d. with law ${\mu}^{\ast L}$. For all $N\in\mathbb{N}$, we associate to
any configuration $\sigma\in\\{-1,1\\}^{\\{0,1,\dots,NL\\}}$ the configuration
$\sigma^{L}\in\\{-1,1\\}^{\\{0,1,\dots,N\\}}$ defined by:

$\sigma^{L}_{n}=\sigma_{nL},\qquad n=0,\dots,N.$ (38)

Multiple configurations $\sigma$ are sent to the same configuration
$\sigma^{L}$, we rewrite the summation in (1) according to this partitioning
and we obtain:

$Z_{NL,h}^{a,b}(J)=\sum_{\sigma\in\\{-1,1\\}^{\\{0,1,\dots,N\\}}:\sigma_{0}=a,\sigma_{N}=b}\prod_{n=1}^{N}Z_{(n-1)L+1,nL,h}^{\sigma_{n-1},\sigma_{n}}(J)\,.$
(39)

Our aim is to compare this expression with

$Z_{N,h^{L}}^{a,b}(J)=\sum_{\sigma\in\\{-1,1\\}^{\\{0,1,\dots,N\\}}:\sigma_{0}=a,\sigma_{N}=b}\prod_{n=1}^{N}\exp(-2J{\mathbf{1}}_{\\{\sigma_{n-1}\neq\sigma_{n}\\}}+\sigma_{n}h_{n}^{L})\,,$
(40)

so naturally we now want to compare the partition function on the _block_
$\\{(n-1)L+1,\dots,nL\\}$, i.e
$Z_{(n-1)L+1,nL,h}^{\sigma_{n-1},\sigma_{n}}(J)$, with
$\exp(-2J{\mathbf{1}}_{\\{\sigma_{n-1}\neq\sigma_{n}\\}}+\sigma_{n}h_{n}^{L})$.
This will be provided by the following lemma, that without loss of generality
we formulate only for the first block $\\{1,\dots,L\\}$. We introduce the
random variable

$H_{L}:=\max_{1\leq n\leq m\leq L}\Big{|}\sum_{i=n}^{m}h_{i}\Big{|}\,.$ (41)

######  Lemma 11.

For every realization of $h$, for every $(a,b)\in\\{-1,+1\\}^{2}$, for every
$L\in{\mathbb{N}}$, $J\geq 0$ and $M\geq 0$, we have

$Z_{L,h}^{a,b}(J)\geq\exp\left(-2J{\mathbf{1}}_{\\{a\neq
b\\}}+bh_{1}^{L}\right)\,,$ (42)

and

$\displaystyle Z_{L,h}^{a,b}(J)$
$\displaystyle\leq\exp\Bigg{[}-2\left(J\\!-\\!M\\!-\\!\frac{1}{2}\\!\log\\!L\right){\mathbf{1}}_{\\{a\neq
b\\}}+bh_{1}^{L}$ (43)
$\displaystyle\qquad\qquad+L\left(4(H_{L}-M)_{+}+e^{2(M-J)}\right)\Bigg{]}\,.$

###### Proof.

For the lower bound, we simply lower bound the sum defining $Z_{L,h}^{a,b}(J)$
in (14) by the term corresponding to $\sigma=(a,b,b,\dots,b)$.

We now turn to the upper bound. For given
$\sigma\in\\{-1,+1\\}^{\\{0,1,\dots,L\\}}$ such that
$\sigma_{0}=a,\sigma_{L}=b$, denoting
$j=\sum_{n=1}^{L}{\mathbf{1}}_{\\{\sigma_{n}\neq\sigma_{n-1}\\}}$ the number
of spin flips in $\sigma$, we observe that:

$\sum_{n=1}^{L}\sigma_{n}h_{n}\leq bh^{L}_{1}+2jH$ (44)

Furthermore, there are at most $\binom{L}{j}$ configurations in
$\\{-1,+1\\}^{\\{0,1,\dots,L\\}}$ having $j$ spin flips. Therefore:

$Z_{L,h}^{a,b}(J)\leq\sum_{j}\binom{L}{j}\exp(-2jJ+bh_{1}^{L}+2jH)\,,$ (45)

where the sum is over $2\mathbb{N}$ if $a=b$, over $2\mathbb{N}+1$ if $a\neq
b$, and with the convention $\binom{L}{j}=0$ if $j>L$. In the first case, let
us upper bound the sum by the sum over all non-negative integers and in the
second case let us bound by the sum over all positive integers and use
$\binom{L}{j}\leq L\binom{L}{j-1}$. Using Newton’s formula, we get:

$Z_{L,h}^{a,b}(J)\leq\exp\left(bh_{1}^{L}-2(J-H_{L}-\frac{1}{2}\log
L){\mathbf{1}}_{\\{a\neq b\\}}\right)\times(1+e^{2(H_{L}-J)})^{L}$ (46)

We use $H_{L}\leq M+(H_{L}-M)_{+}$ to bound $H_{L}{\mathbf{1}}_{\\{a\neq
b\\}}$ and we use it again to get:

$1+e^{2(H_{L}-J)}\leq e^{2(H_{L}-M)_{+}}(1+e^{2(M-J)})$ (47)

Using the fact that $1+x\leq e^{x}$ for all real number $x$, we get:

$Z_{L,h}^{a,b}(J)\leq\exp\left[-2\left(\\!J\\!-\\!M\\!-\\!\frac{1}{2}\\!\log\\!L\\!\right){\mathbf{1}}_{\\{a\neq
b\\}}+bh_{1}^{L}+2(L+1)(H_{L}-M)_{+}+Le^{2(M-J)}\right]\,.$ (48)

We bound $L+1$ by $2L$ to conclude the proof of the upper bound. ∎

From Lemma 11, we derive the following lemma concerning the free energy
densities.

######  Lemma 12.

For every $L\in{\mathbb{N}}$, $J\geq 0$ and $M\geq 0$, we have:

$F_{{\mu}}(J)\geq\frac{1}{L}F_{{\mu}^{\ast L}}(J).$ (49)

and

$F_{{\mu}}(J)\leq\frac{1}{L}F_{{\mu}^{\ast
L}}(J\\!-\\!M\\!-\\!\frac{1}{2}\\!\log
L)+4\mathbb{E}_{\mu}\left[(H_{L}-M)_{+}\right]+e^{2(M-J)}\,.$ (50)

###### Proof of Lemma 12.

The lower bound is immediate, using (49) in (39), taking the $\log$, dividing
by $NL$ and finally sending $N$ to infinity with the help of (2).

For the upper bound, we do the same, using (50) instead of (49) and using in
addition the law of large numbers. ∎

Replacing $J$ by $J\\!+\\!M\\!+\\!\frac{1}{2}\\!\log L$, Lemma 12 yields the
following corollary.

######  Corollary 13.

For every $L\in{\mathbb{N}}$, $J\geq 0$ and $M\geq 0$,

$\frac{1}{L}F_{{\mu}^{\ast L}}(J)\geq
F_{{\mu}}(J\\!+\\!M\\!+\\!\frac{1}{2}\\!\log
L)-4\mathbb{E}_{\mu}\left[(H_{L}-M)_{+}\right]-\frac{e^{-2J}}{L}\,.$ (51)

######  Remark 14.

In the proofs of Theorem 2 of 3, we will take $L$ depending on $J$ and large
(as $J$ goes to infinity). We will also let $M$ depend on $J$, typically
larger than scale $\sqrt{L}$, so as to make the event $\\{H_{L}>M\\}$
unlikely, but also smaller than scale $J$ so that the bounds remain
interesting for our purpose.

###  3.2 Proof of Theorem 2

######  Lemma 15.

Assume that $\mu$ admits some finite exponential moments. Then, there exists a
positive constant $c$ such that, for all $L\in\mathbb{N}$ and all $M>0$
satisfying $M\leq cL$, we have

$\mathbb{E}_{\mu}[(H_{L}-M)_{+}]\leq\frac{4\vartheta^{2}L^{3}}{M}\exp\left(-\frac{M^{2}}{4\vartheta^{2}L}\right)\,.$
(52)

###### Proof.

Let us denote by
$\varphi:\mathbb{R}\to\mathbb{R},t\mapsto\mathbb{E}_{\mu}[\exp(th_{1})]$ the
moment generating function of $\mu$. Developing it around 0, we see that there
exists a positive constant $c^{\prime}$ such that:

$\forall
t\in[-c^{\prime},c^{\prime}]\,,\quad\varphi(t)\leq\exp(\vartheta^{2}t^{2})\,.$
(53)

We pick any $t\in(0,c^{\prime})$. Recalling the definition of $H_{L}$ in (41),
we observe that:

$\exp(tH_{L})\leq\sum_{1\leq n\leq m\leq
L}\exp(\sum_{i=n}^{m}h_{i})+\exp(-\sum_{i=n}^{m}h_{i})$ (54)

which, by taking the expectation under $\mathbb{P}_{\mu}$ and using (53),
yields:

$\mathbb{E}_{\mu}[\exp(tH_{L})]\leq\sum_{1\leq n\leq m\leq
L}\varphi(t)^{(m-n+1)}+\varphi(-t)^{(m-n+1)}\leq
2L^{2}\exp(\vartheta^{2}t^{2}L)$ (55)

Using the remark that $(H_{L}-M)_{+}\leq\frac{1}{t}\exp(t(H_{L}-M))$, we
obtain:

$\mathbb{E}_{\mu}[(H_{L}-M)_{+}]\leq\frac{2L^{2}}{t}\exp(\vartheta^{2}t^{2}L-Mt)$
(56)

Now, we see that the lemma holds with $c:=2\vartheta^{2}c^{\prime}$. Indeed,
when $M\leq cL$, take $t=\frac{M}{2\vartheta^{2}L}$, which is not larger than
$c^{\prime}$, to derive (52) from (56).

∎

We are now ready to prove Theorem 2.

###### Proof of Theorem 2.

We use ${\tilde{M}}$ as a short-hand for $M+\frac{1}{2}\log L$. Using the
lower bound in Lemma 12 for law $\mu$, then Lemma 9 and finally Corollary 13
for law ${\mathcal{N}}_{\vartheta^{2}}$, we derive a lower bound on
$F_{{\mu}}(J)$:

$\displaystyle F_{{\mu}}(J)$ $\displaystyle\geq\frac{1}{L}F_{{\mu}^{\ast
L}}(J)$ (57)
$\displaystyle\geq\frac{1}{L}F_{{\mathcal{N}}_{L\vartheta^{2}}}(J)-\frac{1}{L}W_{1}({\mu}^{\ast
L},{\mathcal{N}}_{L\vartheta^{2}})$ $\displaystyle\geq
F_{{\mathcal{N}}_{\vartheta^{2}}}(J+{\tilde{M}})-4\mathbb{E}_{{\mathcal{N}}_{\vartheta^{2}}}\left[(H_{L}-M)_{+}\right]-\frac{e^{-2J}}{L}-\frac{1}{L}W_{1}({\mu}^{\ast
L},{\mathcal{N}}_{L\vartheta^{2}})\,.$

Recall Theorem 1. Using the fact that $\frac{1}{1+x}\geq 1-x$ for all
$x\in\mathbb{R}^{+}$, we have

$F_{{\mathcal{N}}_{\vartheta^{2}}}(J+{\tilde{M}})\geq\frac{\vartheta^{2}}{2J}-\frac{\vartheta^{2}{\tilde{M}}}{2J^{2}}+O\left(\frac{(\log
J)^{\frac{1}{2}}}{J^{2}}\right)\,.$ (58)

Similarly, we derive an upper bound on $F_{{\mu}}(J)$. Applying the upper
bound in Lemma 12 for law $\mu$, then Lemma 9 and finally the lower bound in
Lemma 12 for law ${\mathcal{N}}_{\vartheta^{2}}$, we have:

$\displaystyle F_{{\mu}}(J)$ $\displaystyle\leq\frac{1}{L}F_{{\mu}^{\ast
L}}(J-{\tilde{M}})+4\mathbb{E}_{\mu}\left[(H_{L}-M)_{+}\right]+e^{2(M-J)}$
(59)
$\displaystyle\leq\frac{1}{L}F_{{\mathcal{N}}_{L\vartheta^{2}}}(J-{\tilde{M}})+\frac{1}{L}W_{1}({\mu}^{\ast
L},{\mathcal{N}}_{L\vartheta^{2}})+4\mathbb{E}_{\mu}\left[(H_{L}-M)_{+}\right]+e^{2(M-J)}$
$\displaystyle\leq
F_{{\mathcal{N}}_{\vartheta^{2}}}(J-{\tilde{M}})+\frac{1}{L}W_{1}({\mu}^{\ast
L},{\mathcal{N}}_{L\vartheta^{2}})+4\mathbb{E}_{\mu}\left[(H_{L}-M)_{+}\right]+e^{2(M-J)}\,.$

Recall Theorem 1. We assume that ${\tilde{M}}\leq\frac{J}{2}$. Using the fact
that $\frac{1}{1-x}\leq 1+2x$ for all $x\in[0,\frac{1}{2}]$, we have

$F_{{\mathcal{N}}_{\vartheta^{2}}}(J-{\tilde{M}})\leq\frac{\vartheta^{2}}{2J}+\frac{\vartheta^{2}{\tilde{M}}}{J^{2}}+O\left(\frac{(\log
J)^{\frac{1}{2}}}{J^{2}}\right)\,.$ (60)

Now, we make a choice for $L$ and $M$. We set, for $J>1$:

$L_{J}=\left\lfloor\frac{J^{\frac{4}{3}}}{(\log
J)^{\frac{1}{3}}}\right\rfloor,\qquad M_{J}=6\vartheta J^{\frac{2}{3}}(\log
J)^{\frac{1}{3}},\qquad$ (61)

With this choice, we have $e^{2(M-J)}=O(J^{-2})$ and, according to Lemma 15,
both
$\mathbb{E}_{{\mathcal{N}}_{\vartheta^{2}}}\left[(H_{L_{J}}-M_{J})_{+}\right]$
and $\mathbb{E}_{\mu}\left[(H_{L}-M)_{+}\right]$ are $O(J^{-2})$. According to
Lemma 10, $W_{1}({\mu}^{\ast L_{J}},{\mathcal{N}}_{L_{J}\vartheta^{2}})=O(1)$.
Theorem 2 follows. ∎

###  3.3 Proof of Theorem 3

Notice that in the proof of the lower bound in Theorem 2, we only use the
bound on
$\mathbb{E}_{{\mathcal{N}}_{\vartheta^{2}}}\left[(H_{L}-M)_{+}\right]$ but not
on $\mathbb{E}_{\mu}\left[(H_{L}-M)_{+}\right]$. Thus, it is possible to carry
this proof in the same way in the case of polynomial integrability. The only
adaptation that is needed is to take into account the disjonction in Lemma 10.

###### Proof of the lower bound in Theorem 3.

We use the lower bounds (57) and (58) establised in the proof of Theorem 2.
For the case $\mu\in L^{3}$, we choose $L_{J}$ and $M_{J}$ as in the proof of
Theorem 2, see (61). For the case $\mu\in L^{p}$, with $p\in[2,3)$, we set,
for $J>1$:

$L_{J}=\left\lfloor\frac{J^{\frac{4}{p}}}{(\log
J)^{\frac{1}{p}}}\right\rfloor,\qquad M_{J}=6\vartheta J^{\frac{2}{p}}(\log
J)^{\frac{p-1}{2p}}\,.$ (62)

With these choices, we have $e^{2(M-J)}=O(J^{-2})$ and, according to Lemma 15,
$\mathbb{E}_{{\mathcal{N}}_{\vartheta^{2}}}\left[(H_{L_{J}}-M_{J})_{+}\right]=O(J^{-2})$.
We also use Lemma 10 to bound $W_{1}({\mu}^{\ast
L_{J}},{\mathcal{N}}_{L_{J}\vartheta^{2}})$. The proof of the lower bound in
Theorem 3 follows. ∎

Now, we turn to the proof of the upper bound in Theorem 3.

######  Lemma 16.

Assume that $\mu\in L^{p}$, with $p\in[2,\infty)$. Then, for every
$L\in\mathbb{N}$ and every $M>0$, we have:

$\mathbb{P}_{\mu}[H_{L}>M]=O\left(\frac{L^{\frac{p}{2}}}{M^{p}}\right)\,,$
(63)

and

$\mathbb{E}_{\mu}[(H_{L}-M)_{+}]=O\left(\frac{L^{\frac{p}{2}}}{M^{p-1}}\right)\,.$
(64)

###### Proof.

Let us introduce $S_{n}:=\sum_{i=1}^{n}h_{i}$ and observe that
$(S_{n})_{n\in\mathbb{N}}$ is a martingale and $H_{L}\leq 2\sup_{1\leq n\leq
L}|S_{n}|$. Doob’s inequality in $L^{p}$ yields:

$\mathbb{E}[H_{L}^{p}]\leq 2^{p}\mathbb{E}[(\sup_{1\leq n\leq
L}|S_{n}|)^{p}]\leq
2^{p}\left(\frac{p}{p-1}\right)^{p}\mathbb{E}[|S_{n}|^{p}].$ (65)

Now, Rosenthal’s inequality for independent centered random variables yields:
$\mathbb{E}[|S_{L}|^{p}]=O(L^{\frac{p}{2}})$, so
$\mathbb{E}[H_{L}^{p}]=O(L^{\frac{p}{2}})$. Use Markov’s inequality for
$H_{L}^{p}$ to derive (63). To show (64), use instead the following
inequality:

$\mathbb{E}[(H_{L}-M)_{+}]\leq\mathbb{E}[H_{L}{\mathbf{1}}_{\\{H_{L}>M\\}}]\leq\mathbb{E}\left[H_{L}\left(\frac{H_{L}}{M}\right)^{p-1}{\mathbf{1}}_{\\{H_{L}>M\\}}\right]\leq\frac{\mathbb{E}[H_{L}^{p}]}{M^{p-1}}\,.$
(66)

∎

With this lemma at hand, it is possible to adapt the proof of Theorem 2 to
obtain an upper bound $F_{\mu}(J)$. However, we present a different approach
that slightly ameliorates the final result. We modify the upper bounds in
Lemmas 11 and 12 in the following way.

######  Lemma 17.

For every realization of $h$, for every $(a,b)\in\\{-1,+1\\}^{2}$, for every
$L\in{\mathbb{N}}$, $J\geq 0$ and $M\geq 0$, we have

$\displaystyle Z_{L,h}^{a,b}(J)$
$\displaystyle\leq\exp\Bigg{[}-2\left(J\\!-\\!M\\!-\\!\frac{1}{2}\\!\log\\!L\right){\mathbf{1}}_{\\{a\neq
b\\}}+bh_{1}^{L}$ (67) $\displaystyle\qquad\qquad+(L\log
2+2\sum_{n=1}^{L}|h_{n}|){\mathbf{1}}_{\\{H_{L}>M\\}}+Le^{2(M-J)}\Bigg{]}\,.$

and, consequently:

$F_{{\mu}}(J)\leq\frac{1}{L}F_{{\mu}^{\ast
L}}(J\\!-\\!M\\!-\\!\frac{1}{2}\\!\log L)+\mathbb{E}_{\mu}\Big{[}(\log
2+\frac{2}{L}\sum_{n=1}^{L}|h_{n}|){\mathbf{1}}_{\\{H_{L}>M\\}}\Big{]}+e^{2(M-J)}\,.$
(68)

###### Proof.

If $H\leq M$, we use (46) to see that (67) holds. If instead $H_{L}>M$, we use
the bound

$\sum_{n=1}^{L}\sigma_{n}h_{n}\leq bh^{L}_{1}+2\sum_{n=1}^{L}|h_{n}|$ (69)

and the fact that if $a\neq b$ then $\sigma$ exhibits at least one spin flip.
We get:

$Z_{L,h}^{a,b}(J)\leq 2^{L}\exp\left(-2J{\mathbf{1}}_{\\{a\neq
b\\}}+bh_{1}^{L}+2\sum_{n=1}^{L}|h_{n}|\right)$ (70)

so (67) holds also in this case.

To derive (68), reason as for the upper bound in Lemma 12. ∎

######  Lemma 18.

Assume that $\mu\in L^{p}$, with $p\in[2,\infty)$. Then:

$\mathbb{E}_{\mu}\Big{[}(\log
2+\frac{2}{L}\sum_{n=1}^{L}|h_{n}|){\mathbf{1}}_{\\{H_{L}>M\\}}\Big{]}=O\left(\frac{L^{\frac{p}{2}}}{M^{p}}+\frac{1}{M^{p-1}}\right)\,.$
(71)

###### Proof.

Recall the bound
$\mathbb{P}[H_{L}>M]=O\left(\frac{L^{\frac{p}{2}}}{M^{p}}\right)$ which
controls the first term. We now bound for each $n=1,\dots,L$ the term
$\mathbb{E}[|h_{n}|{\mathbf{1}}_{\\{H_{L}>M\\}}]$. We set

$H_{L}^{(n)}:=\sup_{1\leq m\leq m^{\prime}\leq
L}\left|\sum_{i\in\\{m,\dots,m^{\prime}\\}\setminus\\{n\\}}h_{i}\right|$ (72)

and we observe that if $|h_{n}|\leq\frac{M}{2}$ and $H_{L}>M$ then
$H_{L}^{(n)}>\frac{M}{2}$, so:

$\mathbb{E}[|h_{n}|{\mathbf{1}}_{\\{H_{L}>M\\}}]\leq\mathbb{E}[|h_{n}|{\mathbf{1}}_{\\{H_{L}^{(n)}>\frac{M}{2}\\}}]+\mathbb{E}[|h_{n}|{\mathbf{1}}_{\\{|h_{n}|\geq\frac{M}{2}\\}}]\,.$
(73)

We handle the first term in the right-hand side of (73) by noticing that
$h_{n}$ and $H_{L}^{(n)}$ are independent, with $H_{L}^{(n)}$ distributed as
$H_{L-1}$, so that (63) can be used. For the second one we write, proceeding
as in (66):

$\mathbb{E}[|h_{n}|{\mathbf{1}}_{\\{|h_{n}|\geq\frac{M}{2}\\}}]\leq\frac{\mathbb{E}[|h_{n}|^{p}]}{\left(\frac{M}{2}\right)^{p-1}}\,.$
(74)

The lemma follows. ∎

###### Proof of the upper bound in Theorem 3.

We proceed as in the proof of the upper bound in Theorem 2, but using Lemma 17
instead of Lemma 12: the error term $4\mathbb{E}_{\mu}[(H_{L}-M)_{+}]$ in (59)
replaced by $\mathbb{E}_{\mu}\Big{[}(\log
2+\frac{2}{L}\sum_{n=1}^{L}|h_{n}|){\mathbf{1}}_{\\{H_{L}>M\\}}\Big{]}$. We
assume that $\mu$ is in $L^{p}$, with $p\in[2,\infty)$ and we make a choice
for $L$ and $M$, depending on $p$. For $p\in[3,\infty)$, we set
$\eta_{p}=\frac{4p}{3p+2}$ and for $J\geq 1$:

$L_{J}=\left\lfloor J^{\eta_{p}}\right\rfloor,\qquad M_{J}=J^{2-\eta_{p}}\,.$
(75)

For $p\in[2,3)$, we set $\eta_{p}=\frac{2p(p-1)}{p^{2}+p-1}$ and for $J\geq
1$:

$L_{J}=\left\lfloor J^{\frac{2\eta_{p}}{p-1}}\right\rfloor,\qquad
M_{J}=J^{2-\eta_{p}}\,.$ (76)

With these choices, we have $e^{2(M_{J}-J)}=O(J^{-2})$, and according to Lemma
18, the _new_ error term $\mathbb{E}_{\mu}\Big{[}(\log
2+\frac{2}{L_{J}}\sum_{n=1}^{L_{J}}|h_{n}|){\mathbf{1}}_{\\{H_{L_{J}}>M_{J}\\}}\Big{]}$
is $O(J^{-\eta_{p}})$, using the identity
$(2-\eta_{p})p-\eta_{p}\frac{p}{2}=\eta_{p}$ for $p\in[3,\infty)$, the
identity $(2-\eta_{p})p-\frac{2\eta_{p}}{p-1}\frac{p}{2}=\eta_{p}$ for
$p\in[2,3)$ and the inequality $(p-1)(2-\eta_{p})\geq\eta_{p}$ in both cases.
We finally use Lemma 10 to bound $W_{1}({\mu}^{\ast
L_{J}},{\mathcal{N}}_{L_{J}\vartheta^{2}})$ and we obtain the upper bound in
Theorem 3. ∎

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