# Fusion-FlowNet: Energy-Efficient Optical Flow Estimation using Sensor Fusion
and Deep Fused Spiking-Analog Network Architectures

Chankyu Lee, Adarsh Kumar Kosta and Kaushik Roy
Purdue University, West Lafayette, IN 47907, USA
{lee2216, akosta<EMAIL_ADDRESS>
###### Abstract

Standard frame-based cameras that sample light intensity frames are heavily
impacted by motion blur for high-speed motion and fail to perceive scene
accurately when the dynamic range is high. Event-based cameras, on the other
hand, overcome these limitations by asynchronously detecting the variation in
individual pixel intensities. However, event cameras only provide information
about pixels in motion, leading to sparse data. Hence, estimating the overall
dense behavior of pixels is difficult. To address such issues associated with
the sensors, we present Fusion-FlowNet, a sensor fusion framework for energy-
efficient optical flow estimation using both frame- and event-based sensors,
leveraging their complementary characteristics. Our proposed network
architecture is also a fusion of Spiking Neural Networks (SNNs) and Analog
Neural Networks (ANNs) where each network is designed to simultaneously
process asynchronous event streams and regular frame-based images,
respectively. Our network is end-to-end trained using unsupervised learning to
avoid expensive video annotations. The method generalizes well across distinct
environments (rapid motion and challenging lighting conditions) and
demonstrates state-of-the-art optical flow prediction on the Multi-Vehicle
Stereo Event Camera (MVSEC) dataset. Furthermore, our network offers
substantial savings in terms of the number of network parameters and
computational energy cost.

##  1 Introduction

Optical flow estimation is a fundamental computer vision problem, allowing us
to visualize the motion field in scenes. It involves estimating the spatio-
temporal motion patterns of pixels and forms the groundwork for more complex
tasks such as motion segmentation [22] and action recognition [29]. Over the
past years, the optical flow estimation has been largely dominated by
conventional computer vision algorithms such as differential [18], phase
correlation and block-based methods [2]. Recently, deep Analog Neural Networks
(ANNs111We refer to the conventional deep learning networks as ANNs owing to
their analog nature of inputs and computations. This nomenclature helps to
distinguish them from Spiking Neural Networks (SNNs) which perform event-based
computations.) based approaches for optical flow estimation have gained
immense popularity [7, 24]. In general, these methods rely on the standard
frame-based cameras as input sensors that capture pixel intensities over the
entire frame at a regular sampling rate. However, the frame-based images
suffer from a variety of issues such as motion blur and temporal aliasing when
capturing high speed motion due to fixed low temporal resolution. They are
also unable to perceive information accurately in high dynamic range scenes
due to uneven exposure [8].

Event-based cameras, such as Dynamic Vision Sensors (DVS) [16], address these
problems by asynchronously sampling intensity changes on each pixel element,
generating a stream of asynchronous events. This grants promising advantages,
namely high temporal resolution (10$\mu$s vs 3ms), high dynamic range (140dB
vs 60dB) and low power consumption (10mW vs 3W) compared to standard frame-
based cameras [8]. Note, event cameras only capture the varying components of
visual signals, generating sparse event streams. Hence, the output prediction
becomes limited only at pixels-points where events exist, adding difficulty
towards encoding the scene context.

As is evident from the above discussions, none of the above sensors by
themselves is able to effectively capture all relevant information of a scene.
The limited applicability of each individual camera gives rise to the need for
an optimal sensor-fusion technique, enabling the sensors to complement the
limitations of each other. Such a technique would provide a practical solution
towards accurately estimating dense pixel-wise motion in challenging scenarios
such as rapid motion and high dynamic range environments.

Conventional computer vision and ANN-based methods are incompatible at
handling the discrete and asynchronous event streams from event-based camera
in their native form. This is due to the fact that these methods are generally
designed for frame-based images, assuming regular frame rate and brightness
consistency over the entire frames. In this regard, Spiking Neural Networks
(SNNs), inspired from biological neuronal mechanisms, show a great promise for
directly handling event-camera outputs. Moreover, SNNs perform event-based
operations by carrying out the computations only at the arrival of the input
events, exploiting the inherent sparsity of spatio-temporal event streams and
thus enabling energy-efficient computations on specialized neuromorphic
hardware such as Loihi from Intel Labs [6] and TrueNorth from IBM [21].

In this work, we propose a method for combining the advantages of regular
frame-based images and a stream of asynchronous events. For this purpose, we
present Fusion-FlowNet, a deep fused spiking-analog architecture for
estimating optical flow that uses sensors of different modalities (standard
frame-based images and asynchronous event streams). Our main contributions are
as follows:

  * •

We propose Fusion-FlowNet architecture composed of a fusion of SNNs and ANNs
for simultaneously processing event streams and frame-based images, leveraging
their complementary sensing capabilities.

  * •

We present a Signed Integrate-and-Fire (SIF) neuron model for SNNs which can
generate spike outputs with polarity (either positive or negative). In
addition, we show that the SIF model coupled with a surrogate gradient method
enables end-to-end learning in SNNs.

  * •

We show that Fusion-FlowNet outperforms the corresponding previous works in
terms of optical flow estimation on the Multi-Vehicle Stereo Event Camera
(MVSEC) dataset. Furthermore, we analyze that Fusion-FlowNet provides
substantial savings in terms of network parameters and computational energy
cost.

##  2 Related Works

Over the past few years, there have been major advancements towards optical
flow estimation using event-cameras. Conventional computer vision algorithms
have been adapted to encompass the asynchronous event stream from these
sensors in [1, 3, 9]. In ANN-based approaches, the event streams are
essentially accumulated for fixed time intervals to generate synchronous
frames. In EV-FlowNet [32], the recent event counts as well as pixel-wise last
timestamp information are encoded in a frame-based representation. However,
this approach heavily suffers during rapid motion and in scenarios with dense
localized events, resulting in loss of rich spatio-temporal information.
Researchers in [34] proposed a 3D input representation of events interpolated
in a 3D volume with time dimension comprising the input channels to retain the
temporal fidelity. Nevertheless, this approach struggled to estimate the dense
predictions in image regions with fewer events.

In general, SNNs provide advantages towards directly handling the asynchronous
events and exploiting the inherent temporal information. Recently, authors in
Spike-FlowNet [15] aimed to overcome a noticeable drawback of SNNs – namely
the “spike vanishing” phenomenon where the number of spikes drastically reduce
in the deeper layers, hindering learning. They proposed to effectively
integrate SNNs and ANNs into a single network with the SNN layers enabling
efficient event stream handling and the ANN layers addressing the spike
vanishing problem. Note, since they used only the event streams as input, the
predictions were limited to only the pixel locations containing non-zero
number of events. Hence, estimating dense motion behavior was greatly limited.

In contrast, researchers in [23] presented a two-step approach to estimate
optical flow by jointly using a set of events and a single frame-based image.
They employed an optimization-based method to restore a sharp intensity image
from the inputs, followed by ANN-based flow estimation methods [17, 28] on the
restored frame-based image to generate an optical flow prediction. Contrary to
[23], we explore an end-to-end learning approach that can directly process
event streams and frame-based images for predicting final outputs while
skipping the image restoration step. Moreover, our proposed method utilizes
all available event streams as well as frame-based images within a time
window, enabling accurate optical flow estimations over longer time windows.

##  3 Method

###  3.1 Sensors and Input representation

####  3.1.1 Frame-based Camera

Frame-based cameras have been widely popular for computer vision applications.
They provide dense and highly accurate pixel intensity information as frames
over regular time intervals. In general, the frame intensity information is
pivotal in various computer vision applications that require the high degree
of accuracy such as face and object recognition [20]. Optical flow estimation
using ANNs requires consecutive frame-based images to pass through separate
input channels to the network. In our work, this input representation is
utilized for the ANN part of Fusion-FlowNet (Sec. 3.3).

####  3.1.2 Event-based Camera

Event-based cameras are novel vision sensors, emulating the functionality of
biological retina cells [19]. Event cameras transmit a stream of asynchronous
events as the outcome of tracking intensity changes ($I$) at each pixel
element, thereby capturing the relative motion of objects in the scene.
Whenever the logarithmic intensity change at a pixel element surpasses a
specified threshold ($\theta$), a discrete event is asynchronously generated
as follows:

$\|\log(I_{t+1})-\log(I_{t})\|\geq\theta$ (1)

Event cameras provide the data in Address Event Representation (AER) format
which incorporates a tuple $\\{x,y,t,p\\}$, comprising the pixel address ($x$
and $y$ locations), timestamp ($t$) and polarity of the intensity change
($p$). Here, each ON/OFF polarity corresponds to the increase or decrease in
intensity of the pixel, respectively.

Event cameras may not be generally suited for vision applications which need
precise intensity information. However, their high temporal resolution and
high dynamic range in addition to having low power consumption, make them
ideal for usage on resource constrained platforms operating in challenging
environments. Optical flow estimation is one such task which heavily suffers
in such environments when realized using standard frame-based cameras and can
greatly benefit with the usage of event-cameras.

In our work, the raw event stream is transformed into two groups (former and
latter) of discretized event frames and are passed as inputs to the SNN part
of Fusion-FlowNet (Sec. 3.3). The input to the SNN encoder-branch consists of
a sequence of event frames with four channels, each from the ON/OFF polarity
of event frames from the former and the latter groups as illustrated in Fig.
1. This representation preserves the spatio-temporal information in the event
stream while displaying superior algorithmic performance and high energy-
efficiency.

Figure 1: ($left$) Asynchronous raw event stream between two consecutive
frame-based images. ($right$) Discretized event-frames between two consecutive
frame-based images to shape the former and latter groups of events.

####  3.1.3 Sensor-fusion

Interestingly, numerous available sensors, including the Dynamic and Active
Vision Sensor (DAVIS) [4], are capable of simultaneously generating the
asynchronous events as well as synchronous grayscale frames, simplifying the
hardware costs of sensor-fusion. In addition, since there is a single camera
coordinate system for both data modalities, the requirements for any expensive
transformation and synchronization between multiple coordinate systems are
eliminated. For this purpose, we employ the DAVIS sensor in this work.

In our work, the frame-based images serve two objectives. First, they are
provided as network inputs and allow for dense optical flow predictions.
Second, they are used for constructing the unsupervised loss required for
training. On the other hand, the event streams are only provided as network
inputs and enable accurate optical flow prediction in challenging environments
as discussed previously. The proposed sensor fusion framework would thus allow
to accurately estimate dense optical flow.

###  3.2 Neuron Models

The primary difference between ANN and SNN operations is the notion of time.
While ANNs feed-forward the dense analog-valued inputs at once, SNNs process
the sparse binary inputs (spikes) as a function of time. Accordingly,
different neuron models are employed in ANNs and SNNs.

####  3.2.1 LeakyReLU Model

In ANNs, LeakyReLU neuron [31] replaces the negative part of the popular ReLU
model by a linear function with a relatively small slope as below:

$y=\begin{cases}x,&\text{if~{}$x>0$}\\\ \alpha x,&\text{otherwise}\\\
\end{cases}$

where $\alpha$ is typically set to 0.01-0.1. Note, ReLU has a “dead neuron”
problem that some neurons could get stuck in the negative side and play no
role in discriminating between inputs. LeakyReLU addresses this problem by
having a non-zero slope in the negative direction. This makes it useful
especially for hard regression tasks such as motion estimation and predicting
pixel-wise and high resolution outputs. In our work, LeakyReLU is employed for
the ANN part of Fusion-FlowNet.

Figure 2: Dynamics of Signed Integrate-and-Fire (SIF) neuron model. Whenever
the membrane potential crosses either positive- or negative-threshold, the
neuron fires a signed spike output and resets its membrane potential. Figure
3: Detailed illustration of Fusion-FlowNet${}_{\text{Early}}$. The network
contains the SNN- and ANN-based encoder-branches to extract features from
asynchronous event streams and synchronized grayscale images, respectively.
The rest of networks, involving residual and decoder blocks, are composed of
ANN layers. The colors represent the types of layers. Best viewed in color.

####  3.2.2 Signed Integrate-and-Fire (SIF) Model

Spiking neurons are inspired by biological models for emulating the efficient
event-based operations in the human brain. In the literature, the Integrate-
and-Fire (IF) neuron model [5] is widely used for building SNNs because of its
simplicity. In an IF neuron, input spikes are modulated by weight ($w$) and
accumulated in an internal state of the neuron, called membrane potential over
time. In the discrete time model, whenever the membrane potential ($v$)
crosses a firing threshold, the neuron emits a binary output ($1$ or $0$) and
resets the membrane potential as follows,

$v^{l}[n+1]=v^{l}[n]+w^{l}o^{l-1}[n]$ (3)

where $o^{l-1}[n]$ indicates the spike output from previous layer at time-step
$n$. However, the IF neuron would also suffer from the “dead neuron” problem.
To address this issue, we propose a Signed Integrate-and-Fire (SIF) neuron
model that can generate signed spike outputs. The SIF neuron is equipped with
positive and negative thresholds that enable the generation of positive- and
negative-valued spike outputs, respectively. This operation is illustrated in
Fig. 2 and formulated as follows:

$o^{l}=\begin{cases}+1,&\text{if~{}$v^{l}>v_{th,pos}$}\\\
-1,&\text{elif~{}$v^{l}<v_{th,neg}$}\\\ 0,&\text{otherwise}\end{cases}$

However, the discontinuous and non-differentiable spike generation function of
SIF model poses a critical challenge for conventional gradient-based learning.
To overcome this challenge, we propose a surrogate gradient method for the SIF
neuron to enable end-to-end backpropagation (discussed in Sec. 3.5).

###  3.3 Fusion-FlowNet Architecture

The Fusion-FlowNet incorporates a deep fused network architecture that
supports an end-to-end learning. It is built upon the U-Net architecture [25]
that contains four encoder layers, two residual blocks and four decoder
layers. The distinctions in our work involve the addition of dual pathways
starting at the encoder, namely the SNN- and ANN-based branches. Each branch
is composed of narrow convolution layers (similar to grouped convolutions used
in AlexNet [14]) containing half the number of intermediate feature maps,
compared to the original wide convolution layers. This is possible because of
the usage of different modalities of input data, leading to reduction in
network parameters without compromising on qualitative performance.

In the SNN-based encoder-branch, the four-channeled input event frames
sequentially pass through the narrow convolution layers consisting of SIF
neurons over time while being downsampled at each layer. At every time-step,
the weighted spike outputs from each layer are integrated into the
corresponding output accumulator. After passing all consecutive event images,
the accumulated output is passed on ahead to subsequent layers.

In the ANN-based encoder-branch, the consecutive frame-based images in the
time window pass through the narrow ANN layers in a single time step. Each ANN
layer comprises of a convolution, batch-norm [11] and a LeakyReLU layer. Here
too, the feature maps are downsampled at each layer.

After completing the forward propagation in both encoder-branches, the outputs
are fused together before passing through the rest of the network. This is
achieved by concatenating the intermediate activations from both the SNN and
ANN branches at the same spatial locations. The fused activations from the
last encoder layer pass through the residual blocks while the fused
intermediate encoder outputs serve as input to corresponding layers in the
decoder block. The four layers in the decoder block perform upsampling using
transposed convolutions as well as produce multi-scale optical flow
predictions. The multi-scale flow predictions, the transposed convolution
outputs and the corresponding activations from the encoder layers are all
concatenated together to construct the input for the next decoder layer.
Finally, a full-scale optical flow prediction having the same dimension as the
input frames is made at the final decoder layer. Fig. 3 showcases the proposed
Fusion-FlowNet architecture, illustrating the discussed operations.

###  3.4 Unsupervised Training Method

Due to the limited availability of event-camera datasets containing ground-
truth labels, we adopt an unsupervised approach to train the optical flow
estimation network [13]. Fusion-FlowNet is trained using unlabeled sequences,
utilizing frame-based images for computing the loss. The overall loss function
is composed of two parts:

$l_{\text{total}}=l_{\text{photo}}+\lambda l_{\text{smooth}}$ (5)

where $l_{photo}$ and $l_{smooth}$ represent photometric loss and smoothness
loss respectively, and $\lambda$ denotes the loss weight factor.

####  3.4.1 Photometric Loss

Photometric loss helps to realize the object motion over time by tracking the
pixel intensities between images. It is computed by using the start and end-
frame grayscale images $(I_{t}(x,y)$, $I_{t+dt}(x,y))$ as well as the
predicted optical flow. A spatial transformer [12] inversely warps the end-
frame image $(I_{t+dt}(x,y))$ using the current estimated optical flow ($u,v$)
to obtain an image prediction $(I_{t+dt}(x+u,y+v))$. Then, the photometric
loss ($l_{photo}$) aims to minimize the discrepancy between the start-frame
image $(I_{t}(x,y))$ and the image prediction $(I_{t+dt}(x+u,y+v))$. The
computation is as follows:

$l_{photo}=\sum_{x,y}\rho(I_{t}(x,y)-I_{t+dt}(x+u,y+v))$ (1)

where $I_{t}$ $(I_{t+dt})$ indicates the pixel intensity of the first (last)
frame-based image, $u$, $v$ are the flow estimates in the $x,y$ directions,
$\rho$ is the robust Charbonnier loss $\rho(x)=(x^{2}+\eta^{2})^{r}$ used for
outlier rejection [27]. We set $r$ = $0.45$ and $\eta$ = $1e^{-3}$ as they
show optimal results in prior works [32, 15].

####  3.4.2 Smoothness Loss

Smoothness loss ($l_{smooth}$) is applied to reduce the optical flow
deviations between neighboring pixels by adding a regularizing effect on the
predicted flow. It is computed as follows:

$l_{smooth}=\sum_{j}\sum_{i}(\|u_{i,j}-u_{i+1,j}\|+\|u_{i,j}-u_{i,j+1}\|\\\
+\|v_{i,j}-v_{i+1,j}\|+\|v_{i,j}-v_{i,j+1}\|)$ (2)

where $u_{i,j}$ and $v_{i,j}$ are the flow estimates at pixel location ($i,j$)
in the $x$ and $y$ directions, respectively.

###  3.5 Backpropagation in Fusion-FlowNet

After forward propagation, the final loss ($l_{total}$) is evaluated and used
to perform the backward propagation of the gradients. In ANN layers, the
LeakyReLU is a differentiable activation that can be represented by the linear
functions where the slope differs in positive and negative parts of input as
shown in $left$ of Fig. 4. The derivative of LeakyReLU activation
($\pderiv{f(x)}{x}$) is unity when input is positive, $\alpha$ when input is
negative, and zero otherwise. Hence, standard backpropagation can calculate
the gradient of the loss function with respect to each weight using chain
rule. The parameter updates for the $l^{th}$ ANN layer are described as
follows:

$\triangle
w^{l}_{ANN}=\pderiv{\text{loss}}{f(x^{l})}\pderiv{f(x^{l})}{o^{l}}\pderiv{o^{l}}{w^{l}}$
(8)

By contrast, the spike generation mechanism of SIF neuron results in a hard
threshold function, making it discontinuous and non-differentiable. Hence,
standard backpropagation cannot be directly applied to SNNs in its native form
as illustrated in $right$ of Fig. 4. To overcome this impediment, we present a
surrogate gradient method for approximately estimating the spike generation
function of SIF neuron. The surrogate gradient of SIF model is herein computed
as follows:

$\pderiv{o[n]}{v[n]}=\begin{cases}\frac{1}{V_{th,pos}},&\text{if~{}$v^{l}>v_{th,pos}$}.\\\
\frac{1}{V_{th,neg}},&\text{if~{}$v^{l}<v_{th,neg}$}.\\\
0,&\text{otherwise.}\end{cases}$

where each threshold ($V_{th,pos},V_{th,neg}$) accounts for the change in the
signed spike outputs with respect to the inputs.

Figure 4: Illustration of activation functions ($left$) LeakyReLU neuron
($right$) Signed Integrate-and-Fire (SIF) neuron.

During the backward pass, the errors ($\pderiv{l_{total}}{o^{l}}$) are
backpropagated through the SNN layers using the surrogate gradient method and
BackPropagation Through Time (BPTT) [30]. In BPTT, the network is unrolled for
all time-steps and the weight update is assessed as the sum of gradients over
each time-step. The parameter updates of the $l^{th}$ SNN layer are described
as follows:

$\triangle
w^{l}_{SNN}=\sum_{n}\pderiv{\text{loss}}{o^{l}[n]}\pderiv{o^{l}[n]}{v^{l}[n]}\pderiv{v^{l}[n]}{w^{l}}$
(10)

Figure 5: Predicted optical flow compared with other state-of-the-art methods.
EV-FlowNet [32] and Spike-FlowNet [15] use only the event stream as input.
Full-fledged ANN uses only grayscale images as input. Fusion-FlowNet uses both
the event stream as well as grayscale images. The samples are taken from
($top$) $outdoor\\_day1$ and ($bottom$) $indoor\\_flying2$. Best viewed in
color.

##  4 Experiments

###  4.1 Dataset and Training Details

We validate Fusion-FlowNet on the MVSEC dataset [33] which contains events as
well as grayscale frame sequences recorded using the DAVIS346 camera [4] in
multiple indoor and outdoor environments. We use three indoor_flying sequences
and two outdoor_driving sequences. The indoor_flying sequences were collected
using a drone flying in a closed room containing a variety of objects and are
used mainly for evaluation. The outdoor_day sequences were recorded from a car
driving on public roads. We employ the outdoor_day2 sequence for training and
outdoor_day1 sequence for evaluation. The training and evaluation are
performed for two different time-window lengths (i.e, dt=1 and dt=4). Every
consecutive pair of grayscale images encapsulate an event volume where dt=1
corresponds to constructing inputs using one such event volume while dt=4
corresponds to using four such volumes.

The event streams and frame-based images from left-camera are used for
training. They are pre-processed by randomly cropping to $256$$\times$$256$
size and flipping horizontally and vertically (with $0.5$ probability). The
learning rate is scaled by $0.7$ every $5$ epochs until $20$ epoch, and every
$10$ epochs thereafter. The number of event frames in each group are set to
$5$ for the $dt=1$ case and $20$ for the $dt=4$ case. In ANN layers, LeakyReLU
model is employed with an $\alpha$ of $0.1$. In SNN layers, the positive and
negative thresholds of the SIF neuron are set to $0.75$ and $7.5$,
respectively. The loss weight factor $\lambda$ is set to $0.0003$.

Table 1: $\text{AEE}_{event}$ comparison with previous works

|  | dt=1 frame |  | dt=4 frame
---|---|---|---|---
$\text{AEE}_{event}$ |  | ind1 | ind2 | ind3 | out1 |  | ind1 | ind2 | ind3 | out1
Zhu et al.’19 |  | 0.58 | 1.02 | 0.87 | 0.32 |  | 2.18 | 3.85 | 3.18 | 1.30
EV-FlowNet |  | 1.03 | 1.72 | 1.53 | 0.49 |  | 2.25 | 4.05 | 3.45 | 1.23
Spike-FlowNet |  | 0.84 | 1.28 | 1.11 | 0.49 |  | 2.24 | 3.83 | 3.18 | 1.09
Fusion-FlowNet |  | 0.56 | 0.95 | 0.76 | 0.59 |  | 1.68 | 3.24 | 2.43 | 1.17

###  4.2 Evaluation of Optical Flow

For evaluation, the center cropped images of $256$$\times$$256$ size are taken
from indoor_flying1,2,3 and outdoor_day1 sequences. For indoor_flying
sequences, events and grayscale frames corresponding to the entire sequences
are used for evaluation. However, for outdoor_day1 sequence, 800 grayscale
frames and the asscociated event streams are used for evaluation as suggested
in [15, 32]. For quantitative results, we calculate the standard Average End-
point Error (AEE) which is the mean Euclidean distance between the estimated
flow ($y_{\text{estim}}$) and the provided ground-truth ($y_{\text{gt}}$). In
our work, we measure the two types of AEE results: (1) over all pixels
($\text{AEE}_{all}$) and (2) over pixels where events are present within the
time-window ($\text{AEE}_{event}$).

$\text{AEE}=\frac{1}{m}\sum_{\text{m}}\left\|(u,v)_{\text{estim}}-(u,v)_{\text{gt}}\right\|_{2}$
(11)

where $m$ indicates the count of active pixels in the event frames for
$\text{AEE}_{event}$ and every pixels of images for $\text{AEE}_{all}$.

Table 2: Average Endpoint Error (AEE) results for ablation studies

|  | indoor1 |  | indoor2 |  | indoor3 |  | outdoor1
---|---|---|---|---|---|---|---|---
|  | dt=1 | dt=4 |  | dt=1 | dt=4 |  | dt=1 | dt=4 |  | dt=1 | dt=4
|  | event | all | event | all |  | event | all | event | all |  | event | all | event | all |  | event | all | event | all
Fusion-FlowNet${}_{\text{Early}}$ |  | 0.56 | 0.62 | 1.68 | 1.81 |  | 0.95 | 0.89 | 3.24 | 2.90 |  | 0.76 | 0.85 | 2.43 | 2.46 |  | 0.59 | 1.02 | 1.17 | 3.06
Fusion-FlowNet${}_{\text{Late}}$ |  | 0.57 | 0.63 | 1.71 | 1.89 |  | 0.99 | 0.92 | 3.26 | 2.93 |  | 0.79 | 0.87 | 2.46 | 2.54 |  | 0.55 | 1.00 | 1.34 | 3.48
Fusion${}_{\text{Early}}$ [IF model] |  | 0.56 | 0.62 | 1.72 | 1.93 |  | 0.97 | 0.90 | 3.36 | 3.07 |  | 0.78 | 0.87 | 2.51 | 2.63 |  | 0.58 | 1.04 | 1.37 | 3.52
Fusion${}_{\text{Late}}$ [IF model] |  | 0.57 | 0.64 | 1.71 | 1.90 |  | 1.00 | 0.93 | 3.41 | 3.08 |  | 0.80 | 0.88 | 2.56 | 2.64 |  | 0.55 | 0.99 | 1.38 | 3.53
Spike-FlowNet |  | 0.84 | 0.91 | 2.24 | 2.94 |  | 1.28 | 1.23 | 3.83 | 4.09 |  | 1.11 | 1.20 | 3.18 | 3.92 |  | 0.49 | 1.42 | 1.09 | 3.28
Full-fledged ANN |  | 0.60 | 0.68 | 1.73 | 1.90 |  | 1.00 | 0.97 | 3.35 | 3.03 |  | 0.83 | 0.97 | 2.52 | 2.62 |  | 0.83 | 1.53 | 1.27 | 3.19

###  4.3 Results

We compare Fusion-FlowNet with previous state-of-the-art works [32, 34, 15] in
terms of the performance of optical flow prediction. As listed in Table 1,
only $\text{AEE}_{event}$ results are compared here since other works do not
provide AEE values for dense optical flow estimation. We observe that Fusion-
FlowNet outperforms other implementations in almost all scenarios. The
outdoor_day1 sequence is known to have suffered from certain issues with its
grayscale images during dataset creation, leading to anomalous results for
$\text{AEE}_{event}$ as well as $\text{AEE}_{all}$. We report the results for
it to maintain completeness in terms of comparison with previous works. Fig. 5
visualizes the predicted flow for this work and compares it with previous
state-of-the-art methods.

###  4.4 Ablation studies

####  4.4.1 Architectural Variations

We perform an ablation study to analyze the effect of architectural variations
on model performance and efficiency. We investigate a second architecture
where the dual pathway branches are extended to residual blocks. As shown in
Fig. 6, we denote the first architecture as Fusion-FlowNet${}_{\text{Early}}$
and the second architecture as Fusion-FlowNet${}_{\text{Late}}$. Rows
$1$$-$$2$ in Table 2 highlight the optical flow prediction capability of both
the architectures. We find that Fusion-FlowNet${}_{\text{Early}}$ outperforms
Fusion-FlowNet${}_{\text{Late}}$ in predicting accurate optical flow outputs.
Fusion-FlowNet${}_{\text{Early}}$ contains comparatively larger number of
parameters and fuses the intermediate features from the ANN/SNN branches in
early layers, leading to better AEE results. On the other hand, Fusion-
FlowNet${}_{\text{Late}}$ performs the fusion at later layers leading to
promising advantages in further reducing the network parameters and
computational energy cost, as shown in Table 3.

Figure 6: Architectures of ($left$) Fusion-FlowNet${}_{\text{Early}}$ and
($right$) Fusion-FlowNet${}_{\text{Late}}$. Best viewed in color.

####  4.4.2 Neuron Model Choice

For investigating the benefits of proposed SIF neuron, we compare the
variations of Fusion-FlowNet with the SNN blocks composed of SIF and IF neuron
models. Rows $3$$-$$4$ in Table 2 provide the AEE results for Fusion-FlowNet
with IF neurons in the SNN layers. A comparison with results in rows $1$$-$$2$
show that networks using SIF model can predict more accurate flow outputs
compared to networks using IF model. This establishes the benefit of the SIF
model towards mitigating the “dead neuron” problem in deep SNN layers.

Table 3: Comparison of number of parameters and computational energy cost for
different architectures for dt=1 and dt=4 cases. Lowest numbers highlighted in
bold. (results averaged over all indoor and outdoor1 sequences)

|  | #Parameters ($\times 10^{6}$) |  |  #OPS${}_{\text{ANN}}$ ($\times 10^{9}$) |  | Spiking Activity ($\%$) |  |  #OPS${}_{\text{SNN}}$ ($\times 10^{6}$) |  |  E${}_{\text{Total}}$ (mJ) |  | Improvement
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | dt=1 | dt=4 |  | dt=1 | dt=4 |  | dt=1 | dt=4 |  | dt=1 | dt=4 |  | dt=1 | dt=4 |  | dt=1 | dt=4
Full-fledged ANN |  | 13.044 | 13.046 |  | 5.339 | 5.367 |  | – | – |  | – | – |  | 24.536 | 24.666 |  | 1.00$\times$ | 1.00$\times$
Spike-FlowNet |  | 13.039 | 13.039 |  | 4.409 | 4.409 |  | 0.480 | 1.008 |  | 15.81 | 195.99 |  | 20.296 | 20.458 |  | 1.21$\times$ | 1.21$\times$
Fusion-FlowNet$\\_{\text{Early}}$ |  | 12.269 | 12.270 |  | 4.648 | 4.648 |  | 0.173 | 0.174 |  | 1.03 | 4.18 |  | 21.381 | 21.384 |  | 1.15$\times$ | 1.15$\times$
Fusion-FlowNet$\\_{\text{Late}}$ |  | 7.549 | 7.550 |  | 2.849 | 2.849 |  | 0.147 | 0.179 |  | 5.24 | 6.44 |  | 13.113 | 13.114 |  | 1.87$\times$ | 1.88$\times$

####  4.4.3 Sensor Fusion

We study the usefulness of sensor fusion approach against single sensor
approaches using inputs as either the event streams or frame-based images. For
the event only approach, we investigate Spike-FlowNet [15], a hybrid neural
architecture where the initial layers are composed of SNNs and the deeper
layers are composed of ANNs. Note, Spike-FlowNet utilizes the similar event-
based input representation scheme and unsupervised learning method, providing
a fair comparison. For the frame-based image only approach, we implement a
custom full-fledged ANN architecture that resembles the U-Net [25]
architecture, and train it with the equivalent unsupervised method as Fusion-
FlowNet.

Rows $5$$-$$6$ of Table 2 summarize the results for the single sensor
approaches. Unsurprisingly, both Fusion-FlowNet${}_{\text{Early}}$ and Fusion-
FlowNet${}_{\text{Late}}$ achieve better AEE performances in dt=1 and dt=4
scenarios compared to single sensor approaches. This verifies that the
proposed fusion approach benefits from utilizing the complementary
characteristics of event- and frame-based images, leading to better
performance in both slow- and fast-motion scenarios. Furthermore, in
comparison to prior works as listed in Table 1, both fusion options provide
superior AEE results.

###  4.5 Computational Efficiency

We validate the efficiency of Fusion-FlowNet in terms of the number of network
parameters and computational energy cost for inference. Table 3 provides a
detailed analysis on computational efficiency along with the comparison with
previously discussed alternate architectures.

We observe that both Fusion-FlowNet${}_{\text{Early}}$ and Fusion-
FlowNet${}_{\text{Late}}$ contain fewer number of parameters compared to a
full-fledged ANN architecture and Spike-FlowNet. This is due to the usage of
narrow convolution layers which greatly reduce the number of parameters and
computations. In particular, Fusion-FlowNet${}_{\text{Late}}$ contains the
least number of network parameters ($\sim 58\%$ compared to full-fledged ANN)
as the residual blocks contain the majority of the parameters and utilizing
narrow convolutional layers for them helps reduce the total network parameters
drastically.

For estimating the computational energy cost for different architectures, we
first describe how computations in SNNs and ANNs differ from each other.
Conceptually, SNNs perform highly sparse asynchronous ACcumulate (AC)
operations over time. These synaptic operations are executed only at the
arrival of input spikes due to the nature of binary-valued inputs. In
contrast, ANNs perform expensive Multiply-and-ACcumulate (MAC) operations for
computing dense Matrix-Vector Multiplications (MVMs). Based on the findings in
[10], a MAC operation requires a total of $E_{MAC}$=$4.6pJ$ of energy while an
AC operation requires only $E_{AC}$=$0.9pJ$ for a 32-bit floating-point
computation (45nm CMOS technology). This leads to the AC operation being
$~{}5.1\times$ more energy-efficient compared to the MAC operation. These
findings coupled with the number of synaptic operations are commonly used to
benchmark the computational energy cost of SNNs. [21, 26, 15].

Next, we calculate the total number of synaptic operations for every layer. In
the SNN layers, the number of synaptic operations are obtained by multiplying
the pre-spike activities, the number of synaptic connections and the number of
time-steps. Also, the computational energy of AC and MAC computations are
taken into consideration for SNNs and ANNs, respectively. The total
computational energy cost can be formalized as:

$\text{\\#OPS}_{\text{SNN}}=N\sum_{l}M_{l}C_{l}F_{l},\;\text{\\#OPS}_{\text{ANN}}=\sum_{l}M_{l}C_{l}\\\
$ (12)
$\text{E}_{\text{Total}}=\text{\\#OPS}_{\text{SNN}}\times\text{E}_{\text{AC}}+\text{\\#OPS}_{\text{ANN}}\times\text{E}_{\text{MAC}}\\\
$ (13)

where $M$ is the number of neurons, $C$ is the number of synaptic connections,
$F$ represents mean spiking activity, $N$ is the number of timesteps,
$\text{\\#OPS}_{\text{SNN}}$/$\text{\\#OPS}_{\text{ANN}}$ indicate the number
of operations for SNN/ANN portions, and $\text{E}_{\text{Total}}$ denotes the
total computational energy cost.

The last column in Table 3 provides the overall improvement in computational
energy cost. We observe that Fusion-FlowNet${}_{\text{Late}}$ demonstrates the
highest improvement in energy ($\sim 1.88\times$) compared to full-fledged
ANN. This is because more layers, including encoder and residual blocks,
utilize narrow convolutions, leading to reduction in the number of parameters
and consequently reduction in the total computational energy cost.
Furthermore, the SNN pathway contributes negligibly to the total computational
energy cost compared to ANN pathway.

##  5 Conclusion

We propose a sensor/architecture fusion framework for accurately estimating
optical flow in challenging environments. We leverage the complementary
characteristics of event- and frame-based sensors as well as ANNs and SNNs.
Our framework (Fusion-FlowNet) reports state-of-the-art optical flow
prediction results, while substantially reducing network parameters and
computational energy cost. This work contributes two different deep fused
architectures (Fusion-FlowNet${}_{\text{Early}}$ and Fusion-
FlowNet${}_{\text{Late}}$), having different applications of interest. Fusion-
FlowNet${}_{\text{Early}}$ provides highly accurate dense optical flow,
proving to be appropriate for safety-critical applications. While, Fusion-
FlowNet${}_{\text{Late}}$ promises immense benefits in terms of computational
efficiency, making it suitable for the edge applications on resource-
constrained hardware.

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# Anisotropic Ising model in $d+s$ dimensions

Estevão F. Borel, Aldo Procacci∗, Rémy Sanchis∗, Roger W. C. Silva
Departamento de Matemática, Universidade Federal de Minas Gerais, Belo
Horizonte, Brazil.
Departamento de Estatística, Universidade Federal de Minas Gerais, Belo
Horizonte, Brazil.

###### Abstract

In this note, we consider the asymmetric nearest neighbor ferromagnetic Ising
model on the $(d+s)$-dimensional unit cubic lattice ${\mathbb{Z}}^{d+s}$, at
inverse temperature $\beta=1$ and with coupling constants $J_{s}>0$ and
$J_{d}>0$ for edges of ${\mathbb{Z}}^{s}$ and ${\mathbb{Z}}^{d}$,
respectively. We obtain a lower bound for the critical curve in the phase
diagram of $(J_{s},J_{d})$. In particular, as $J_{d}$ approaches its critical
value from below, our result is directly related to the so-called dimensional
crossover phenomenon.

Keywords: random current representation; anisotropic Ising model; dimensional
crossover

AMS 1991 subject classification: 82B20; 82B26

##  1 Introduction

In this paper, we consider an asymmetric nearest neighbor ferromagnetic Ising
model in the $(d+s)$-dimensional unit cubic lattice $\mathbb{Z}^{d+s}$, with
coupling constants $J_{d}>0$ and $J_{s}>0$ in the hyperplanes $\mathbb{Z}^{d}$
and $\mathbb{Z}^{s}$, respectively. Anisotropic lattice spin systems have been
the subject of great interest within the physics community since the sixties.
The study of such models has been tackled both numerically, mainly via Monte
Carlo simulations (see e.g. [11, 12, 13, 23] and references therein), and
theoretically, via mean-field, Bethe approximation, truncated high-temperature
expansion of the susceptibility, etc… (see e.g. [14, 15, 17, 18, 20, 21, 23,
25]). A strong motivation to study anisotropic systems is to investigate
finite-size effects in realistic materials modeled by quasi-two-dimensional
(thin films) and quasi-one-dimensional spin systems. Furthermore, exploring
these systems could provide valuable insights into isotropic models, notably
the three-dimensional Ising model (see. e.g. [24] and references therein).

Rigorous results on the asymmetric Ising model on $\mathbb{Z}^{d+s}$ have been
obtained mainly in the case $\mathbb{Z}^{1+s}$ with strong coupling in one
dimension and small coupling in the remaining directions. In particular, in a
well-known article (see [10]), Fisher derived an asymptotic bound on the
critical temperature of the $\mathbb{Z}^{1+s}$ anisotropic Ising model in the
limit $\frac{J_{s}}{J_{1}}\to 0$. It has also been shown rigorously (see [16])
that the free energy of the Ising model on $\mathbb{Z}^{1+s}$ is analytic for
any inverse temperature $\beta$ if $J_{s}$ is small enough (depending on
$J_{1}$ and the inverse temperature). These rigorous results rely heavily on
the fact that when $d=1$ the one-dimensional system is in the gas phase at all
temperatures so that the standard high-temperature expansion can be used with
effectiveness. On the other hand, to obtain the same kind of results of
references [10, 16] in the case $\mathbb{Z}^{d+s}$, $d\geq 2$, is expected to
be trickier since the $d-$dimensional system exhibits a phase transition and
the usual high-temperature expansion turns to be much more difficult to
control.

In this paper, we somehow extend the results obtained in [10] and [16] for the
case $\mathbb{Z}^{1+s}$ to the case $\mathbb{Z}^{d+s}$ with $d\geq 1$. Namely,
we show that for all $J_{d}$ below the critical reduced temperature
$J_{d}^{c}$, the susceptibility of the $(d+s)$-dimensional system is finite
when the coupling $J_{s}$ is sufficiently small (inversely proportional to the
susceptibility $\chi(J_{d})$ of the $d$-dimensional system). A similar result
was obtained by two of us in [19] for the Bernoulli anisotropic bond
percolation model on $\mathbb{Z}^{d+s}$, in which edges in the
$\mathbb{Z}^{d}$ hyperplane are open with probability $p<p_{c}(d)$ and edges
parallel in the $\mathbb{Z}^{s}$ hyperplane are open with probability $q$. In
[19], probabilistic arguments were applied and in particular, a crucial use of
the van den Berg–Kesten (BK) inequality has been made. In the present paper,
to get the analogous result for the anisotropic Ising model in
$\mathbb{Z}^{d+s}$ (for which BK inequalities are not available), we use an
alternative (w.r.t. the high temperature) expansion of the Ising partition
function, namely the so-called random current representation. This powerful
technique, introduced in the eighties by Aizenman [2], was widely used by
several authors in the following decades, and recently it has employed as a
crucial tool in several remarkable papers, e.g., [3], [4], [5].

As mentioned earlier, our results may be of interest in the study of realistic
quasi-two-dimensional magnets which can be modeled by a two-dimensional sub-
critical Ising bilayer. When the interaction between the two two-dimensional
layers is weak and the transverse interaction is sub-critical, the bilayer
system is expected to still exhibit sub-critical behavior. However, as the
coupling between the layers strengthens, the overall system may exhibit
spontaneous magnetization. Our result implies rigorously that as long as the
inter-layer interaction between layers is below the inverse two-dimensional
susceptibility (with a constant factor 1/(2s)), the global system remains sub-
critical. The understanding of how several layers of 2D slightly sub-critical
systems with small interactions between them can start to behave as a
$(2+s)$-dimensional system, is the so-called dimensional crossover phenomenon
(see e.g. Sec. VI in [15]). This phenomenon is characterized by a critical
exponent, which is believed to depend on the original dimension of the layers,
but not on the target dimension. Moreover, numerical simulations and formal
calculations (see e.g. [15, 17]), suggest that this critical exponent equals
the exponent of the susceptibility of the original dimension. Our results
imply rigorously an inequality between the two exponents.

##  2 The model and results

Let ${\mathbb{Z}}^{d+s}={\mathbb{Z}}^{d}\times{\mathbb{Z}}^{s}$ be the
$(d+s)$-dimensional unit cubic lattice. We will denote by $\mathbb{E}^{d+s}$
the set of nearest neighbor pairs of $\mathbb{Z}^{d+s}$ so that
$\mathbb{G}^{d+s}$ is the graph with vertex set $\mathbb{Z}^{d+s}$ and edge
set $\mathbb{E}^{d+s}$. Given two vertices $x,y\in\mathbb{Z}^{d+s}$, we denote
by $|x-y|$ the usual graph distance between $x$ and $y$ (i.e. the edge-length
of the shortest path between $x$ and $y$). We will suppose that
$\mathbb{Z}^{d+s}$ is equipped with the usual operation of sum. We represent
hereafter a site $x\in\mathbb{Z}^{d+s}$ as $x=(u,t)$, where
$u\in\mathbb{Z}^{d}$ and $t\in\mathbb{Z}^{s}$.

Given an integer $N$, we denote by $\Lambda_{N}\subset\mathbb{Z}^{d+s}$ the
hypercube with side length $2N+1$, centered at the origin, so that
$\Lambda_{N}\to\infty$ means that $N\to\infty$. We denote by $E_{N}$ the set
of edges of $\mathbb{E}^{d+s}$ with both endpoints in $\Lambda_{N}$, so that
$\mathbb{G}^{d+s}|_{\Lambda_{N}}=(\Lambda_{N},E_{N}$) is the restriction of
$\mathbb{G}^{d+s}$ to $\Lambda_{N}$. Note that
$\Lambda_{N}=\bar{\Lambda}_{N}\times\hat{\Lambda}_{N}$ where
$\bar{\Lambda}_{N}$ denotes the $d$-dimensional hypercube in
${\mathbb{Z}}^{d}$ of size $2N+1$ centered at the origin and
$\hat{\Lambda}_{N}$ denotes the $s$-dimensional hypercube in
${\mathbb{Z}}^{s}$ of size $2N+1$ centered at the origin. Given
$w\in\hat{\Lambda}_{N}$, we set
$\Lambda^{w}_{N}=\\{(u,t)\in{\mathbb{Z}}^{d+s}:t=w\\}$. Namely
$\Lambda^{w}_{N}$ is the subset of $\Lambda_{N}$ formed by sites of
$\Lambda_{N}$ with $w$ as the second coordinate. Similarly, $E^{w}_{N}$ will
denote the set of edges with both endpoints in $\Lambda_{N}^{w}$. Observe that
$\Lambda_{N}^{w}$ is a $d$-dimensional box of side length $2N+1$ centered at
$(0,w)$.

To each vertex $x\in\Lambda_{N}$ we associate a random variable $\sigma_{x}$
taking values in the set $\\{+1,-1\\}$. A spin configuration in $\Lambda_{N}$
is a function $\bm{\sigma}:\Lambda_{N}\to\\{+1,-1\\}:x\mapsto\sigma_{x}$. The
energy of a configuration $\bm{\sigma}$ is given by the (free boundary
condition) Hamiltonian

$H_{\Lambda_{N}}(\bm{\sigma})=-\sum_{\\{x,y\\}\in
E_{N}}J_{\\{x,y\\}}\sigma_{x}\sigma_{y},$

where

$J_{\\{x,y\\}}=\begin{cases}J_{s}&{\rm if}\;\;x=(u,t)\;\;{\rm
and}\;\;y=(u,t^{\prime})\;\;{\rm with}\;\;|t-t^{\prime}|=1,\\\ J_{d}&{\rm
if}\;\;x=(u,t)\;\;{\rm and}\;\;y=(u^{\prime},t)\;\;{\rm
with}\;\;|u-u^{\prime}|=1,\end{cases}$

with $J_{s}>0$ and $J_{d}>0$. In what follows, an edge $\\{x,y\\}\in E_{N}$ is
called vertical if $x=(u,t)$, $y=(u,t^{\prime})$ with $|t-t^{\prime}|=1$, and
called planar if $x=(u,t)$, $y=(u^{\prime},t)$ with $|u-u^{\prime}|=1$. So
$J_{\\{x,y\\}}=J_{s}$ if $\\{x,y\\}$ is vertical and $J_{\\{x,y\\}}=J_{d}$ if
$\\{x,y\\}$ is planar.

The partition function of the system is given by

$Z_{\Lambda_{N}}(J_{d},J_{s})=\int
d\mu_{\Lambda_{N}}(\bm{\sigma})e^{-H_{\Lambda_{N}}(\bm{\sigma})}=\int
d\mu_{\Lambda_{N}}(\bm{\sigma})\prod_{b\in
E_{N}}e^{J_{b}\sigma_{x}\sigma_{y}},$

where $\int d\mu_{\Lambda_{N}}(\bm{\sigma})$ is a short notation for
$\prod_{x\in\Lambda_{N}}\frac{1}{2}\sum_{\sigma_{x}=\pm 1}$ (a product
probability measure). Moreover, without loss of generality, we have set the
inverse temperature $\beta=1$.

The two-point correlation function of the $(d+s)$-system is then defined as

$\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}=\frac{\int
d\mu_{\Lambda_{N}}(\bm{\sigma})\sigma_{x}\sigma_{y}e^{-H_{\Lambda_{N}}(\sigma_{\Lambda})}}{Z_{\Lambda_{N}}(J_{d},J_{s})}=\frac{\int
d\mu_{\Lambda_{N}}(\bm{\sigma})\sigma_{x}\sigma_{y}\prod_{\\{x,y\\}\in
E_{N}}e^{J_{b}\sigma_{x}\sigma_{y}}}{Z_{\Lambda_{N}}(J_{d},J_{s})}.$ (1)

In general, for any set $U\subset\Lambda_{N}$, letting
$E_{U}=\\{\\{x,y\\}\in\Lambda_{N}:\\{x,y\\}\subset U\\}$, we set

$H_{U}(\bm{s})=-\sum_{\\{x,y\\}\in E_{U}}J_{\\{x,y\\}}\sigma_{x}\sigma_{y},$
$Z_{U}(J_{d},J_{s})=\int
d\mu_{\Lambda_{N}}(\bm{\sigma})e^{-H_{U}(\bm{\sigma})},$

and, for any $x,y\in U$,

$\langle\sigma_{x}\sigma_{y}\rangle_{U}=\frac{\int
d\mu_{\Lambda_{N}}(\bm{\sigma})\sigma_{x}\sigma_{y}e^{-H_{U}(\sigma_{\Lambda})}}{Z_{U}(J_{d},J_{s})}.$

According to the above notations, for any $t\in\hat{\Lambda}_{N}$, we have
that

$\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda^{t}_{N}}=\frac{\int
d\mu_{\Lambda_{N}}(\bm{\sigma})\sigma_{x}\sigma_{y}e^{-H_{\Lambda^{t}_{N}}(\sigma_{\Lambda})}}{Z_{\Lambda_{N}^{t}}(J_{d},J_{s})},$

The finite volume susceptibility function of the system is defined as

$\chi_{\Lambda_{N}}(J_{d},J_{s})\coloneqq\sup_{x\in\Lambda_{N}}\Bigg{\\{}\sum_{y\in\Lambda_{N}}\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}\Bigg{\\}},$

so that

$\chi_{d+s}(J_{d},J_{s})=\lim_{N\to\infty}\chi_{\Lambda_{N}}(J_{d},J_{s})$ (2)

is the susceptibility of the anisotropic $(d+s)$-dimensional Ising model.

For any $t\in\hat{\Lambda}_{N}$, let

$\chi_{\Lambda_{N}^{t}}(J_{d})=\sup_{x\in\Lambda_{N}^{t}}\Bigg{\\{}\sum_{y\in\Lambda_{N}^{t}}\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}^{t}}\Bigg{\\}},$
(3)

so that

$\chi_{d}(J_{d})=\lim_{N\to\infty}\chi_{\Lambda_{N}^{t}}(J_{d})$ (4)

is the susceptibility of the $d$-dimensional Ising model with ferromagnetic
interaction $J_{d}$.

We are now in a position to state our main result.

######  Theorem 1.

Take any $J_{d}$ such that

$\chi_{d}(J_{d})<+\infty,$

and let $J_{s}$ such that

$\tanh(J_{s})<\frac{1}{2s\chi_{d}(J_{d})}.$

Then

$\chi_{d+s}(J_{d},J_{s})<+\infty.$

Remark. As mentioned in the introduction, Theorem 1 is related to the so-
called dimensional crossover phenomenon. Denoting by $J_{d}^{c}$ the critical
inverse reduced temperature of the $d$-dimensional system, one can define, for
any $J_{d}>0$, the function
$J_{s}^{c}(J_{d}):[0,\infty)\rightarrow[0,\infty)$, where

$J_{s}^{c}(J_{d})=\sup\\{J_{s}:\chi_{d+s}(J_{d},J_{s})<+\infty\\}.$ (5)

By definition, $J_{s}^{c}(J_{d})=0$ if $J_{d}\geq J^{c}_{d}$. Theorem 1
determines a region in the $(J_{d},J_{s})$ plane where no phase transition
occurs in the $(d+s)$-system and it also gives an upper bound for the function
$J_{s}^{c}(J_{d})$ as $J_{d}$ varies in the interval $[0,J_{d}^{c})$. There is
a strong interest in understanding the behavior of the function
$J_{s}^{c}(J_{d})$ when $J_{d}<J_{d}^{c}$, and in particular it is widely
believed that there exists a constant $\phi_{d}>0$ such that

$J_{s}^{c}(J_{d})\approx|J_{d}-J_{d}^{c}|^{\phi_{d}},~{}~{}{\rm
as}~{}~{}J_{d}\uparrow J_{d}^{c},$

where the symbol $\approx$ stands for log equivalence, that is, $f(x)\approx
g(x)$ if $\displaystyle\lim_{\beta\uparrow\beta_{c}}\frac{\log f(x)}{\log
g(x)}=1$. The constant $\phi_{d}$ is the so-called crossover critical
exponent. On the other hand, the $d$-dimensional Ising susceptibility
$\chi_{d}(J_{d})$ is known to behave like

$\chi_{d}(J_{d})\approx|J_{d}-J_{d}^{c}|^{-\gamma_{d}},\mbox{ as
}J_{d}\uparrow J_{d}^{c},$

where $\gamma_{d}>0$ is the susceptibility $d$-dimensional critical exponent.
As an immediate corollary, Theorem 1 implies that $\phi_{d}\leq\gamma_{d}$ for
all $d\geq 2$, and a lower bound for $J_{s}^{c}(J_{d})$ still inversely
proportional to $\chi_{d}(J_{d})$ would imply that $\phi_{d}=\gamma_{d}$. The
conjectured equality of the crossover critical exponent $\phi_{d}$ and
susceptibility critical exponent $\gamma_{d}$ has been discussed in the
physics literature by several authors, see for instance [1, 20, 13, 14, 15].

##  3 The random current representation

As mentioned in the introduction, in order to prove Theorem 1 we will use the
so-called random-current representation for the Ising model introduced by
Aizenman in [2]. We will describe this technique here below following mainly
reference [6].

Let $\mathfrak{F}(E_{N})$ be the set of all functions
$\eta:E_{N}\to\mathbb{N}:b\mapsto\eta_{b}$. Then we can expand the exponential
inside the product and rewrite the partition function
$Z_{\Lambda_{N}}(J_{d},J_{s})$ as

$Z_{\Lambda_{N}}(J_{d},J_{s})=\sum_{\eta\in\mathfrak{F}(E_{N})}W(\eta)\int
d\mu_{\Lambda_{N}}(\bm{\sigma})\prod_{x\in\Lambda_{N}}(\sigma_{x})^{\sum_{b\ni
x}\eta_{b}},$

where we have denoted shortly

$\sum_{b\ni x}\eta_{b}=\sum_{b\in E_{N}\atop x\in b}\eta_{b}$

and we have set

$W(\eta)=\prod_{b\in E_{N}}\frac{(J_{b})^{\eta_{b}}}{\eta_{b}!}.$

Observe that the integral $I(\eta)=\int
d\mu_{\Lambda_{N}}(\bm{\sigma})\prod_{x\in\Lambda_{N}}(\sigma_{x})^{\sum_{b\ni
x}\eta_{b}}$ is zero, unless $\sum_{b\ni x}\eta_{b}$ is even for all
$x\in\Lambda$, in which case $I(\eta)=1$. Hence

$Z_{\Lambda_{N}}(J_{d},J_{s})=\sum_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N})\\\
\partial\eta=\emptyset\end{subarray}}W(\eta),$

where we have set

$\partial\eta=\\{x\in\Lambda:\;\sum_{\begin{subarray}{c}b\ni
x\end{subarray}}\eta_{b}\;\mbox{is odd}\\}.$

In general, given any $\eta\in\mathfrak{F}(E_{N})$, the vertices in
$\partial\eta$ are called sources of $\eta$, and if $\partial\eta=\emptyset$,
then $\eta$ is called sourceless. Proceeding similarly we have that the
random-current expansion for the two-point function is

$\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}=\sum_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N})\\\
\partial\eta=\\{x,y\\}\end{subarray}}\frac{W(\eta)}{Z_{\Lambda_{N}}(J_{d},J_{s})}.$

We now rewrite the function $\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}$
as a sum of edge-self-avoiding walks from $x$ to $y$. Given an edge
$\\{x,y\\}\in E_{N}$, the ordered pair $(x,y)$ will be called a step from $x$
to $y$. For any $x\in\Lambda_{N}$, we establish an arbitrary order (denoted by
$\preceq$) for the set of steps emerging from $x$ (i.e. for those $(x,y)$ such
that $|x-y|=1$). For each site $x$, and each step $(x,z)$, we consider the set
$\Gamma_{(x,z)}$ formed by the edges $b=\\{x,y\\}$ such that
$(x,y)\preceq(x,z)$. This set will be referred to as the set of edges canceled
by $(x,z)$. In particular, since $(x,y)\preceq(x,y)$, a step $\\{x,y\\}$
cancels itself.

We recall that a path in $\Lambda_{N}$ is a sequence
$p=\\{x_{0},x_{1},...,x_{n}\\}$ of vertices of $\Lambda_{N}$ such that
$\\{x_{i-1},x_{i}\\}\in E_{N}$ for all $i=1,\cdots,n$. We say that a path
$p=\\{x_{0},x_{1},...,x_{n}\\}$ is consistent if, for each $k=1,...,n$, we
have that $\\{x_{k-1},x_{k}\\}\notin\cup_{i=1}^{k-1}\Gamma_{(x_{i-1},x_{i})}$.
That is, each step used in this path is not associated with an edge that was
canceled by the previous steps. If $p=\\{x_{0},x_{1},...,x_{n}\\}$ is a
consistent path, we denote by ${p}^{*}$ the set of all edges canceled by $p$,
that is $p^{*}=\cup_{i=1}^{n}\Gamma_{(x_{i-1},x_{i})}$. Clearly, by
construction, a consistent path is always edge-self-avoiding. We denote by
$C_{xy}(\Lambda_{N})$ the set of all consistent paths in $\Lambda_{N}$ from
$x$ to $y$.

We now define a function $\Omega$, which associates to each current
configuration $\eta$ with $\partial\eta=\\{x,y\\}$, a consistent path
${\omega}=\Omega(\eta)$ from $x$ to $y$, which belongs to
$C_{xy}(\Lambda_{N})$. As in [6], such a consistent path will be called the
backbone of $\eta$.

Given $\eta$ with $\partial\eta=\\{x,y\\}$, let $\Gamma_{\eta}$ be the set of
edges $b\in E$ such $\eta_{b}$ is odd. Then, $\Gamma_{\eta}$ forms a subgraph
of $(\Lambda_{N},E_{N})$ (in general not connected) such that every vertex has
degree either even or zero, except on $x$ and $y$, whose degrees are odd. The
graph $\Gamma_{\eta}$ necessarily contain a connected component, say
$\gamma_{\eta}^{x,y}$, which contains $x$ and $y$. Therefore, we can look at
this connected component $\gamma_{\eta}^{x,y}$ (seen as a set of edges in
$E_{N}$), uniquely determined by $\eta$, and associate to it a consistent path
${\omega}=\Omega(\eta)$. This is the path
${\omega}=\\{z_{0}=x,z_{1}\\},\\{z_{1},z_{2}\\},\dots,\\{z_{k-1},z_{k}=y\\}$
for some $k\geq|x-y|$ such that for any $i=1,2,\dots,k$, $(z_{i-1},z_{i})$ is
the minimal step according to the order established among the steps emerging
from $z_{i-1}$ associated to edges of
$\gamma_{\eta}^{x,y}-\\{z_{0},z_{1}\\}\cup\\{z_{1},z_{2}\\}\cup\dots\cup\\{z_{1-2},z_{i-1}\\}$.

Once the function $\Omega$ is defined, we now can rewrite
$\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}$ as

$\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}=\sum_{{\omega}\in
C_{xy}(\Lambda_{N})}\sum_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N})\\\
\partial\eta=\\{x,y\\},\Omega(\eta)={\omega}\end{subarray}}\frac{W(\eta)}{Z_{\Lambda_{N}}(J_{d},J_{s})}.$

Note that if $\Omega(\eta)={\omega}$, then $\eta$ is odd on the edges of the
set ${\omega}$ and is even on the edges of ${\omega}^{*}\setminus{\omega}$.
Also, $\eta$ restricted to $E_{N}\setminus{\omega}$, as well as to
$E_{N}\setminus{\omega}^{*}$, is such that $\partial\eta=\emptyset$.
Therefore, setting shortly $Z_{N}=Z_{\Lambda_{N}}(J_{d},J_{s})$, we have that

$\displaystyle\sum_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N})\\\
\partial\eta=\\{x,y\\},\Omega(\eta)={\omega}\end{subarray}}\frac{W(\eta)}{Z_{N}}$
$\displaystyle=\prod_{b\in{\omega}}\sinh(J_{b})\prod_{b\in{\omega}^{*}\setminus{\omega}}\cosh(J_{b})\sum_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N}\setminus{\omega}^{*})\\\
\partial\eta=\emptyset\end{subarray}}\frac{W(\eta)}{Z_{N}}$
$\displaystyle=\prod_{b\in{\omega}}\tanh(J_{b})\prod_{b\in{\omega}^{*}}\cosh(J_{b})\sum_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N}\setminus{\omega}^{*})\\\
\partial\eta=\emptyset\end{subarray}}\frac{W(\eta)}{Z_{N}}$
$\displaystyle=\prod_{b\in{\omega}}\tanh(J_{b})\left[\prod_{b\in{\omega}^{*}}\cosh(J_{b})\sum_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N}\setminus{\omega}^{*})\\\
\partial\eta=\emptyset\end{subarray}}\frac{W(\eta)}{Z_{N}}\right]$
$\displaystyle=\prod_{b\in{\omega}}\tanh(J_{b})\sum_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N}):\;\partial\eta=\emptyset\\\
\eta\;{\rm even\;on}\;{\omega}^{*}\end{subarray}}\frac{W(\eta)}{Z_{N}},$

where the last summation is over all sourceless current configurations $\eta$
on $E$ with the additional restriction that $\eta_{b}$ is even on all edges
$b$ canceled by $\omega$. Hence, we can rewrite
$\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda}$ as

$\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}=\sum_{{\omega}\in
C_{xy}(\Lambda_{N})}\rho_{E_{N}}(\omega),$ (6)

where

$\rho_{E_{N}}(\omega)=\prod_{b\in{\omega}}\tanh(J_{b})\sum\limits_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N}):\;\partial\eta=\emptyset\\\
\eta\;{\rm even\;on}\;{\omega}^{*}\end{subarray}}\frac{W(\eta)}{Z_{N}}.$ (7)

Observing that

$\sum\limits_{\begin{subarray}{c}\eta\in\mathfrak{F}(E_{N}):\;\partial\eta=\emptyset\\\
\eta\;{\rm
even\;on}\;{\omega}^{*}\end{subarray}}\frac{W(\eta)}{Z_{N}}\leq\frac{\sum_{\eta\in\mathfrak{F}(E_{N}):\,\partial\eta=\emptyset}W(\eta)}{Z_{N}}=1,$

we obtain straightforwardly the following upper bound

$\rho_{E}(\omega)\leqslant\prod_{b\in\omega}\tanh(J_{b}).$ (8)

##  4 Proof of Theorem 1

To prove Theorem 1 we shall use two properties of the weights
$\rho_{E_{N}}(\omega)$ defined in (7). The interested reader can check their
proofs in Section 4.2 of [6].

  * a)

Let $U\subset E_{N}$ be a set of edges of $\Lambda_{N}$ and let and
$\omega\subset U$ be a consistent path. Then

$\rho_{E_{N}}(\omega)\leqslant\rho_{U}(\omega).$ (9)

  * b)

If $\omega_{1}\circ\omega_{2}$ is a consistent path, where $\circ$ denotes the
usual concatenation of paths, then

$\rho_{E_{N}}(\omega_{1}\circ\omega_{2})=\rho_{E_{N}}(\omega_{1})\rho_{E_{N}\setminus{\omega^{*}_{1}}}(\omega_{2}).$

As shown in Section 3, the backbone expansion (6) for the two-point function
on $\Lambda_{N}$ is given by

$\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}=\sum_{\omega\in
C_{xy}(\Lambda_{N})}\rho_{E_{N}}(\omega),$

where $x=(u_{0},t_{0})$, $y=(u,t)$ and $C_{xy}(\Lambda_{N})$ is the set of all
consistent paths $\omega$ with extremes $\partial\omega=\\{x,y\\}$ using edges
of $E_{N}$.

Let $\omega$ be a consistent path connecting $x$ to $y$. It is possible to
break this path into $n+1$ “planar” pieces $\omega_{i}$, and $n$ “vertical”
steps $s_{i}$ (i.e. such that $|s_{i}|=1$) connecting two $d$-dimensional
hyperplanes (note that there are $2s$ possibilities for the choice of
$s_{i}$). Namely, we can write

$\omega=\omega_{1}\circ s_{1}\circ\omega_{2}\circ s_{2}\circ...\circ
s_{n}\circ\omega_{n+1}.$ (10)

We are denoting by ${\omega}_{1}$ the initial piece of the path ${\omega}$ all
contained in $\Lambda_{N}^{t_{0}}$. This initial piece ${\omega}_{1}$ of the
path ${\omega}$ is a “planar” path connecting the site $(u_{0},t_{0})$ to the
site $(u_{1},t_{0})$, which is the last site of $\Lambda_{N}^{t_{0}}$ visited
by $\omega$ before leaving $\Lambda_{N}^{t_{0}}$; this path only uses edges of
$E_{N}^{t_{0}}$. Then $s_{1}$ is the first vertical step, that is, the edge
connecting $(u_{1},t_{0})$ to $(u_{1},t_{1})$ (where $t_{1}=t_{0}+s_{1}$),
which is the first site visited by the path $\omega$ after reaching a new
hyperplane. Similarly, for each $k=1,...,n$, we denote by $\omega_{k}$ the
consistent piece of $\omega$ that connects $(u_{k-1},t_{k-1})$ to
$(u_{k},t_{k-1})$, using only edges of $E(\Lambda_{N}^{t_{k-1}})$. Here,
$(u_{k-1},t_{k-1})$ is the first site of $\Lambda_{N}^{t_{k-1}}$ visited after
the last vertical step $s_{k-1}$ and $(u_{k},t_{k-1})$ is the last site of
this hyperplane visited by $\omega$ before it makes another jump, that is,
before it reaches another hyperplane. Also, we denote by $s_{k}$ the vertical
jump, that is, the single bond connecting $(u_{k},t_{k-1})$ to
$(u_{k},t_{k})$, the first site visited by the path $\omega$ in a hyperplane
different from $\Lambda_{N}^{t_{k-1}}$. Finally, the last piece $\omega_{n+1}$
of the path ${\omega}$ connects $(u_{n},t_{n})$ to $(u_{n+1},t_{n})=(u,t)=y$,
using only edges of $E_{N}^{t_{n}}$. Note that
$t_{k}=t_{0}+\sum_{j=1}^{k}s_{j}$, for any $k=1,\dots,n$. See Figure 1 for a
sketch of this construction.

$\omega_{1}$$s_{1}$$(u_{0},t_{0})$$(u_{1},t_{0})$$\omega_{2}$$s_{2}$$(u_{1},t_{1})$$(u_{2},t_{1})$$(u_{2},t_{2})$$(u_{3},t_{2})$$\omega_{3}$$(u_{3},t_{3})$$(u_{4},t_{3})$$\omega_{4}$$\omega_{5}$$(u_{4},t_{4})$$(u_{5},t_{4})$$s_{3}$$s_{4}$
Figure 1: A sketch of a possible consistent path $\omega$.

We stress that since $\omega$ is consistent, each one of its pieces
$\omega_{i}$ is also consistent. Let us set $F_{1}=\emptyset$ and, for
$k=2,\dots,n+1$, we set $F_{k}={\omega}_{1}\circ
s_{1}\circ\dots\circ{\omega}_{k-1}\circ s_{k-1}$ so that $F^{*}_{k}$ is the
set of edges of $\Lambda_{N}$ canceled by the steps preceding $\omega_{k}$ and
$s_{k}$.

By definition, the piece ${\omega}_{k}$ of the path ${\omega}$ is in the
$d$-dimensional hypercube $\Lambda_{N}^{t_{k-1}}$. This hypercube may have
already been visited by some piece ${\omega}_{i}$ of the path ${\omega}$ with
$i<k-1$ (e.g. in Figure 1, ${\omega}_{4}$ is in the same hyperplane as
${\omega}_{2}$). Since the path $\omega$ is consistent, $\omega_{k}$ must
avoid edges of the set $F_{k}$. Therefore, $\omega_{k}$ is a consistent path
which is a subset of $E_{N}^{t_{k-1}}\setminus F_{k}$, and we denote by
$C_{k}$ the set of all such paths with these properties.

$\omega_{k}$$s_{k}$$\omega_{k+1}$$(u_{k}{}_{-1},t_{k}{}_{-1})$$(u_{k}{},t_{k}{}_{-1})$$(u_{k},t_{k})$$(u_{k+1}{},t_{k})$$\Lambda_{t_{k}{}_{-1}}$$\Lambda_{t_{k}}$
Figure 2: A sketch of a transition between hyperplanes.

Now, for $n=0,1,2,\dots$, set $\mathscr{U}_{n}=(u_{1},...,u_{n+1})$ and
$\mathscr{S}_{n}=(s_{1},...,s_{n})$, with the convention that
$\mathscr{S}_{0}=\emptyset$. Then, we can write

$\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}=\sum_{n\geq
0}\sum_{\begin{subarray}{c}\mathscr{U}_{n},\mathscr{S}_{n}\\\ u_{n+1}=u\\\
t_{n}=t\end{subarray}}\sum_{\omega_{1}\in C_{1}}...\sum_{\omega_{n+1}\in
C_{n+1}}\rho_{E_{N}}({\omega}),$

with ${\omega}$ given by (10). Summing over $y\in\Lambda_{N}$, we get

$\sum_{y\in\Lambda_{N}}\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}=\sum_{n\geq
0}\sum_{\mathscr{U}_{n},\mathscr{S}_{n}}\sum_{\omega_{1}\in
C_{1}}...\sum_{\omega_{n+1}\in C_{n+1}}\rho_{E_{N}}(\omega).$ (11)

By Property b) given at the beginning of this section, we get

$\rho_{E_{N}}(\omega)=\rho_{E_{N}-F_{n+1}}(\omega_{n+1})\prod_{k=1}^{n}\rho_{E_{N}-F_{k}}(\omega_{k})\rho_{E_{N}-F^{*}_{k}}(s_{k}),$

where we recall $F_{1}\coloneqq\varnothing$.

Now, using the bound (8), and recalling that $s_{k}$ is a single vertical edge
(and thus with $J_{s_{k}}=J_{s}$), it holds that

$\rho_{E_{N}-F^{*}_{k}}(s_{k})\leq\tanh(J_{s}),$

for any $k=1,\cdots,n$. Therefore,

$\rho_{E_{N}}(\omega)\leqslant[\tanh(J_{s})]^{n}\prod_{k=1}^{n+1}\left[\rho_{E_{N}-F_{k}}(\omega_{k})\right].$

Observe that $E_{N}^{t_{k-1}}-F_{k}\subset E_{N}-F_{k}$. Moreover, since
$\omega$ is consistent, $\omega_{k}$ only uses edges of
$E_{N}^{t_{k-1}}-F_{k}$. Hence, we can apply Property a) and inequality (9) to
obtain

$\rho_{E_{N}-F_{k}}(\omega_{k})\leq\rho_{E_{N}^{t_{k-1}}-F_{k}}(\omega_{k}),$

for any $k=1\cdots,n+1$, yielding

$\rho_{E_{N}}(\omega)\leqslant[\tanh(J_{s})]^{n}\prod_{k=1}^{n+1}\rho_{E_{N}^{t_{k-1}}-F_{k}}(\omega_{k}).$
(12)

Plugging (12) in (11), we write

$\sum_{y\in\Lambda_{N}}\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}\leqslant\sum_{n\geq
0}[\tanh(J_{s})]^{n}\sum_{\mathscr{U}_{n},\mathscr{S}_{n}}S_{1}\cdot...\cdot
S_{n+1},$

where, for $k=1,...,n+1$, we have set

$S_{k}=S_{k}(u_{k-1},u_{k},t_{k-1})=\sum_{\omega_{k}\in
C_{k}}\rho_{\Lambda_{N}^{t_{k-1}}-F_{k}}(\omega_{k}).$

Then, we can write

$\sum_{\mathscr{U}_{n},\mathscr{S}_{n}}S_{1}\cdot...\cdot
S_{n+1}=\sum_{u_{1}}S_{1}\sum_{s_{1}}\sum_{u_{2}}S_{2}\dots\sum_{u_{n-1}}S_{n-1}\sum_{s_{n-1}}\sum_{u_{n}}S_{n}\sum_{s_{n}}\sum_{u_{n+1}}S_{n+1}.$

Observe now that, for any $k=1,\cdots,n$,

$\begin{split}S_{k}&=\sum_{\omega_{k}\in
C_{k}}\rho_{\Lambda_{N}^{t_{k-1}}-F_{k}}(\omega_{k})\\\
&=\langle\sigma_{(u_{k-1},t_{k-1})}\sigma_{(u_{k},t_{k-1})}\rangle_{\Lambda_{N}^{t_{k-1}}-F_{k}}\\\
&\leq\langle\sigma_{(u_{k-1},t_{k-1})}\sigma_{(u_{k},t_{k-1})}\rangle_{\Lambda_{N}^{t_{k-1}}}.\end{split}$

where the last line follows by the GKS inequalities. Therefore, for any fixed
$u_{1},\dots,u_{n}$, and any fixed $s_{1},\dots,s_{n-1}$, we get

$\begin{split}\sum_{s_{n}}\sum_{u_{n+1}}S_{n+1}&\leq\sum_{s_{n}}\sum_{u_{n+1}}\langle\sigma_{(u_{n},t_{n})}\sigma_{(u_{n+1},t_{n})}\rangle_{\Lambda_{N}^{t_{n}}}\\\
&\leqslant\sum_{s_{n}}\sup_{u\in\Lambda_{N}^{t_{n}}}\sum_{u_{n+1}}\langle\sigma_{(u,t_{n})}\sigma_{(u_{n+1},t_{n})}\rangle_{\Lambda_{N}^{t_{n}}}\\\
&=\sum_{s_{n}}\chi_{\Lambda_{N}^{t_{n}}}(J_{d})\\\
&=2s\chi_{\Lambda_{N}^{t_{n}}}(J_{d})\\\ &=2s\chi_{\Lambda_{N}^{0}}(J_{d}).\\\
\end{split}$

Proceeding iteratively for the sums $\sum_{s_{k-1}}\sum_{u_{k}}S_{k}$ (with
$k=n,n-1,\dots,1$), and with the convention that $\sum_{s_{0}}=1$, we obtain

$\sum_{\mathscr{U}_{n},\mathscr{S}_{n}}S_{1}\cdot...\cdot
S_{n+1}\leq(2s)^{n}[\chi_{\Lambda_{N}^{0}}(J_{d})]^{n+1},$

whence, for any $x\in\Lambda_{N}$,

$\sum_{y\in\Lambda_{N}}\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda_{N}}\leqslant\sum_{n\geq
0}(2s\tanh(J_{s}))^{n}[\chi_{\Lambda_{N}^{0}}(J_{d})]^{n+1}.$

Finally, taking the limit $N\to\infty$ and recalling the definitions of
$\chi_{d+s}(J_{d},J_{s})$ and $\chi_{d}(J_{d})$ given in (2) and (4),
respectively, we get

$\chi_{d+s}(J_{d},J_{s})\leq\sum_{n\geq
0}(2s\tanh(J_{s}))^{n}[\chi_{d}(J_{d})]^{n+1}.$

The r.h.s. of the inequality above is finite provided that

$\tanh(J_{s})<\frac{1}{2s\chi_{d}(J_{d})},$

and thus the proof of Theorem 1 is concluded.

## Acknowledgements

Estevão Borel was partially supported by Fundação Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES). Aldo Procacci has been
partially supported by the Brazilian science foundations Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq), CAPES and Fundação de Amparo
à Pesquisa do Estado de Minas Gerais (FAPEMIG). Rémy Sanchis was supported by
CNPq, and by FAPEMIG, grants APQ-00868-21 and RED-00133-21. Roger Silva was
partially supported by FAPEMIG, grant APQ-00774-21.

## Conflict of interest statement

The authors have no conflicts of interest to declare. All co-authors have seen
and agree with the contents of the manuscript and there is no financial
interest to report. We certify that the submission is original work and is not
under review at any other publication.

## Data availability statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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