# From Compton Scattering of photons on targets to Inverse Compton Scattering
of electron and photon beams

Luca Serafini, Vittoria Petrillo  INFN Sez. Milano and Universita’ degli Studi
di Milano, Dipartimento di Fisica, Via Celoria 16, 20133, Milano (Italy)

###### Abstract

We revisit the kinematics of Compton Scattering (electron-photon interactions
producing electrons and photons in the exit channel) covering the full range
of energy/momenta distribution between the two colliding particles, with a
dedicated view to statistical properties of secondary beams that are generated
in beam-beam collisions. Starting from the Thomson inverse scattering, where
electrons do not recoil and photons are back-scattered to higher energies by a
Lorentz boost effect (factor $4\gamma^{2}$), we analyze three transition
points, separating four regions. These are in sequence, given by increasing
the electron recoil (numbers are for transition points, letters for regions):
a) Thomson back-scattering, 1) equal sharing of total energy in the exit
channel between electron and photon, b) deep recoil regime where the
bandwidth/energy spread of the two interacting beams are exchanged in the exit
channel, 2) electron is stopped, i.e. taken down at rest in the laboratory
system by colliding with an incident photon of $mc^{2}/2$ energy, c) electron
back-scattering region, where incident electron is back-scattered by the
incident photon, 3) symmetric scattering, when the incident particles carry
equal and opposite momenta, so that in the exit channel they are back-
scattered with same energy/momenta, d) Compton scattering ($a^{\prime}$ $la$
Arthur Compton, see ref.4), where photons carry an energy much larger than the
colliding electron energy. For each region and/or transition point we discuss
the potential effects of interest in diverse areas, like generating mono-
chromatic gamma ray beams in deep recoil regions with spectral purification,
or possible mechanisms of generation and propagation of very high energy
photons in the cosmological domain.

##  I Introduction

After the formulation of special relativity theory and the derivation of the
relativistic Doppler effect Einstein , predicting a blu-shift of e.m. field
frequency seen by an observer in relativistic motion counter-propagating with
respect to the direction of propagation of the electro-magnetic wave (see Ref.
Thomson ), confirmed by experimental measurements (Stilwell ), the theoretical
frame of Classical Electro-Dynamics was ready to explain and quantitatively
calculate the characteristics of radiation generated by charged particles
moving in space at relativistic velocities. Such a theoretical frame has been
made available in the early ’900 by A. Einstein, J.J. Thomson and J. Larmor,
to predict a broad class of phenomena of radiation emission mechanisms where
the quantum nature of e.m. fields was not needed nor critical. Only after A.
Compton’s fundamental work Compton  the quantum nature of e.m. fields, i.e.
the photon, was experimentally demonstrated: its existence was a game changer
and basically showed through A.Compton’s measurements that the effect of
electron recoil produced by the X-ray photon in the collision of X-rays with
atomic electrons couldn’t be explained by classical electro-dynamics and
special relativity. The red-shift of back-scattered photons could be explained
only invoking a photon-electron collision where the electron recoils and takes
out some of the photon energy in form of kinetic energy. After this discovery
the electron recoil became the fundamental parameter, i.e. the continental
divide, between the classical picture of radiation emission and the quantum
model of photon emission by charged particles propagating in space at
relativistic velocities. In this paper we revisit the full range of values
that the recoil parameter can assume, from the very small values associated to
Thomson scattering (or Thomson back-scattering depending on the reference
frame where the electron is observed) up to the deep recoil region, defined as
when the incident photon has an energy much larger than $mc^{2}$ in the
electron rest frame (step A. Compton!), underlying four different domains and
three transition points separating them. The recoil factor $X$ is a
dimensionless parameter given by :

$X=4E_{ph}E_{e}/(mc^{2})^{2}=4\gamma E_{ph}/mc^{2}=4\gamma^{2}E_{ph}/E_{e}$
(1)

where $E_{e}$ is the (total) electron energy and $E_{ph}$ is the photon energy
before the interaction, and the electron Lorentz factor gamma is given by
$\gamma=E_{e}/mc^{2}$. The recoil factor $X$ is linked to the energy available
in the center of mass reference system $E_{cm}$ by
$E_{cm}=mc^{2}\sqrt{(1+X)}$. The scope of this paper is to underline the
specific behaviors and statistical properties of photon beams generated in
correspondence of such transition points and regions. Several applications and
investigation areas can benefit from these properties: we anticipate here the
spectral purification mechanism active in the deep recoil region, the
surprising property of X-rays with energy equal to $mc^{2}/2$ to stop any
colliding relativistic electron of any arbitrarily high kinetic energy, and
the cancellation of the $\gamma^{2}\theta^{2}$ angular correlation, inherent
in any relativistic Doppler based radiation source, occurring at the Symmetric
Compton Scattering transition point Serafini . Historically, the progress in
understanding Compton scattering in its whole dynamical range, following the
fundamental work of A. Compton, who studied the phenomenon in the frame of
direct scattering of energetic (X-ray) photons interacting with electrons at
rest in the laboratory frame, has been initially pursued by researchers active
in astrophysics, trying to explain the behaviour of high energy charged
particles propagating throughout the universe, interacting with photons of the
cosmic back-ground (microwave, infrared, optical). J.W. Follin in 1947 firstly
studied, in the course of his PhD thesis project Follin , the inverse
kinematics of a relativistic electron colliding with a low energy (< 1 eV)
photon where the term inverse defines a specular mode of interaction w.r.t
A.Compton’s one, i.e. a configuration in the interaction where the photon
gains energy in the collision at expense of the electron kinetic energy, the
opposite of what happens in direct Compton scattering. Follin’s study has been
re-issued one year later by Feenberg and Primakoff Primakoff  still in the
frame of astrophysical phenomena, where inverse Compton scattering plays a
crucial role in determining the universe opacity to high energy photons and
electrons 111Previous literature on inverse Compton attributed to ref.
Primakoff  the first interpretation of inverse Compton - we discovered from
reference 2 of Feenberg’s paper that Follin developed and published his study
one year earlier - Feenberg and his coauthor Primakoff were prompted by J.R.
Oppenheimer to read Follin’s work that anticipated their study. Therefore, the
first studies of Inverse Compton Scattering (ICS) dated back to the late ’40s
and had the aim to explain astrophysical phenomena related to the first
observations of cosmic rays carried out just after W.W.II. It’s been only
fifteen years later, with the advent of high energy (GeV-class) electron
synchrotrons and the invention of the ruby-laser, that R.H. Milburn Milburn
firstly conceived the possibility to perform a laboratory-based experiment of
ICS using the relativistic electron beams generated by particle accelerators
making them collide with the intense optical photon beams made available by
lasers. Milburn was quickly followed by Arutyunian and Tumanian’s theoretical
analysis Arutyunian . The crucial role of recoil factor $X$ was fully
recognized since the initial stages of these theoretical studies, both in the
astrophysical domain and in the accelerator-laser based scenario: while in the
astrophysical domain Follin and Feenberg classified just two main regimes of
the electron-photon collision kinematics, i.e. the low relativistic regime
$X\ll 1$ and the high relativistic regime $X\gg 1$ , both accessible in the
cosmic ray contest, Milburn and Arutyunian focused mostly on the low to mild
relativistic regime ($X<1$) that was only accessible in laboratory
experiments. The first measurements of ICS at a laboratory scale followed a
few years after Milburn’s publication Telnov ; Bemporad ; Ladon  at a number
of particle accelerator laboratories. We organize this paper as follows: in
Section 2 we discuss the full range of inverse kinematics reporting a synoptic
table listing all ICS regions and transition points. In Section 3 we show how
the dependence on the collision angle is also vanishing in large recoil
regimes, and in Section 4 we show some quantitative examples of spectral
purification, the relevant aspect of large recoil electron-photon collisions,
where the properties of the incident primary beams are mapped into the
secondary beams with an exchange of entropy, i.e. the scattered photon beam is
cooled and the exit electron beam after collision is heated, a strategic way
to generate mono-chromatic gamma rays using mono-energetic incident electron
beams and broad-band white-spectrum photon beams We should remark that in this
paper we are not discussing non-linear effects due to intensity of the
incident photon beam, i.e. those effects who become important when the laser
parameter $a_{0}$ associated to the optical photon beam carried by an incident
laser pulse becomes close or larger than 1, inducing multi-photon or dressed-
electron effects.

##  II Different Compton Scattering regimes

In the synoptic table 1 the different Compton Scattering regimes are
summarized. Three transition points are listed in the table rows, separating
four regions, each characterized by specific range of collision kinematics.

Table 1:  Regimes of Compton Scattering.  Regime | Recoil Factor | Inc. ph. en. | Max scatt. ph. en. | Min scatt. e-en. | Cent. of mass en. | El. rest frame
---|---|---|---|---|---|---
Symbol | $X$ | $E_{ph}$ | $E^{\prime}_{ph}$ | $E^{\prime}_{e}$ | $E_{cm}$ | ${E_{ph}}^{ERF}$
Definition | $X=\frac{4\gamma E_{ph}}{mc^{2}}$ | $k_{x}=k_{y}=0$ | $E^{\prime}_{ph}=\frac{4\gamma^{2}E_{ph}}{1+X}$ |  | $E_{cm}=mc^{2}\sqrt{1+X}$ | $2\gamma E_{ph}$
ITS | $X\ll 1$ | $\ll E_{e}$ | $4\gamma^{2}E{ph}$ | $\simeq E_{e}$ | $\simeq mc^{2}$ | $<<mc^{2}$
DICS | X=1 | $\frac{mc^{2}}{4\gamma}$ | $2\gamma^{2}E_{ph}$ | $2\gamma^{2}E_{ph}$ | $\sqrt{2}mc^{2}$ | $mc^{2}/2$
DRCS | $1\ll X\ll 2\gamma$ | $\gg\frac{mc^{2}}{4\gamma}$ | $\simeq(1-\frac{1}{X})E_{e}$ | $\ll E_{e}$ | $>\sqrt{2}mc^{2}$ | >$mc^{2}/2$
FICS | $X=2\gamma$ | $mc^{2}/2$ | $E_{e}-mc^{2}/2$ | $mc^{2}$ | $\sqrt{2\gamma}mc^{2}$ | $\gamma mc^{2}$
EBS | $2\gamma<X<4\gamma^{2}$ | <$\gamma mc^{2}$ | $\sim E_{e}$ | $<E_{e}$ | >$\sqrt{2\gamma}mc^{2}$ | >$\gamma mc^{2}$
SYCS | $X=4\gamma^{2}$ | $\gamma mc^{2}$ | $E_{ph}$ | $E_{e}$ | $2\gamma mc^{2}$ | $2\gamma^{2}mc^{2}$
RDCS | $X>4\gamma^{2}$ | $>\gamma mc^{2}$ | $<E_{ph}$ | $>E_{e}$ | >$2\gamma mc^{2}$ | >$2\gamma^{2}mc^{2}$

Table columns report all relevant parameters characterizing the electron-
photon collision quantities: recoil factor $X$, energy of incident photon
$E_{ph}$, maximum energy $E^{\prime}_{ph}$ of scattered photon at $\theta=0$
scattering angle (full back-scattering), the corresponding minimum energy
$E^{\prime}_{e}$ of the scattered electron, the energy in the center of mass
reference frame $E_{cm}$ and the energy of the incident photon as seen in the
electron rest frame $E_{ph}^{ERF}$ . Note that the incident electron is
assumed to be relativistic (i.e. $\gamma\gg 1$) and the electron-photon
collision is assumed to be head-on (i.e. vanishing transverse momentum of the
incident photon and electron). Besides the recoil factor $X$, a second
dimensionless parameter governing the kinematics of Compton scattering is the
asymmetry parameter $A$, defined as in ref. Serafini

$A=\gamma^{2}-X/4$ (2)

(in case of relativistic electrons). $A$ represents the divide between direct
Compton scattering (characterized by $A<0$, where the energy of the incident
photon is larger than the energy of the incoming electron), and inverse
Compton scattering (region defined by $A>0$) where the electron energy before
interaction is larger than that of the incident photon. $A=0$ is the
transition point where the scattering is fully symmetric (actually the center
of mass reference system is steady in the laboratory system because the total
momentum is null) and was denominated Symmetric Compton Scattering in Ref.
Serafini .

For simplicity in the discussion we will consider in this section an ideal
head-on collision between an electron of energy $E_{e}$ and momentum aligned
with the z-axis, directed towards positive z, and a photon of energy $E_{ph}$,
counter-propagating w.r.t. the electron towards negative z. The generalization
to an arbitrary value of the collision angle $\alpha$ between the incoming
electron and the incident photon will be given in Section 3. The electron
momentum before collision is $\textbf{P}=(p_{x}=0,p_{y}=0,p_{z}=\beta\gamma
mc)$ while the photon momentum is $\hbar\textbf{k}=(\hbar k_{x}=0,\hbar
k_{y}=0,\hbar k_{z}=-E_{ph}/c)$. Following references Ranjan ; Curatolo  and
invoking the conservation of total energy and momentum, if the scattered
photon is propagating at a scattering angle $\theta$ w.r.t. the z-axis, its
energy will be given by the well known formula:

$E^{\prime}_{ph}=\frac{(1+\beta)E_{ph}E_{e}}{(1-\beta
cos\theta)E_{e}+(1+cos\theta)E_{ph}}$ (3)

that is valid for any arbitrary value of colliding electron and photon energy
and momentum. Actually, the direct Compton effect is represented by Eq. 3 just
setting $\beta=0$ and $E_{e}=mc^{2}$ , that applies to an electron at rest in
the laboratory system. In this way, Eq.3 transforms into the well know
Compton’s formula :

$E^{\prime}_{ph}=\frac{E_{ph}}{1+(1+cos\theta)E_{ph}/mc^{2}}$ (4)

By knowing the value of the scattered photon energy $E^{\prime}_{ph}$, and
calling $E_{tot}=E_{e}+E_{ph}$, it is straightforward to derive the electron
energy after scattering as $E^{\prime}_{e}=E_{e}+E_{ph}-E^{\prime}_{ph}$ =
$E_{tot}-E^{\prime}_{ph}$ . Nevertheless, our focus is on relativistic
electrons, so Table 1 illustrates the behavior of electron and photon energy
after scattering in case $\gamma>>1$.

Figure 1: Different regimes of emission versus the initial photon energy (in
MeV) for $E_{e}=50MeV$. All energies (namely $E^{\prime}_{e}$ (in
blue),$E^{\prime}_{ph}$ (in red) and $E_{ph}$ (in black) are dimensionless,
normalized to $E_{tot}$. The recoil $X$ (on the same left ordinate) is in
green. (a) indicates the regime ITS, (b) indicates DRCS, (c) EBS and (d) RDCS;
(1) represents the point DICS ,(2) represents the point FICS and (3) SYCS

Therefore the fourth region named RDCS refers to Direct Compton effect with
electrons in relativistic motion in the laboratory system, though of energy
smaller than that of incident photons. In order to better illustrate the 4
regions and 3 transition points of generalized Compton scattering, we
specialize in the following the discussion considering those photons that are
totally back-scattered and propagate at a scattering angle $\theta=0$ after
colliding with the electron. These are photons achieving maximum energy gain
or loss depending on the value of the Asymmetry parameter $A$ (photons gain
energy when $A>0$ and loose energy when $A<0$). If $\theta=0$, the energy of
the back-scattered photon is simply given by
$E^{\prime}_{ph}=4\gamma^{2}E_{ph}/(1+X)$. We plot in Fig. 1 the relevant
quantities characterizing the kinematics as a function of the incident photon
energy $E_{ph}$ for a fixed incident electron energy, namely $E_{e}$ = 50 MeV.
All energies are normalized to the total energy of the system
$E_{tot}=E_{e}+E_{ph}$. The recoil parameter $X$ is also plotted. The
normalized energies of the electron and photon after scattering can be
expressed in terms of only the recoil parameter $X$ and the Lorentz factor
$\gamma$ as follows:

$\frac{E^{\prime}_{ph}}{E_{tot}}=\frac{X}{(1+X)(1+\frac{X}{4\gamma^{2}})}$ (5)
$\frac{E^{\prime}_{e}}{E_{tot}}=1-\frac{X}{(1+X)(1+\frac{X}{4\gamma^{2}})}$
(6)

These are very useful expressions to explain how to classify the various
regimes and transition points of Compton inverse-direct scattering. Fig.2
shows the total momentum $p_{tot}$, as well as the final electron and photon
momenta, respectively $p^{\prime}_{e}$ and $p^{\prime}_{ph}$. The first
region, named Inverse Thomson Scattering (label (a) in Figs.1 and 2), is
basically defined as that of "negligible momentum exchange" between the
incident photon and the colliding electron: as a matter of fact the momentum
exchange in the Electron Rest Frame (ERF) is given by $2E_{ph}^{ERF}=4\gamma
E_{ph}$ (note that $X=2E_{ph}^{ERF}/mc^{2}$). If the momentum exchange is
negligible w.r.t. $mc^{2}$ then the condition $X\ll 1$ holds, and the back-
scattered photon in ERF will have same energy as $E_{ph}^{ERF}$ : Lorentz
transforming back to the laboratory system will give $E^{\prime}_{ph}=2\gamma
E_{ph}^{ERF}=4\gamma^{2}E_{ph}$. Hence, Inverse Thomson Scattering can be
simply explained as inverse photon scattering with negligible momentum
exchange (note that in this regime the total energy-momentum conservation is
violated by the expression $4\gamma^{2}E_{ph}$ unless $X\ll 1$). ITS regime
and its peculiar $4\gamma^{2}$ scaling of the back-scattered photon energy can
be also explained going through a classical electro-dynamics description of
undulator radiation emitted by the electron oscillating in the em. field of
the counter-propagating laser pulse (see ref. Tomassini ). The majority of
Inverse Compton Sources under design or operation belong to this ITS regime
(see Petrillo  ), with the exception of XFELO (Hajima ). Looking at Eq. 5 and
6, it is interesting noticing that the two particles after scattering have
equal energy, i.e. $E^{\prime}_{e}=E^{\prime}_{ph}$ when the recoil parameter
becomes equal to either $X=1$ or $X=4\gamma^{2}$. We denominate the transition
point characterized by $X=1$ , labelled (1) in Fig.s 1-3 , Democratic Inverse
Compton Scattering, since the two particles emerging after scattering are
sharing democratically the total energy of the system $E_{tot}$ and the total
momentum $p_{tot}$ (while before scattering most of the energy is carried by
the electron and the photon carries an energy still much smaller than
$mc^{2}$, namely $mc^{2}/(4\gamma)$). The second solution $X=4\gamma^{2}$ ,
that is valid only when $\gamma\gg 1$, corresponds actually to Symmetric
Compton Scattering, very well discussed and analyzed in ref. Serafini , that
is characterized by interacting electron and photon carrying equal energy and
momentum before and after scattering. In SYCS (point (3) in the Figures) what
happens is basically a flip-over of the incoming momentum of the two
scattering particles, that just flip the direction of their momentum without
modifying their energy, with a uniform angular distribution after scattering.

Region (b) in Figs. 1 and 2, denominated Deep Recoil Compton Scattering, is
characterized by an increasing value of the recoil factor, in the range $1\ll
X\ll 2\gamma$. In this region the incident electron is going to lose most of
its energy/momentum in favour of the back-scattered photon: a saturation
effect starts to appear in the energy of the back-scattered photon, given
approximately by $(1-1/X)E_{e}$ , which depends only perturbatively on the
energy of the incident photon (through the recoil factor $X$). This is the
basis of spectral purification mechanism that is further discussed in section
4. Still in region (b) the scattered electron emerging after scattering keeps
propagating towards the incidental positive z-axis direction, as clearly shown
in Fig.2. Instead, in region (c), denominated Electron Back-Scattering,
characterized still by a large value of the recoil factor, the electron starts
to be back-scattered and propagating back in the negative direction of the
z-axis after scattering. Both in region (b) and region (c), the emerging
photon after scattering is carrying almost all the total energy of the system,
since $E^{\prime}_{ph}/E_{tot}\simeq 1$ .

Table 2:  Parameters of simulations  Regime | $E_{e}$ | $E_{p}h$ | X | A | Max scatt ph. En. | Min scatt. e-En. | Cent. of mass En. | Figure
---|---|---|---|---|---|---|---|---
Unit | MeV | MeV |  |  | MeV | MeV | MeV |
ITS | 50 | 1.5$\times 10^{-6}$ | 0.0011 | 9574 | 0.057 | 50 | 0.511 | Fig.4
DICS | 50 | 1.3$\times 10^{-3}$ | 1 | 9574 | 24.88 | 24.88 | 0.722 | Fig.4
DRCS | 50 | $10^{-2}$ | 7.65 | 9572 | 43.46 | 6.5 | 1.5 | Fig.4
FICS | 50 | 0.2555 | 195.7 | 9525 | 50 | 0.2555 | 7.1 | Fig.5
EBS | 50 | 15 | 1.15$\times 10^{4}$ | 6700 | 50 | 15 | 54.7 | Fig.6
SYCS | 50 | 50 | 3.8$\times 10^{4}$ | 74 | 50 | 50 | 99.99 | Fig.6
RDCS | 50 | 150 | 1.15$\times 10^{5}$ | -9676 | 50 | 150 | 173 | Fig.6
Figure 2: Momenta versus the initial photon energy (in MeV) for left
$E_{e}=5MeV$ and right $E_{e}=50MeV$. All momenta (namely $p^{\prime}_{e}$ (in
blue),$p^{\prime}_{ph}$ (in red) and $p_{tot}$ (in green) are in MeV/c. (a)
indicates the regime ITS, (b) indicates DRCS, (c) EBS and (d) RDCS; (1)
represents the point DICS ,(2) represents the point FICS and (3) SYCS

This is the region denominated "extremely relativistic" in Ref. Primakoff  and
Arutyunian , whose authors discuss the amount of energy lost by the electron
in a single collision: the electron looses a large fraction of its kinetic
energy in favour of the photon when collisions take place in regions (b) and
(c). The way Inverse Compton Scattering in deep recoil regime can be realized
is double-fold: either considering electrons with increasing initial energy
$E_{e}$ before scattering, as in ref. Primakoff ; Arutyunian  (dealing with
astrophysical phenomena where incident photons are in the microwave or
infrared or optical range), or increasing the energy of the incident photon,
as discussed here. What matters in order to keep $X$ large is the product of
the incident photon energy and the incoming electron energy: just as a side
comment we note that the maximum value achieved in experiments operated in
laboratories with accelerators, has been about $X=1.9$ so far (Bamber ,
reporting the data of experiment E144 at SLAC) using about 45 GeV electron
beams colliding with IR (1.2 eV photons) and green (2.4 eV photons) laser
pulses - DICS has been observed in single photon linear Compton scattering in
this experiment, as clearly reported in the photon spectra published in the
paper. There is however a special transition point, denominated Full Inverse
Compton Scattering (point (2) in the Fig.s 1-3), that is characterized by a
unique absolute value of the incident photon energy, i.e. $E_{ph}=mc^{2}/2$,
that is the divide between electrons emerging after scattering with still
positive momentum (region (a) and (b)) and electrons back-scattered with
negative momentum (region (c) and (d)) by the collision with incident photons.
At the FICS transition point, the electron is taken down to rest in the
laboratory system by the interaction with a 255.5 keV ($mc^{2}/2$) photon: its
energy after scattering is just $mc^{2}$. Its kinetic energy is totally
transferred to the back-scattered photon, that will propagate back towards
positive z-axis direction with energy
$E^{\prime}_{ph}=E_{e}-mc^{2}/2=T_{e}+mc^{2}/2$. Interesting to note that
$mc^{2}/2$ is a value playing an important role also in direct Compton
scattering, with a dual behavior than FICS: as a matter of fact if we consider
an incident photon on a target, as predicted by Eq.4, with an energy much
larger than the electron rest mass energy, i.e. $E_{ph}>>mc^{2}$, then the
back-scattered photon at $\theta=0$ will have an energy equal to
$E^{\prime}_{ph}=mc^{2}/2$ independently on the energy $E_{ph}$ of the
incident photon. At FICS point the recoil factor is $X=2\gamma$. We believe
that FICS can play an important role in astro-physical gamma-ray sources,
where the flux of 255.5 keV X-rays can be intense and can interact with very
high energy electrons generating in turns very high energetic photons in a
single collision with extremely large energy transfer. At the end, FICS is the
Compton scattering modality maximizing the energy/momentum transfer from the
electron to the photon: this characteristic was not specifically underlined in
previous literature. The impressive characteristic of FICS interaction point
is the capability of 255.5 keV photons to stop "any" relativistic electron of
whatsoever energy. Fig. 3 shows the different emission regimes for
$E_{e}=50MeV$. $E^{\prime}_{e}$ (in blue),$E^{\prime}_{ph}$ (in red) and
$E_{ph}$ (in black) in MeV are represented versus $E_{ph}$ in MeV. The stars
represent few typical values chosen for spectral simulations. In Table 2 the
values assumed by some relevant quantities are reported. The first of these
points falls in the Inverse Thomson Scattering regime (a 50 MeV electron beam
collides with a optical radiation pulse of 1.5 eV energy) and is an example of
the various Compton sources operating worldwide Petrillo . Characterized by a
small recoil $X=1.1\times 10^{-3}$, this working point generates hard X-rays
of edge energy $E^{\prime}_{ph}$=57.4 keV, in a range useful for medical
imaging applications. The radiation spectrum, shown in Fig. 4, window (i) in
red, has the typical bowl shape, while the electrons (in blue) lose in the
interaction a negligible amount of energy. The emitted photons are all in the
$1/\gamma$ cone, while the electrons maintain their straight trajectories.

Figure 3: Different regimes of emission versus the initial photon energy (in
MeV) for $E_{e}$= 50 MeV. $E^{\prime}_{e}$ (in blue),$E^{\prime}_{ph}$ (in
red) and $E_{ph}$ (in black) in MeV are represented versus $E_{ph}$ in MeV.
The recoil $X$ (on the same left ordinate) is in green. Differently from Fig.
1 the quantities (except $X$) are not dimensionless. The stars represent the
values chosen for the spectrum simulations. Figure 4: (i) Inverse Thomson
Scattering (ITS). Left: energy distribution, right: angular distribution. Red:
scattered photons, blue:scattered electrons. $E_{ph}$=1.5 eV, $E_{e}$=50 MeV,
$bw_{ph}=5\%$. (ii) Democratic Inverse Compton Scattering (DICS). $E_{ph}$=1.3
keV, $E_{e}=50MeV$, $bw_{ph}=5\%$. (iii) Deep Recoil Scattering (DRS).
$E_{ph}$=10 keV, $E_{e}$=50 MeV, $bw_{ph}=5\%$. Figure 5: Full Inverse Compton
Scattering (FICS). Left: energy distribution, right: angular distribution.
Red: scattered photons, blue:scattered electrons.$E_{ph}=255.5keV$, $E_{e}$=50
MeV, $bw_{ph}=5\%$

When the recoil is $X=1$, corresponding to $E_{ph}$= 1.3 keV for $E_{e}$=50
MeV (Fig. 4, window (ii)), the emitted photon and the emerging electron
spectra maintain the broad bowl shape, while the Compton edge and the minimum
electron energy limit coincide at about $E=E_{e}/2$. The photon angular
distribution still covers the $1/\gamma$ aperture and the emerging electrons
are all close to the axis.

Figure 6: (i) Electron Back-Scattering (EBS). Left: energy distribution,
right: angular distribution. Red: scattered photons, blue:scattered electrons.
$E_{ph}=15MeV$, $E_{e}=50MeV$, $bw_{ph}=5\%$.(ii) Symmetric Compton Scattering
(SYCS): $E_{ph}=50$ MeV, $E_{e}$=50 MeV, $bw_{ph}=5\%$. (iii) Relativistic
Direct Compton Scattering: $E_{ph}$=150 MeV, $E_{e}$=50 MeV, $bw_{ph}=5\%$

Increasing further the recoil, penetrating the deep recoil regime (DRS), (see
Fig. 4, window(iii)), the radiation spectrum develops a sharp forward peak,
already described in Ref.s Hajima ; Curatolo . In a symmetric way, the
electrons lose energy. The angular distributions begin to enlarge. This
situation achieves its maximum expression in correspondence to the Full
Inverse Compton Scattering (FICS) point, where the electron beam, hit by
photons at 255.5 keV, stops whatever its energy. As shown in the simulation
(Fig. 5), there is an almost full exchange of energy between the photons and
the electrons. The photon spectrum appears strongly peaked at the initial
electron energy at 50 MeV, while the electrons lose almost all their kinetic
energy, stabilizing at 511 keV total energy $E^{\prime}_{e}$. While the
photons still propagate in a cone around the positive z-axis, the electrons
scatter in the whole solid angle and a consistent fraction revert. Increasing
the incoming photon energy beyond 255.5 keV (Fig. 6 windows (i),(ii) and
(iii)), the emitted photon energy remains attested at the initial electron
energy value (namely 50 MeV), the remaining energy being acquired by the
electrons which now move faster and faster in the reverse direction. The
radiation spectrum shrinks, reaching the full monochromaticity in
correspondence of the Symmetric Inverse Compton Scattering (SYCS) point (6,
window (ii)), defined by the condition $E_{e}=E_{ph}$ and widely analyzed in
Ref. Serafini  also in general to non-relativistic electrons, for which the
SYCS condition actually applies to the electron and photon momentum, i.e.
$p_{e}=\hbar k$. We note that a relatively broad bandwidth of the initial
photon beam does not change the thinness of the final spectrum. In Fig. 6, the
initial photon relative bandwidth is assumed 5%, while in the symmetric
Compton condition the final bandwidth is under $0.2\%$. Conversely, the
electrons, even if monoenergetic at the beginning, exit the collision
dispersed in energy.

Figure 7: $E^{\prime}_{ph}$ as function of $E_{ph}$ for various values of
initial energy $E_{e}$, namely 100 GeV,1,10,100 TeV, 1,100 PeV. Note that all
electrons stop after collision when $E_{ph}=mc^{2}/2$=0.2555 MeV,
independently on their initial energy $E_{e}$. The incident photon energy
region $E_{ph}=10^{-10}$ to $10^{-9}$ MeV corresponds to the Cosmic Microwave
Background Radiation (CMBR), while the region $E_{ph}=10^{-7}$ to $10^{-5}$
MeV corresponds to Extra-Galactic Light (EBL) and Inter-Stellar Radiation
Field (ISRF), ref. Vernetto .

The unbelievable properties of the FICS working point are highlighted in Fig.
7. Spanning a very large range of incident photon energies, from 1 $\mu eV$ up
to 1 TeV (in order to cover the energy range of intra-galactic and extra-
galactic photon background, spanning from CMBR to EBL and ISRF, see ref.
Vernetto ), and considering interacting electron energies of 100 GeV,1, 10,
100 TeV, and 1, 100 PeV, all curves plotted in Fig.7 show the energy of the
electron after scattering $E^{\prime}_{e}$ : this quantity exhibits a common
minimum at $mc^{2}/2$ incident photon energy with a common value of
$E^{\prime}_{e}=mc^{2}$ independent on the initial electron energy $E_{e}$. It
is also interesting to notice that the behaviour of $E^{\prime}_{e}$ as
function of $E_{ph}$ becomes independent on the initial electron energy
$E_{e}$ as soon as the recoil factor $X$ becomes much larger than 1, as well
shown in Fig. 7. This is clearly due to the fact that for large recoil the
energy of the back-scattered photon asymptotically tends to the energy of the
incident electron, therefore $E^{\prime}_{e}$ asymptotically tends to
$E_{ph}$. Nevertheless, we need to underline that increasing the energy
available in the center of mass $E_{cm}$, as clearly inherent in data shown in
Fig.7, other reactions become competitive and statistically more likely than
Compton scattering, like triplet production (above $E_{cm}$= 1.5 MeV) , muon
pair production (above $E_{cm}$=210 MeV) and other hadronic reactions
involving pion/meson production, that become dominant only above $E_{cm}=$ 600
MeV (see ref. Burk ), that implies $E_{e}=$360 GeV for 255.5 keV incident
photons. A specific discussion on cross section behaviors of different QED
processes in this energy range can be found in ref. EXMP . The behavior shown
in Fig.7 is strictly relative to the kinematics of inverse Compton scattering
in exotic regions of the incident particle energy range never considered
previously in the literature under such specific details. In all regimes, the
number of events is proportional to the beam luminosity and weighted by the
cross section KleinNishina . The total unpolarized Compton cross section is:

$\begin{split}\sigma=\frac{2\pi
r_{e}^{2}}{X}\left[\frac{1}{2}+\frac{8}{X}-\frac{1}{2(1+X)^{2}}+\right.\\\
\left.\left(1-\frac{4}{X}-\frac{8}{X^{2}}\right)\log(1+X)\right].\end{split}$
(7)

The values of $\sigma$ vary between the classical limit $X\to 0$ and the
ultra-relativistic limit $X\to\infty$ as:

$\left\\{\begin{aligned} &\lim_{X\to 0}\sigma=\frac{8\pi
r_{e}^{2}}{3}(1-X)=\sigma_{T}(1-X)\\\ &\lim_{X\to\infty}\sigma=\frac{2\pi
r_{e}^{2}}{X}\left(\log{X}+\frac{1}{2}\right)\end{aligned}\right.$ (8)

where $r_{e}$ is the classical electron radius and $\sigma_{T}=0.67$ barn is
the total Thomson cross section Jackson . In particular, when the recoil
factor is very large the relationship $X=(E_{cm}/mc^{2})^{2}$ holds, so the
cross section scales roughly like the inverse square of the center of mass
energy, i.e.

$\lim_{X\to\infty}\sigma\simeq(E_{cm}/mc^{2})^{2}(0.5+\log(E_{cm}/mc^{2})^{2}),$
(9)

decreasing the probability to undergo Compton scattering between the incoming
electron and the incident photon. Nevertheless, as extensively discussed in
ref. Serafini , the angular cross section peaks at $\theta=0$, implying that
the most likely scattering is back-scattering of the photon, condition of
maximum energy/momentum exchange between the electron and the photon, at which
the values reported in Table I correspond. This may be the peculiarity of
Compton scattering w.r.t. other two-particle interaction, like for example
Bhabha scattering or Moeller scattering, as is well shown in the evolution of
the spectral distribution visible from Fig.4 to Fig. 6.

##  III Deep recoil and Symmetric Compton Scattering cancel all angular
correlations

All radiation emission by relativistic charged particles that involves a blue-
shift based on relativistic Doppler effect is characterized by an angular
correlation of the emitted photon energy vs. the angle of emission w.r.t. the
propagation axis of the emitting particle. This is represented in all formulas
by the well know $\gamma^{2}\theta^{2}$ term present in synchrotron radiation,
ICS, betatron radiation, etc. Photons emitted at a non-zero $\theta$ angle are
red-shifted w.r.t. that emitted at $\theta=0$. On the other side the radiation
emission is concentrated within a $\theta=1/\gamma$ cone. It is only SYCS that
cancels completely such an angular correlation: we rewrite Eq.3 by expressing
the formula only in terms of the recoil parameter $X$, the asymmetry factor
$A$ and the Lorentz factor $\gamma$. $A$ is generally defined, for any value
of the electron velocity $\beta$ and Lorentz factor $\gamma$ as
$A=\beta\gamma^{2}-X/4$. We obtain

$E^{\prime}_{ph}=\frac{\gamma^{2}+A+X/4}{\gamma^{2}-Acos\theta+X/4}E_{ph}$
(10)

that is at all equivalent to Eq. 3. Stating that at the SYCS transition point
($A=0$) the dependence of the back-scattered photon energy $E^{\prime}_{ph}$
on the scattering angle $\theta$ is cancelled (see ref. Serafini  for further
discussion). While SYCS truly cancels the angular correlation with the
scattering angle, any large recoil regime $X\gg 1$ is characterized by a
strong damping of the angular correlation $\gamma^{2}\theta^{2}$. We can re-
write Eq.3 by assuming both $\gamma>>1$ and $\theta<1/\gamma$ (see Ref.
Curatolo ) as

$E^{\prime}_{ph}=\frac{4\gamma^{2}E_{ph}}{1+\gamma^{2}\theta^{2}+X}$ (11)

that can be asymptotically expressed for large values of $X\gg 1$ as

$E^{\prime}_{ph}\simeq E_{e}(1-\frac{1+\gamma^{2}\theta^{2}}{X})$ (12)

clearly showing the damping of angular correlation at large recoils. As
extensively discussed in ref. Curatolo  and Serafini , such a damping applies
also to the angular spread at collision due to the electron beam emittance,
making a deep recoil ICS insensitive to beam emittance, a great advantage in
designing ICS sources for very mono-chromatic X and $\gamma$ rays.

Another interesting effect of the large recoil regime is the cancellation of
the dependence of back-scattered photon energy on the collision angle. This is
another quite surprising effect, that can be appreciated by recalling the
general formula (see, for instance, twocolor )

$E^{\prime}_{ph}=E_{ph}\frac{\gamma^{2}(1-\beta cos\alpha)}{\gamma^{2}(1-\beta
cos\theta)+\frac{X}{4}(1-cos(\alpha+\theta))}$ (13)

that reduces to Eq.3 when the collision angle $\alpha$ is equal to
$\alpha=\pi$, i.e. at head-on collision.

When the recoil factor is very large the previous formula asymptotically tends
to

$E^{\prime}_{ph}\simeq\frac{4\gamma^{2}E_{ph}(1-cos\alpha)/2}{1+\gamma^{2}\theta^{2}+X(1-cos\alpha)/2}$
(14)

Considering photons fully back-scattered at $\theta=0$ previous equation
reduces to

$E^{\prime}_{ph}\simeq E_{e}(1-\frac{2}{X(1-cos\alpha)})$ (15)

clearly showing the de-sensitivity to collision angle $\alpha$ induced by
large recoils. This peculiar property opens the way to designing sources of
mono-chromatic gamma-rays based on the interaction of mono-energetic low
energy electron beams with photon hohlraums, where large X-ray photon
densities are stored (see ref. Pike ). If the electron-photon collision inside
the hohlraum is taking place at large recoil values, then the energy of the
secondary gamma-ray beam generated (leaving the hohlraum in the same direction
of propagation of the electron beam) does not depend on the large dispersion
of the collision angle inside the thermal X-ray photon bath. The dependence on
the thermal photon energy spread is also cancelled by large recoil
interactions, through a spectral purification mechanism, as discussed in next
section.

##  IV Spectral purification induced by deep recoil

Figure 8: Left window: Spectrum of the incident photon similar to channelling
radiation in crystals, with colliding electron beams of energy
$E_{e}=2MeV,3.5MeV,5MeV,$ and $6MeV$. Central window: Momentum spectrum of
scattered photons. Right window: Momentum spectrum of scattered electrons
Figure 9: Collimated energy spectrum. $E_{e}=1.9MeV$, curve
magenta:$\theta_{max}=0.25rad$, red:$\theta_{max}=0.5rad$,blue:
$\theta_{max}=1rad$

The peculiar spectral behavior around the SYCS point can be exploited to
conceive an extremely mono-chromatic source of $\gamma$ rays. Starting from
broadband radiation sources ($\Delta E_{ph}/E_{ph}\simeq 30\%$), as channeling
radiation Bandiera  or betatron emission from plasma accelerated beams
EupraXia , a highly monochromatic gamma-ray beam can be produced by colliding
these photons with electron beams of similar energy .

Once again large recoils reduce the dependence of the scattered photon beam
bandwidth on the bandwidth of the incident photon beam, similarly to what
happens with angular correlations (as discussed in previous section).
Recalling that for large recoil factors $X\gg 1$ the asymptotic expression of
the back-scattered photon at $\theta=0$ is given by
$E^{\prime}_{ph}\simeq(1-1/X)E_{e}$ , we can easily derive the relative
bandwidth of the back-scattered photon beam $\Delta
E^{\prime}_{ph}/E^{\prime}_{ph}$ as:

$\frac{\Delta E^{\prime}_{ph}}{E^{\prime}_{ph}}\simeq\frac{1}{X}\frac{\Delta
E_{ph}}{E_{ph}}$ (16)

as a function of the incident photon beam relative bandwidth $\Delta
E_{ph}/E_{ph}$. In agreement with ref. Curatolo  as far as the condition $X\gg
1$ is assumed.

This damping of the incident photon beam bandwidth, that is compressed by a
factor $1/X$ into the back-scattered photon beam bandwidth, is very clearly
illustrated in Fig. 8, showing the initial energy spectrum (left window) and
the emitted photon (central window) and electron (right window) momentum
spectra, for the case of an incident broad-band photon beam generated by
channeling radiation in a crystal (ref. Bandiera ) and a colliding mono-
energetic electron beam with variable energy, spanning the 2 - 6 MeV energy
range. The photon beam spectra with a highly monochromatic peak shown in the
central diagram clearly demonstrate the spectral purification mechanism
enabled by the deep recoil inverse Compton scattering regime. Here what
matters to achieve spectral purification is just a large value of the recoil
factor $X$, which vary in a range from 64 (minimum value at 2 MeV colliding
electron beam) up to about 200 (maximum value in case of 6 MeV electrons). The
very peculiar point is that this example of beam collision spans several
regions of ICS, since the electron beam Lorentz factor $\gamma$ varies from
about 5 up to nearly 12, implying that the 2 MeV case is fully in RDCS regime,
while 3.5 MeV is spanning across RDCS, SYCS and EBS, 5 MeV covers EBS and FICS
and 6 MeV FICS and DRCS. As a consequence, the spectrum of the electron
momentum (Fig. 8, right window) resembles the initial photon spectrum, but at
the turn of the zero with an electron ensemble going forward and another going
backward. No matter what regime is in action, as far as the recoil factor $X$
is a large number, high spectral purification applies. Fig. 9 presents instead
the collimated energy spectrum, taken by selecting the radiation inside the
angles $\theta_{max}$=0.25 rad$(\simeq 1/\gamma)$, $\theta_{max}$=0.5 rad
$\theta_{max}$=1 rad, where the spectral purification manifests itself
evidently. Let us also underline the great advantage of using these deep
recoil regimes to generate highly mono-chromatic gamma-rays: the energy of the
colliding electron beam needed to generate gamma-ray beams in the MeV energy
range is not much larger than a few MeV. Unlike typical ICS sources aimed at
gamma-ray beam generation in the ITS regime ELI  where GeV-class electron
beams are needed to enable applications in the photo-nuclear physics and
photonics. Compactness and sustainability are certainly in favour of the deep
recoil regime.

##

##  V Conclusions

We revisited the kinematics of Compton Scattering, the electron-photon
interactions producing electrons and photons in the exit channel, covering the
full range of energy/momenta distribution between the two colliding particles,
with a dedicated view to statistical properties of secondary beams that are
generated in beam-beam collisions. Starting from the Thomson inverse
scattering, where electrons do not recoil and photons are back-scattered to
higher energies by a Lorentz boost effect (factor $4\gamma^{2}$), we analyze
three transition points, separating four regions. These are in sequence, given
by increasing the electron recoil (numbers are for transition points, letters
for regions): a) Thomson back-scattering, with a recoil factor $X\ll 1$ 1)
equal sharing of total energy in the exit channel between electron and photon
(recoil factor X=1, b) deep recoil regime where the bandwidth/energy spread of
the two interacting beams are exchanged in the exit channel (here $1\ll X\ll
2\gamma$) 2) electron is stopped, i.e. taken down at rest in the laboratory
system by colliding with an incident photon of $mc^{2}/2$ energy and the
recoil factor is $X=2\gamma$, c) electron back-scattering region, where
incident electron is back-scattered by the incident photon, with a recoil
$2\gamma<X<4\gamma^{2}$ 3) symmetric scattering, when the incident particles
carry equal and opposite momenta, so that in the exit channel they are back-
scattered with same energy/momenta and $X=4\gamma^{2}$, d) Compton scattering
($a^{\prime}$ $la$ Arthur Compton, see ref.4), where photons carry an energy
much larger than the colliding electron energy, occurring when
$X>4\gamma^{2}$. For each region and/or transition point we discussed the
potential effects of interest in diverse areas, like generating mono-chromatic
gamma ray beams in deep recoil regions with spectral purification, or possible
mechanisms of generation and propagation of very high energy photons in the
cosmological domain.

Acknowledgements: we acknowledge useful discussions with Laura Bandiera and
Gianfranco Paterno’.

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# Component sizes in the supercritical percolation on the binary cube

Michael Krivelevich  School of Mathematical Sciences, Tel Aviv University, Tel
Aviv 6997801, Israel. Email<EMAIL_ADDRESS>
###### Abstract

We present a relatively short and self-contained proof of the classical result
on component sizes in the supercritical percolation on the high dimensional
binary cube, due to Ajtai, Komlós and Szemerédi (1982) and to Bollobás,
Kohayakawa and Łuczak (1992).

The purpose of this expository note is to present a fairly short and
essentially self-contained proof of the classical result of Ajtai, Komlós and
Szemerédi [1] and of Bollobás, Kohayakawa and Łuczak [4] about typical
component sizes in the supercritical percolation on the binary cube. The
argument relies on several beautiful ideas presented in these two papers.

Recall that the $d$-dimensional binary cube $Q^{d}$ is defined as follows: its
vertex set is $V=\\{0,1\\}^{d}$, and two vertices $\bar{x},\bar{y}\in V$ are
connected by an edge in $Q^{d}$ if they differ in exactly one coordinate.
Thus, $Q^{d}$ is a $d$-regular graph on $n:=2^{d}$ vertices. Throughout this
note we assume that $d$ is an asymptotic parameter tending to infinity.

For $p=p(d)\in[0,1]$, form a random subgraph $Q^{d}_{p}$ of $Q^{d}$ by
retaining every edge of $Q^{d}$ independently and with probability $p$.

We start with the much easier to handle subcritical case; peculiarly enough,
this case will turn out to be useful for our argument in the supercritical
regime. This case is immediately settled by the following general statement.

######  Theorem 1.

Let ${\varepsilon}>0$ be a small enough constant. Let $G$ be a graph on $n$
vertices of maximum degree $d\geq 3$. Let $p=\frac{1-{\varepsilon}}{d-1}$, and
form a random subgraph $G_{p}$ of $G$ by retaining every edge of $G$
independently and with probability $p$. Then whp all connected components of
$G_{p}$ are of size at most $\frac{9\ln n}{{\varepsilon}^{2}}$.

Remark. In the above statement we purportedly put the edge probability at
$p=\frac{1-{\varepsilon}}{d-1}$ instead of $p=\frac{1-{\varepsilon}}{d}$, to
make this statement more applicable to the case of constant degree $d$, where
one expects the critical probability $p^{*}$ to be close to $1/(d-1)$. For
constant ${\varepsilon}$ and growing $d$, the difference between $1/(d-1)$ and
$1/d$ is insignificant.

###### Proof.

Let $m=|E(G)|\leq\frac{dn}{2}<n^{2}$. We run the Breadth First Search (BFS)
Algorithm on $G$, feeding it with the sequence of random bits
$\bar{X}=(X_{i})_{i=1}^{m}$, where $X_{i}$’s are independent Bernoulli$(p)$
random variables. Whenever the algorithm queries the existence of the $i$-th
random edge (in its count of exposed edges) $e\in E(G)$, we reach to the bit
$X_{i}$ and assume that $e\in E(G_{p})$ if and only if $X_{i}=1$.

Suppose towards a contradiction that $G_{p}$ has a component $K$ with more
than $k$ vertices, the value of $k$ will be chosen later. Consider the moment
in the algorithm’s execution where it is in the midst of revealing $K$ and has
just discovered the $(k+1)$-th vertex of $K$. In the phase of discovering $K$
up until this moment, only the edges of $G$ incident to the set $K_{0}$ of the
first $k$ discovered vertices of $K$ have been queried, and $k-1$ edges of
$G_{p}$ induced by this set of vertices have been revealed, implying that the
number of edges of $G$ spanned by $K_{0}$ satisfies: $e_{G}(K_{0})\geq k-1$.
Recalling our assumption on the maximum degree of $G$, we conclude that the
number of edges of $G$ queried so far in this phase is at most

$k\cdot d-e_{G}(K_{0})\leq kd-(k-1)=k(d-1)+1\,.$

It follows that the sequence $\bar{X}$ contains $k(d-1)+1$ consecutive bits
$X_{i}$, out of which at least $k$ are equal to 1. For a given interval of
length $k(d-1)+1$, the probability of having at least $k$ ones is

$\operatorname{Pr}\left[\operatorname{Bin}\left(k(d-1)+1,\frac{1-{\varepsilon}}{d-1}\right)\geq
k\right]\leq e^{-\frac{{\varepsilon}^{2}k}{4}}$

by the standard Chernoff-type bounds. Union bounding, we have that the
probability of having an interval of length $k(d-1)+1$ in $[m]$ with at least
$k$ ones is at most $m\cdot e^{-\frac{{\varepsilon}^{2}k}{4}}$. Taking
$k=\frac{9\ln n}{{\varepsilon}^{2}}$, we see that whp $\bar{X}$ has no such
interval, meaning that whp $G_{p}$ has no component larger than $k$. ∎

We now proceed to the main part of this note, the supercritical regime. For
$c>1$, define $y=y(c)$ to be the unique solution in $(0,1)$ of the equation

$y=1-\exp\\{-cy\\}\,.$ (1)

Note that for $c=1+\epsilon$ and ${\varepsilon}>0$ small enough, we can
estimate $y=(1+o_{{\varepsilon}}(1))2{\varepsilon}$.

The following is a fundamental statement on the typical component sizes in the
random subcube $Q^{d}_{p}$ for the supercritical case $p=\frac{c}{d}$, $c>1$ a
constant.

######  Theorem 2.

[1, 4] Let $c>1$ be a constant, and let $p=\frac{c}{d}$. Form a random
subgraph $Q^{d}_{p}$ of the $d$-dimensional binary cube $Q^{d}$ by retaining
every edge of $Q^{d}$ independently and with probability $p$. Then, whp the
graph $Q^{d}_{p}$ has a connected component $L_{1}$ whose size is asymptotic
to $yn$, where $y=y(c)$ is as defined in (1), and all remaining connected
components $L_{i},i\geq 2$, satisfy: $|L_{i}|\leq\frac{d}{c-1-\ln c}$.

We will need the following facts.

######  Lemma 1.

For a constant $c>1$ and $d\rightarrow\infty$, the probability of survival of
the Galton-Watson tree with offspring distribution
$\operatorname{Bin}(d,\frac{c}{d})$ is asymptotic to $y$, where $y=y(c)$ is
defined by (1).

This is very standard, see, e.g., Theorem 4.3.12 of [5] for a proof.

The next statement is the famed edge isoperimetric inequality for the binary
cube due to Harper (1964).

######  Lemma 2.

[6] Let $S\subset V(Q^{d})$, $|S|\leq 2^{d-1}$. Then the number of edges
$e_{Q_{d}}(S,\bar{S})$ between $S$ and its complement $\bar{S}$ in $Q^{d}$
satisfies: $e_{Q^{d}}(S,\bar{S})\geq|S|(d-\log_{2}|S|)$. In particular,
$e_{Q^{d}}(S,\bar{S})\geq|S|$.

The next lemma bounds from above the number of trees of a given size
containing a given vertex in a base graph of bounded degree.

######  Lemma 3.

Let $G$ be a graph of maximum degree $d$, and let $k>0$ be an integer. Then
for every $v\in V(G)$, the number of subtrees of $G$ on $k$ vertices
containing $v$ is at most $(ed)^{k-1}$.

For completeness, we present its short proof here, as given by Beveridge,
Frieze and McDiarmid ([3], Lemma 2).

###### Proof.

Denote by ${\cal T}(v,k)$ the set of all subtrees of $G$ of size $k$ rooted at
$v$, the quantity in question is then $t(v,k):=|{\cal T}(v,k)|$. Given a tree
$T\in{\cal T}(v,k)$, label $v$ with $k$, and choose a labeling
$f:V(T)\setminus\\{v\\}\rightarrow[k-1]$ of the remaining vertices of $T$.
Clearly, every $T\in{\cal T}(v,k)$ is in $(k-1)!$ many such pairs $(T,f)$.
Furthermore, each such pair defines a unique spanning tree $T^{\prime}$ of
$K_{k}$, in which $(i,j)$ is an edge of $T^{\prime}$ if and only if the
vertices $x,y\in V(T)$ with $f(x)=i$, $f(y)=j$ are connected by an edge of
$T$.

Now, we need to see in how many ways a spanning tree $T^{\prime}$ of $K_{k}$
can be obtained under such labelings. Fix, say, a BFS order on $T^{\prime}$
starting from $k$, and upon reaching vertex $\ell\in[k-1]$ for the first time,
define $f^{-1}(\ell)$. For this, we need to allocate a neighbor in $G$ of the
preimage of the already embedded father of $\ell$, and this can be done in at
most $d$ ways. Hence, using Cayley’s formula to count the number of spanning
trees $T^{\prime}$ of $K_{k}$, we obtain:

$\mbox{\\# of pairs }(T,f)=t(v,k)\cdot(k-1)!\leq d^{k-1}k^{k-2}\,,$

implying $t(v,k)\geq\frac{k^{k-2}}{(k-1)!}d^{k-1}$. It is easy to verify that
$\frac{k^{k-2}}{(k-1)!}<e$ for every positive integer $k$, and the lemma
follows. ∎

We now proceed to the proof of Theorem 2. From now on we assume $c>1$ is a
constant. We also assume that $t>0$ is a fixed integer, whose specific value
will be set later in the proof.

######  Lemma 4.

Set $p=\frac{c}{d}$, and form a random subgraph $Q^{d}_{p}$. Then whp
$Q^{d}_{p}$ has no components of size between $\frac{d}{c-1-\ln c}$ and
$d^{t}$.

###### Proof.

Let $v\in V(Q^{d})$, and denote by $C(v)$ the connected component of
$Q^{d}_{p}$ containing $v$. For $|C(v)|=k$ to happen, some tree $T$ of order
$k$ in $Q^{d}$ containing $v$ should have all of its $k-1$ edges open in
$Q^{d}_{p}$, yet all edges of $Q^{d}$ between $V(T)$ and its complement are to
stay close in $Q^{d}_{p}$. By Lemma 2, the number of such edges is at least
$k(d-\log_{2}k)$. Hence, using Lemma 3 and the union bound, we derive:

$\displaystyle\Pr\left[\frac{d}{c-1-\ln c}\leq|C(v)|\leq d^{t}\right]$
$\displaystyle\leq$ $\displaystyle\sum_{k=\frac{d}{c-1-\ln
c}}^{d^{t}}(ed)^{k-1}p^{k-1}(1-p)^{dk-k\log_{2}k}$ $\displaystyle\leq$
$\displaystyle\sum_{k=\frac{d}{c-1-\ln
c}}^{d^{t}}(ed)^{k-1}\left(\frac{c}{d}\right)^{k-1}e^{-\frac{c}{d}(dk-k\log_{2}k)}\leq\sum_{k=\frac{d}{c-1-\ln
c}}^{d^{t}}\left(ece^{-c+\frac{c\log_{2}k}{d}}\right)^{k}$ $\displaystyle=$
$\displaystyle\sum_{k=\frac{d}{c-1-\ln c}}^{d^{t}}\left(e^{-c+1+\ln
c+o(1)}\right)^{k}=o(1/n)\,.$

Applying the union bound over all $n$ vertices of $Q^{d}$, we establish the
lemma. ∎

######  Lemma 5.

Set $p=\frac{c}{d}$, and form a random subgraph $Q^{d}_{p}$. Let $v\in
V(Q^{d})$, and denote by $C(v)$ the connected component of $Q^{d}_{p}$
containing $v$. Then:

$\Pr[|C(v)|\geq d^{t}]=(1-o_{d}(1))y\,,$

where $y=y(c)$ is defined by (1).

###### Proof.

First, we estimate the probability of $|C(v)|\geq d^{1/2}$. We run the BFS
algorithm on $Q^{d}$, starting from $v$ and feeding it with independent
Bernoulli$(p)$ bits, one for each queried edge of $Q^{d}$. For as long as
$|C(v)|\geq d^{1/2}$, every vertex $u\in V(Q^{d})$, queried for neighbors
outside of the current component $C(v)$, has at least $d-d^{1/2}$ potential
neighbors to query. Hence the exploration process can be coupled with the
Galton-Watson tree rooted at $v$ with offspring distribution
$\operatorname{Bin}(d-d^{1/2},p)$. Since $(d-d^{1/2})p=c-o_{d}(1)$, by Lemma 1
the component $C(v)$ grows to $d^{1/2}$ with probability asymptotic to $y$.

Now we estimate $\Pr[d^{1/2}\leq|C(v)|\leq d^{t}]$. The argument here is
nearly identical to that of Lemma 4, and hence we allow ourselves to be brief.
We have:

$\displaystyle\Pr[d^{1/2}\leq|C(v)|\leq d^{t}]$ $\displaystyle\leq$
$\displaystyle\sum_{k=d^{1/2}}^{d^{t}}(ed)^{k-1}p^{k-1}(1-p)^{dk-k\log_{2}k}\leq\sum_{k=d^{1/2}}^{d^{t}}(ed)^{k-1}\left(\frac{c}{d}\right)^{k-1}e^{-\frac{c}{d}(dk-k\log_{2}k)}$
$\displaystyle\leq$
$\displaystyle\sum_{k=d^{1/2}}^{d^{t}}\left(ece^{-c+\frac{c\log_{2}k}{d}}\right)^{k}=o(1)\,,$

since $ece^{-c}<1$ for $c>1$. We thus conclude:

$\Pr[|C(v)|\geq d^{t}]=(1-o_{d}(1))y\,,$

as desired. ∎

######  Lemma 6.

Set $p=\frac{c}{d}$, and form a random subgraph $Q^{d}_{p}$. Let
$W=\\{v:|C(v)|\geq d^{t}\\}$. Then whp,

$|W|=(1+o(1))yn\,.$

###### Proof.

By the previous lemma, we have $\operatorname{\mathbb{E}}[|W|]=(1+o(1))yn$. In
order to argue about concentration, apply edge exposure martingale to
$Q^{d}_{p}$ (see, e.g., Chapter 7 of [2]). Adding or deleting an edge can
change the value of $|W|$ by at most $2d^{t}$. Hence, by the standard edge
exposure martingale inequality:

$\Pr\left[||W|-\operatorname{\mathbb{E}}[|W|]|\geq n^{2/3}\right]\leq
2\exp\left\\{-\frac{n^{4/3}}{2\cdot\frac{nd}{2}\cdot(2d^{t})^{2}}\right\\}=o(1).$

The lemma follows. ∎

We now argue that typically in $Q^{d}_{p}$ “largish” components are well
spread.

######  Lemma 7.

Set $p=\frac{c}{d}$, and form a random subgraph $Q^{d}_{p}$. Let
$W=\\{v:|C(v)|\geq d^{t}\\}$. Then whp every vertex $v\in V(Q^{d})$ is at
distance at most two (in $Q^{d}$) from $W$.

###### Proof.

It is enough to prove that the all-zero vertex $\bar{0}$ is at distance at
most two from $W$ with probability $1-o(1/n)$. Set
${\varepsilon}=\frac{c-1}{c}$. Let $I$ be the set of first
$\left\lfloor\frac{{\varepsilon}d}{2}\right\rfloor$ coordinates, and for
$i\neq j\in I$, look at the subcube

$H_{ij}=\\{v\in Q^{d}:v_{i}=v_{j}=1,v_{k}=0\mbox{ for all }k\in
I\setminus\\{i,j\\}\\}\,.$

Let $u_{ij}$ be the vertex with 1 in coordinates $i$ and $j$ and 0 in all
other coordinates. Clearly, $u_{ij}$ is at distance two from $\bar{0}$, and
$u_{ij}$ belongs to $H_{ij}$.

The subcubes $H_{ij}$ are vertex disjoint and have dimension
$d^{\prime}=d-\left\lfloor\frac{{\varepsilon}d}{2}\right\rfloor\geq\left(1-\frac{{\varepsilon}}{2}\right)d$.
Performing percolation on $H_{ij}$ with $p=\frac{c}{d}$ and observing that
$d^{\prime}p\geq\left(1-\frac{{\varepsilon}}{2}\right)dp=\left(1-\frac{{\varepsilon}}{2}\right)c=\frac{c+1}{2}$,
we derive from Lemma 5 that $\Pr[|C(u_{ij})|\geq
d^{t}]=(1+o(1))y(d^{\prime}p)=(1+o(1))y\left(\frac{c+1}{2}\right)\geq\delta$
for some $\delta=\delta(c)>0$. These events are independent for distinct pairs
$\\{i,j\\}$. Hence the probability that $\bar{0}$ is not at distance at most
two (in $Q^{d}$) from $W$ is at most
$\Pr\left[\operatorname{Bin}\left(\binom{\left\lfloor\frac{{\varepsilon}d}{2}\right\rfloor}{2},\delta\right)=0\right]=e^{-\Theta(d^{2})}=o(1/n)$.
∎

Now it is time for the final attack. Consider $Q^{d}_{p}$, $p=\frac{c}{d}$. We
define probabilities $p_{1},p_{2}$ as follows:

$p_{2}=\frac{1}{d^{5}},\quad 1-p=(1-p_{1})(1-p_{2})\,,$

we have $p_{1}=\frac{c-o(1)}{d}$. Let $G_{1}\sim Q^{d}_{p_{1}},G_{2}\sim
Q^{d}_{p_{2}}$ be independent random subgraphs of $Q_{p}$, their union is
distributed exactly as $Q^{d}_{p}$. Set

$t=31\,,$

and define

$W_{1}=\\{v\in V(Q^{d}):|C_{G_{1}}(v)|\geq d^{t}\\}\,.$

By Lemma 6, whp $|W_{1}|=(1+o(1))yn$. By Lemma 4, whp components of $G_{1}$
outside $W_{1}$ are $O(d)$ in sizes.

We now expose the edges of $G_{2}$.

######  Lemma 8.

Whp all components of $G_{1}[W_{1}]$ merge into a single component in
$Q^{d}_{p}$.

###### Proof.

The proof follows the pioneering idea of [1]. Recall that the edges of $Q^{d}$
appear as edges of $G_{2}$ independently and with probability $p_{2}=d^{-5}$.
By Lemma 7, we can assume that every $v\in V(Q^{d})$ is at distance at most
two from $W_{1}$. If the statement does not hold, then we can partition the
components of $G_{1}[W_{1}]$ into two families ${\cal A}$ and ${\cal B}$ such
that there are no paths in $G_{2}$ between sets $A:=\cup_{C\in{\cal A}}V(C)$
and $B:=\cup_{C\in{\cal B}}V(C)$. Let $s\leq n$ be the total number of
components in $G_{1}[W_{1}]$, and let $\ell\leq s/2$ be the number of
components in a family with fewer components. Then $|A|,|B|\geq\ell\cdot
d^{31}$.

Now, since every vertex of $Q^{d}$ is at distance at most two from $A\cup B$,
we can partition $V(Q^{d})$ into sets $A^{\prime},B^{\prime}$, where
$A\subseteq A^{\prime}$ and $A^{\prime}$ contains all vertices of
$V(Q^{d})\setminus B$ at distance at most two from $A$, and every vertex in
$B^{\prime}:=V(Q^{d})\setminus A^{\prime}$ is at distance at most two from
$B$. By Harper’s edge isoperimetric inequality (Lemma 2), $Q^{d}$ has at least
$\min\\{|A^{\prime}|,|B^{\prime}|\\}\geq\ell d^{31}$ edges between
$A^{\prime}$ and $B^{\prime}$. Since every vertex in $A^{\prime}$ is at
distance at most two from $A$ and every vertex in $B^{\prime}$ is at distance
at most two from $B$, we can extend each edge crossing between $A^{\prime}$
and $B^{\prime}$ to a path of length at most five between $A$ and $B$. As
every edge of $Q^{d}$ is contained in less than $5d^{4}$ paths of length at
most five, by applying a simple greedy argument we obtain a family of
$\frac{\ell d^{31}}{5\cdot 5d^{4}+1}\geq\frac{\ell d^{27}}{30}$ edge-disjoint
paths of length at most five between $A$ and $B$. The probability that none of
these paths is in $G_{2}$ is at most $(1-p_{2}^{5})^{\frac{\ell
d^{27}}{30}}\leq e^{-\frac{\ell d^{2}}{30}}$. Also, $s\leq n$, and hence the
number of ways to partition the components of $G_{1}[W_{1}]$ into ${\cal
A},{\cal B}$ with $\ell$ components in one of the families is at most
$\binom{s}{\ell}\leq\binom{n}{\ell}$. Thus, by the union bound the probability
that the lemma’s statement fails is most

$\sum_{\ell=1}^{s/2}\binom{s}{\ell}e^{-\frac{\ell
d^{2}}{30}}\leq\sum_{\ell=1}^{s/2}\left(\frac{en}{\ell}\cdot
e^{-\frac{d^{2}}{30}}\right)^{\ell}=o(1)\,.$

∎

Proof of Theorem 2. By Lemma 6, whp $|W_{1}|=(1+o(1))yn$. By Lemma 8, whp all
components in $W_{1}$ merge into a single component $L_{1}$, whose size is
then at least $(1+o(1))yn$. Let us now look at the situation with components
outsize of $W_{1}$. Define an auxiliary graph $\Gamma$ whose vertices are the
components of $G_{1}$ outside of $W_{1}$, where two components are connected
by an edge in $\Gamma$ if $G_{2}$ contains at least one edge between them. By
Lemma 4, whp all components of $G_{1}$ outside of $W_{1}$ are of size $O(d)$,
and thus, using a very crude estimate, there are whp $O(d^{2})$ edges of
$Q^{d}$ between every pair of connected components. Hence every pair of
components is connected by an edge of $\Gamma$ independently and with
probability at most $1-(1-p_{2})^{d^{2}}\leq p_{2}d^{2}=O(d^{-3})$. By the
same reasoning, the maximum degree of the underlying graph of $\Gamma$ is whp
$O(d^{2})$. Hence, applying Theorem 1 we derive that whp all components of
$\Gamma$ are $O(d)$. Recalling that whp every component of $G_{1}$ outside of
$W_{1}$ has $O(d)$ vertices, this yields that whp all components of
$Q^{d}_{p}$ outside of $W_{1}$ are $O(d^{2})$ in sizes. Invoking Lemma 4 once
again, we conclude that in fact whp all such components $L_{i}$ satisfy:
$|L_{i}|\leq\frac{d}{c-1-\ln c}$. Finally, applying Lemma 6, this time to
$Q^{d}_{p}$, we can give a likely upper bound on $|L_{1}|$:
$|L_{1}|\leq(1+o(1))yn$. The theorem follows. $\Box$

Remark. The same proof, with minimal changes, works also for (some part of)
the slightly supercritical regime $p=\frac{1+{\varepsilon}}{d}$, for
$d^{-a}\leq{\varepsilon}={\varepsilon}(d)=o_{d}(1)$, for some constant $a>0$.

Acknowledgment. The author wishes to thank his frequent coauthors of recent
papers on percolation on finite graphs Sahar Diskin, Joshua Erde and Mihyun
Kang for the fruitful and enjoyable cooperation, which spurred this note.

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