# Extreme-value statistics and super-universality in critical percolation?

Mohadeseh Feshanjerdi<EMAIL_ADDRESS>Department of Condensed
Matter Physics, Faculty of Physics, Alzahra University, P. O. Box 1993893973,
Tehran, Iran  Peter Grassberger<EMAIL_ADDRESS>JSC, FZ
J$\ddot{u}$lich, D-52425 J$\ddot{u}$lich, Germany  Max Planck Institute for
the Physics of Complex Systems, Nöthnitzer Str.38, 01187 Dresden, Germany

###### Abstract

Recently, the number of non-standard percolation models has proliferated. In
all these models, there exists a phase transition at which long range
connectivity is established, if local connectedness increases through a
threshold $p_{c}$. In ordinary (site or bond) percolation on regular lattices,
this is a well understood second-order phase transition with rather precisely
known critical exponents, but there are non-standard models where the
transitions are in different universality classes (i.e. with different
exponents and scaling functions), or even are discontinuous or hybrid. It was
recently claimed that certain scaling functions are in all such models given
by extreme-value theory and thus independent of the precise universality
class. This would lead to super-universality (even encompassing first-order
transitions!) and would be a major break-through in the theory of phase
transitions. We show that this claim is wrong.

##  I Introduction

Percolation Stauffer ; Sahimi  is one of the paradigmatic topics in
statistical physics. It is concerned with the appearance or breakdown of long-
range connectivity, as local connections are established or removed. This can
be the formation of a gel if bonds between molecules are increased, the
spreading of a disease if contact between people is increased above a
threshold, or even the acceptance of a scientific theory as the number of its
supporters surpasses a critical level. It is thus not surprising that
percolation is a topic discussed in all branches of science, from condensed
matter physics to astrophysics, biology, finance, and sociology Saberi ;
Araujo .

What makes percolation an interesting object of research is that the
transition to long-range connectivity is, in the large system limit, abrupt
with all features usually attached to a phase transition. In what we shall
call in the following “ordinary percolation” (OP), this is a well-understood
continuous (‘second order’) transition, whose critical exponents are exactly
known in $d=2$ and for $d>6$, and known to high precision in dimensions three
to five. The best known examples of OP are bond and site percolation on
regular lattices Stauffer ; Sahimi .

Just like other critical phenomena, also OP enjoys what is known as
universality. This means that properties like critical exponents and scaling
functions are strictly independent of details like the type (but not the
dimension) of the lattice, and whether local connections are broken by
removing sites or bonds. A set of microscopic models which thus have identical
scaling behavior is called a universality class.

But the very wide conceptual range of the notion of percolation implies that
there are systems which fall outside the standard universality classes or,
otherwise said, there are different universality classes. Examples are
rigidity rigidity , bootstrap Adler ; di_Muro , correlated Weinrib ; Schrenk
(e.g. pacman Abete  and aftermath aftermath ) and explosive Achlioptas
percolation, percolation on random (Erdös-Renyi) networks Newman ,
agglomerative percolation Lau , percolation driven by random walks Feshanjerdi
, systems where the ‘local’ bonds can be arbitrarily long and have power-law
distributions Grassberger2012 ; Grassberger2013 ; Gori , and systems where
several local elements have to cooperate in order to establish connections
Dodds ; Dorogovtsev2006 ; Bizhani ; Cai . In some of these cases Adler ;
di_Muro ; Dodds ; Dorogovtsev2006 ; Bizhani ; Cai  the differences to OP are
so big that the transition is discontinuous (‘first order’).

Even when two systems are in different universality classes, it may happen
that for some reason one or more critical exponents (or scaling functions) are
identical, which is then called super-universality. The best known examples
are provided by surface critical phenomena Cardy . Take e.g. site percolation
on a semi-infinite square lattice with $-\infty<x,y<\infty,0<z<\infty$. In
this case clusters growing from a surface site with $z=0$ have different
fractal dimensions than clusters deep in the bulk, but correlations are
described by the usual (bulk) correlation length exponent $\nu$, even very
close to the surface surface-percol . The same happens for the Ising model and
for self-avoiding random walks. Another example is random interfaces within
the so-called Kardar-Parisi-Zhang KPZ  universality class. Here the critical
exponents are independent of the genus of the surface (asymptotically flat,
circular, …), but scaling functions do in general depend on the genus Takeuchi
. Finally, we might mention polymer networks Duplantier  where the Flory index
controlling the size is independent of the topology, but the exponents
controlling the free energy are not.

Recently, it was claimed Fan  that such super-universality is for percolation
much more common than previously thought. The reason, it is claimed in Fan ,
is the following. Usually, a critical point $p_{c}$ is defined as the point in
control parameter space where observables or partition sums of infinite
systems would develop singularities. One can define effective transition
points also for finite systems, but in the standard treatment such definitions
are always ad hoc and heuristic. For percolation problems, a more principled
way to define transition points for finite systems consists in looking at
jumps in the mass of the largest cluster, as local connections are added one
by one Manna  (see also Nagler ; Lee ). If the system is far subcritical,
these jumps are small, since clusters are anyhow small. They are also small in
far supercritical systems, since there the giant cluster can grow only by
‘eating’ much smaller ones. But near the critical point, giant clusters can
grow by merging two or more large clusters, and thus jumps will be large. Thus
one can define a transition point $p_{\rm max}({\cal E})$ for each (finite)
event ${\cal E}$ as the point where the largest jump (called the ‘gap’ in the
following) occurs in ${\cal E}$. Here, an ‘event’ ${\cal E}$ is the entire
evolution of one realization of the system.

In standard finite size scaling (FSS) analyses, observables are considered as
functions of $p-p_{c}$. In addition, one can study the distributions of
observables at $p=p_{c}$, like the masses of the largest and second-largest
clusters. The claim made in Fan  is that several scaling functions become
super-universal in a very wide class of non-standard percolation models, if
FSS analysis is based on $p_{\rm max}({\cal E})$. To be more precise, consider
the distribution of gap sizes over the entire ensemble. By definition, the gap
size of ${\cal E}$ is the size of the largest jump in the entire history of
the event, and $p_{\rm max}({\cal E})$ is its location in the history. Thus a
gap is defined as an extremal value in a set of jumps (all jumps in the given
history). In the limit of large systems this set becomes infinite, and the
jumps in one history are basically unbounded and uncorrelated. Thus the gap
(the extremal value in this set) should be distributed according to a Gumbel
(Fisher-Tippett type 1) distribution Gumbel . Since this should hold for all
(generalized) percolation models, the gap distribution should be super-
universal. The weak point in this argument is, of course, that jumps are
unbounded and uncorrelated only for infinite systems, while we are dealing
with large but finite systems.

It was indeed claimed in Fan  that not only the gap sizes obey a Gumbel
distribution, but also the masses of the largest cluster at $p_{\rm max}({\cal
E})$. Finally, the transition points $p_{\rm max}({\cal E})$ themselves were
claimed to obey a Gauss distribution.

The latter two claims were made without strong mathematical arguments, but
also in case of the gap distribution the argument is very weak indeed. Since
the sum of all jumps in an event is constrained to be the total mass of the
system, it is clear that the jumps in an event cannot be uncorrelated. It
might be a priori possible that these correlations are sufficiently weak that
the distribution is still Gumbel, but this is by no means guaranteed.

Indeed, Fan et al. were not the first ones to claim that the Gumbel
distribution is relevant for percolation. Bazant Bazant  showed that the
masses of the largest cluster in subcritical OP follow a Gumbel distribution,
in the infinite system limit. Although not rigorously provable Bazant , this
makes perfect sense, since in this case the variables are effectively
uncorrelated and unlimited. A similar but less convincing claim was made by
Bramwell et al. Bramwell , who considered the masses of the largest cluster in
nearly critical OP. Precisely at the critical point, the distribution of the
largest cluster is well known to follow a power law for small masses, and to
have a rather sharp peak at large masses Stauffer . As one decreases $p$, the
peak first becomes a shoulder, and then the two merge to form something
resembling superficially a Gumbel distribution. But this is far from an exact
identity, unless one strictly follows Bazant’s strategy of keeping a fixed
$p<p_{c}$ and taking $N\to\infty$. While it is crucial for Bazant  that
different regions in a large subcritical percolation problem are essentially
uncorrelated, Bramwell  claim that their finding holds in an “inertial range”,
and stress that “One of the anomalous statistical properties of inertial
systems is that, whatever their size, they can never be divided into
mesoscopic regions that are statistically independent”. They are right that
this excludes the validity of Gauss’ central limit theorem, but it also
prevents the cluster size distributions from being Gumbels.

In the rest of this paper we shall investigate the claim made by Fan et al.
numerically, by studying six different universality classes. Of these six,
five show continuous transitions, while one (the cluster aggregation (DCA)
model of Cho ) has a discontinuous transition. For the latter, it is
immediately seen that the claim of Fan et al. is wrong, and we thus show only
data for two system sizes. Among the other five models, three are on lattices
(2-d and 3-d site percolation and aftermath percolation in $d=3$ aftermath ),
while two ( Erdös-Renyi (ER) and product-rule explosive percolation (PR_EP))
are off-lattice. Simulation data for five of these universality classes
(DCA,ER,PR_EP, 2-d and 3-d OP) had been used by Fan et al. to support their
claim. We find that in all cases (except possibly PR_EP) the distributions of
$p_{c}({\cal E})$ are wider than Gaussians, although the differences are not
dramatical. Much more significant are the differences among the gap
distributions. Except possibly for ER, they are in all cases significantly
more narrow than Gumbels, and differ widely from model to model. The same is
true, although to a smaller amount, for mass distributions. While the sizes of
the systems simulated in the present paper are comparable to those in Fan  or
even slightly smaller, it seems that our statistics are vastly superior (for
the largest systems we made $>4\times 10^{5}$ simulations for each model, for
the smallest ones $>4\times 10^{6}$).

##  II Simulations and Finite Size System Analysis

All six models were simulated by means of the Newman-Ziff (NZ) algorithm
Newman-Ziff , because in this way we had the fastest and easiest access to
jumps in the masses of the largest cluster. In this algorithm, one starts with
an ‘empty’ system, and sites or bonds are inserted one after another. The mass
of the largest cluster after the $n-$th insertion is $S_{n}$, and the $n-$th
jump is defined as $\Delta_{n}=S_{n}-S_{n-1}$. For each model we denote the
control parameter by $p=n/N$, where $N$ is the total number of sites and $n$
is the number of already inserted sites/bonds. For the lattice models (2-d and
3-d site and aftermath) $p$ is the density of inserted sites, while $p$ is the
fraction of established bonds for the off-lattice models (ER, PR_EP, and DCA).
For the latter (DCA), a rather complicated algorithm was described in the
supplementary material suppl  of Fan . We did not use this, but we used the
simple algorithm described in Cho . Nevertheless, our data seem to agree
perfectly with the simulation data of Fan , even though our interpretation of
these data differs radically.

For the lattice models, we used helical boundary conditions, i.e. sites were
(irrespective of the dimension $d$) indexed by a single integer $i$, neighbors
were $i\pm 1,i\pm L,\ldots i\pm L^{d-1}$, and $i+N\equiv i\;{\rm mod}\;N$ with
$N=L^{d}$. System sizes were $L=512,1024,2048$, and 4096 for 2-d site OP,
$L=32,64,128$, and 256 for 3-d lattices, and $N=2^{16},2^{18},2^{20}$, and
$2^{22}$ for ER and PR_EP. For DCA we show results only for $N=2^{16}$ and
$N=2^{18}$, since the conclusions from this were obvious anyhow.

The gap $\Delta_{\rm max}$ is defined the largest jump during a run of the NZ
algorithm. Assume that it occurs at the $n-$th insertion. This defines the
event-based transition point as $p_{\rm max}({\cal E})=n/N$, and $S_{\rm max}$
is the mass of the largest cluster after the $n-$th insertion.

For each model (and for each system size) we show the three distributions
$P_{p}(p_{c}),P_{\Delta}(\Delta_{\rm max})$, and $P_{S}(S_{\rm max})$.
According to the standard FSS scaling assumptions, they are supposed to
satisfy for finite-dimension models the scaling laws

$P_{S}(S_{\rm max})=L^{-d_{f}}f_{S}(S_{\rm max}/L^{d_{f}}),$ (1) $P_{p}(p_{\rm
max})=L^{1/\nu}f_{p}[(p_{\rm max}-p_{c})L^{1/\nu}],$ (2)

and

$P_{{}_{\Delta}}(\Delta_{\rm max})=L^{-d_{f}}f_{{}_{\Delta}}(\Delta_{\rm
max}/L^{d_{f}}),$ (3)

where $d_{f}$ is the fractal dimension, $\nu$ is the correlation length
exponent, and the functions $f_{S},f_{p}$, and $f_{{}_{\Delta}}$ are universal
scaling functions. For ER and PR_EP, we use the same ansatzes but with $L$
replaced by $N$ and with $d_{f}$ replaced by the scaling exponent $\eta$ in
the relationship $S_{\rm max}\sim N^{\eta}$, and $\nu$ must be defined by
$p_{\rm max}-p_{c}\sim N^{-\nu}$.

Equations (1-3) correspond to Eqs. (11-13) of Fan , but some comments are in
order:
(i) Eqs. (1) and (3) follow immediately from the assumption that scaling
functions $f_{i}$ must have dimensionless arguments and that $S_{\rm max}$ and
$\Delta_{\rm max}$ both are masses. Similarly, Eq. (2) follows from the fact
$(p_{\rm max}-p_{c})L^{1/\nu}$ is essentially the unique dimensionless
variable involving $p_{\rm max}$. The arguments put forward in Fan  seem more
complicated.
(ii) In Eqs. (11) and (13) of Fan , the scaling functions are assumed to be
functions of $S_{\rm max}-\langle S_{\rm max}\rangle$ and $\Delta_{\rm
max}-\langle\Delta_{\rm max}\rangle$, respectively. Since $\langle S_{\rm
max}\rangle$ and $\langle\Delta_{\rm max}\rangle$ are also masses, this
complication has no consequences, except that fitting procedures become less
straight forward.
(iii) The critical exponents in Eqs. (11-13) of Fan  look much more
complicated. But although the notation “$\beta_{2}$” in Eq.(11) suggests a
relation with the order parameter $\beta$, actually $\beta_{2}=d-d_{f}$. The
reason why we have $-d_{f}$ in our Eq.(3) instead of $d-d_{f}$ is that we
defined $\Delta_{n}=S_{n}-S_{n-1}$, while $\Delta_{n}=(S_{n}-S_{n-1})/S_{n}$
in Fan .
(iv) The subscript ‘2’ on all critical exponents in Eqs. (11-13) of Fan
should indicate that variances of observables for finite systems and shifts of
these observables from their infinite system values might follow different
power laws, if standard FSS does not hold – as was indeed found in Christensen
for correlation lengths in explosive percolation. As shown in Fan ; Li2023 ,
this is still true if one uses an event-based FSS analysis as in the present
paper (whence the title of Li2023  is somewhat misleading). Thus Eq. (2) must
be changed for explosive percolation to allow for two different exponents
differents instead of the single $\nu$.

##  III Results

###  III.1 Discontinuous Transition

We first show results for the discontinuous percolation cluster aggregation
model (DCA) of Cho . In this model, one starts with $N$ nodes and zero links.
Links are inserted in 2 steps:
(i) choose two clusters, with probabilities $q_{i}\propto s_{i}^{\omega}$
($i=1,2$), where $s_{i}$ is the mass of cluster $i$ and $\omega\in[0,1]$ is a
constant;
(ii) on each cluster choose a random point, and link these two points, if they
are not yet linked. If they are already linked, choose another pair and
repeat.
If the two chosen clusters are the same, this has no effect on the cluster
distribution. If they are different, however, they are joined by the new link.
This is repeated until all clusters are joined into one big cluster.

Notice that this would be exactly the cluster growth in the ER model, if
$\omega=1$. Interesting new results are found for $\omega<1$. We and Fan et
al. Fan  simulated only the extreme case $\omega=0$. In this case the
algorithm can be simplified drastically – in particular, if the NZ algorithm
is used: In each NZ step, two different clusters are chosen randomly, and
joined by adding a pointer from the root of one of them to the root of the
other. Notice, however, that this simplification – which does not change
$P_{S}$ and $P_{\Delta}$ – changes the definition of the control parameter, as
no links are inserted which do not join two clusters.

It turns out that this leads in many events to a percolation transition
(largest jump) when the last two clusters get joined, and since each of these
two clusters has in general $O(N)$ sites, the transition is discontinuous.
Compared to discontinuous transitions in models with cooperativity Dodds ;
Janssen ; Dorogovtsev2006 ; Bizhani ; Cai  or with dependency links Parshani ;
Son , where the transitions usually happen at $p_{c}<1$, this transition is a
bit trivial (like the OP transition in $d=1$ at $p_{c}=1$), but it is formally
a genuine discontinuous transition.

Figure 1: Distributions of $S_{\rm max}/N$ (panel a), of $p_{\rm max}N$ (panel
b), and of $\Delta_{\rm max}/N$ (panel c) for the DCA model of Cho  at
$\omega=0$. Because we used an optimized algorithm where bonds are inserted
only when needed to join two clusters, $P_{p}(p_{\rm max})$ differs from the
distribution shown in Fan , but the other two distributions should be the
same.

Distributions $P_{S}(S_{\rm max}),P_{p}(p_{\rm max})$, and
$P_{{}_{\Delta}}(\Delta_{\rm max})$ for $N=2^{16}$ and $N=2^{18}$ are shown in
Fig.1. The mass distribution $P_{S}(S_{\rm max})$ shows a delta-peak at
$S_{\rm max}$ due to the fact that there is a finite chance that the maximal
jump occurs exactly at $p=1$. Accordingly, $P_{p}(p_{\rm max})$ and
$P_{{}_{\Delta}}(\Delta_{\rm max})$ show jumps at $p_{\rm max}=1$ and at
$\Delta_{\rm max}=1/2$. The latter is maybe the most obvious indication that
the claim of Fan  that this is a Gumbel distribution (which has an exponential
right hand-tail) is wrong. It is not clear why Fan et al. made this claim,
since our Fig.1c agrees perfectly with their Fig.4a. The distribution of
$S_{\rm max}$ shown in Fig.4c of Fan  lacks the delta-peak, but shows the same
sharp cut-off as our Fig. 1a. Also from this, it should be clear that the
universality claim of Fan et al. is untenable. Finally, our distribution of
$p_{\rm max}$ differs markedly from that of Fan et al., but as we pointed out
above, this was to be expected.

Figure 2: Distributions of $S_{\rm max}/N$ (panel a), of $p_{\rm max*N}$
(panel b), and of $\Delta_{\rm max}/N$ (panel c) for ordinary site percolation
in $d=2$. In contrast to Fig. 1, normalization is now arbitrary, and log-
scales are used on the y-axis in all panels. In panels (b) and (c) we draw
also best-fit Gauss (panel (b)) and Gumbel (panel (c)) distributions. Because
we could not produce even a modestly acceptable Gumbel fit for panel (a), we
plot there the best-fitting Gumbel distribution with the $\beta$-parameter
obtained in Fan  for 2-d bond percolation (due to universality, site and bond
percolation should have identical scaling functions).

Since the transition in the DCA model is not hybrid, the critical exponents
are not anomalous. We see clearly $\langle S_{\rm
max}\rangle\sim\langle\Delta_{\rm max}\rangle\sim N$ in agreement with Fan et
al., and $1-\langle p_{\rm max}\rangle\sim 1/N$. The latter disagrees with the
claim of Fan et al. that the model shows two different correlation length
exponents, and might be due to our optimized algorithm.

###  III.2 Ordinary Site Percolation in $d=2$

In the previous subsection we saw that the claim of Fan et al. is wrong in a
case where the distributions $P_{S}(S_{\rm max})$ and
$P_{{}_{\Delta}}(\Delta_{\rm max})$ could not be Gumbel simply by the fact
that they were sharply cut off at finite arguments, while the Gumbel
distribution has an exponential tail. This could have been a simple oversight.

The next case, OP in $d=2$, is more serious. In Fig.2 we plot $P_{S}(S_{\rm
max}),P_{p}(p_{\rm max})$, and $P_{{}_{\Delta}}(\Delta_{\rm max})$ for
$N=2^{16},2^{18},2^{20}$, and $2^{22}$. To obtain a data collapse, we used on
the $x$-axes the variables $S_{\rm max}/L^{d_{f}},(p_{\rm
max}-p_{c})L^{1/\nu}$, and $\Delta_{\rm max}/L^{d_{f}}$ with known values of
$d_{f},\nu$, and $p_{c}$. Normalization is arbitrary.

Figure 3: Distributions of $S_{\rm max}/N$ (panel a), of $p_{\rm max*N}$
(panel b), and of $\Delta_{\rm max}/N$ (panel c) for ordinary site percolation
in $d=3$. Normalization is arbitrary as in Fig. 2, and log-scales are again
used on the y-axis in all panels. In all panels, the continuous curves
represent best eyeball fits to the data, in which the tails of the
distributions are neglected.

Our data in Fig.2 agree perfectly with the data for 2-d bond percolation shown
in Fig. 2 of Fan . But in none of the panels, the fits with Gumbel resp. Gauss
distributions are acceptable. For panel (b) the disagreement is least severe,
and it would have gone unnoticed if we had not used a log scale on the y-axis.
The disagreement with a Gumbel fit is much worse for panel (c), but for panel
(a) it is so large that it is unclear how this could have gone unnoticed in
Fan . Notice that the Gumbel $\beta$-parameter obtained in Fan  (called there
$\omega_{2}$, see suppl ) is indeed the one which gives the best fit in the
left half of the plot, but it produces nonsense on the right half.
Nevertheless, in Fig. 2f of Fan  a supposed Gumbel distribution is drawn which
gives a nearly perfect fit.

Figure 4: The same three distributions for aftermath percolation in $d=3$.
Normalization is arbitrary as in the previous figures, and log-scales are
again used on the y-axis in all panels. In panel (a) one sees a clear
exponential tail, therefore the (eyeball) Gumbel fit is made so that this tail
gets a substantial weight.

###  III.3 Three-Dimensional Lattice Models

The next models also show that the claim of Fan et al. is untenable, but the
deviations from extreme value distributions are less pronounced than in the
previous models. Data for site percolation in $d=3$ are shown in Fig. 3.
Again, the raw data agree fairly well with those of Fan  (see Fig. S2 in suppl
), but the fits to Gauss and Gumbel distributions are definitely unacceptable.
Finally, we should point out that the right-hand tails of $P_{S}$ and
$P_{{}_{\Delta}}$ slightly rise with increasing $L$. But this rise seems too
slow to cure the problem for large $L$, and there is no similar rise of the
left-hand tails.

The second $3-d$ lattice universality class studied here is aftermath
percolation, introduced in aftermath . It is in the universality class of
pacman percolation Abete , but is easier to simulate with the NZ algorithm.
The critical exponents were conjectured in aftermath  to be exactly
$\beta=1,\nu=2,$ and $d_{f}=5/2$. The three distributions for this
universality class are shown in Fig. 4. This time, $P_{S}$ shows a rather
clear exponential tail, and can thus be fitted fairly well by a Gumbel
distribution (although such a fit is statistically unacceptable). On the other
hand, $P_{{}_{\Delta}}$ is very poorly represented by a Gumbel distribution.

Figure 5: The same three distributions for explosive percolation with the
product rule. Normalization is arbitrary as in the previous figures, and log-
scales are again used on the y-axis in all panels. $P_{S}$ (shown in panel
(a)) is so far from any Gumbel distribution that we refrained from showing a
fit, the fits in (b) and (c) are as before eyeball fits.

###  III.4 Explosive Percolation

There are a large number of explosive percolation models, which all use as a
core ingredient the Achlioptas trick Achlioptas  to delay the formation of
large clusters. The critical scaling of four of them was studied in detail in
Christensen , where it was found that they all had vastly different order
parameter exponents. Thus they belonged to different universality classes –
although $\nu$ (called $\Theta$ in Christensen ) was the same in the three
off-lattice models studied in Christensen .

Figure 6: The same three distributions for critical ER networks. Normalization
is arbitrary as in the previous figures, and log-scales are again used on the
y-axis in all panels.

In the following we shall only show results for the “product rule”, the
original model proposed in Achlioptas . The three distributions for this model
are shown in Fig. 5. We see that $P_{S}$ is so far from a Gumbel that we
didn’t try to fit one. On the other hand, $P_{p}$ is perfectly compatible with
a Gaussian, as was conjectured in Li2023  foot1 . Also, this Gaussian is
nearly perfectly centered at $p-p_{c}=0$, i.e. statistical fluctuations of the
effective transition point $p_{c}(N)$ at large but finite $N$ are much bigger
than its deviation from $p_{c}$, and scale with different exponents
Christensen ; Fan ; Li2023 . This is in sharp contrast to what is found in
most other second-order phase transitions, where both scale with the same
exponent, $\sigma(p_{c})\sim p_{c}(N)-p_{c}\sim N^{-1/\nu}$.

###  III.5 Erdös-Renyi Networks

Finally, we show the three distributions also for critical ER-networks. There
the critical point is at $p_{c}=1/2$, where $p$ is the number of links divided
by $N$, $\nu=3$, and the cluster masses scale as $N^{2/3}$. All these are
clearly seen in Fig. 6. Again we see that $P_{p}$ is not Gaussion, and $P_{S}$
is not Gumbel. The situation is different for $P_{{}_{\Delta}}$. There, we see
a strong finite size dependence in the right hand tail, so that this tail
could well be fitted by a Gumbel for $N\to\infty$. On the other hand, there is
no such $N-$ dependence in the left hand side of $P_{{}_{\Delta}}$, so that a
Gumbel can be again excluded.

##  IV Discussion

We have performed extensive simulations of six different universality classes
of (generalized) percolation. It had been claimed in Fan  that certain
distributions should in all such universality classes follow extreme value
distributions. More precisely, this claim was made for what is called the
“event-based ensemble” in Li2023 , and the distributions are those of the
largest cluster mass at the (event-based) pseudo-critical point, the “gap”
(largest jump in the giant cluster mass), and the control parameter at the
pseudo-critical point. It was claimed in Fan  that the first two should be
Gumbel distributions, and the third Gaussian.

This was claimed to be true even if the transition is discontinuous, which
would establish a novel instance of super-universality extending over
continuous and discontinuous transitions. But we found that the claim was
untenable in all eighteen distributions we looked at, except for one: In
explosive percolation with the product rule, the pseudo-critical control
parameter values showed a Gauss distribution, as had been conjectured in
Li2023 .

Apart from this negative result, we obtained very precise universal
distributions, which could be very useful in future studies. We also verified
in all cases the known and/or conjectured values of critical exponents. In
particular this holds for aftermath percolation aftermath , which is in the
universality class of pacman percolation, and where $\beta=1$ seems to hold
for all dimensions between $d=3$ and $d=6$ Fesh-pacman .

Acknowledgements: M.F. acknowledges the support from the research council of
the Alzahra University. P.G. thanks Youjin Deng for discussions about
explosive percolation.

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