ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P4.05
https://doi.org/10.26493/1855-3974.3114.d47
(Also available at http://amc-journal.eu)
Complexity function of jammed configurations
of Rydberg atoms*
Tomislav Došlić
Department of Mathematics, Faculty of Civil Engineering,
University of Zagreb, Zagreb, Croatia and
Faculty of Information Studies, Novo Mesto, Slovenia
Mate Puljiz , Stjepan Šebek † , Josip Žubrinić
Department of Applied Mathematics, Faculty of Electrical Engineering and Computing,
University of Zagreb, Zagreb, Croatia
Received 3 May 2023, accepted 4 March 2024, published online 25 September 2024
Abstract
In this article, we determine the complexity function (configurational entropy) of
jammed configurations of Rydberg atoms on a one-dimensional lattice. Our method con-
sists of providing asymptotics for the number of jammed configurations determined by
direct combinatorial reasoning. In this way we reduce the computation of complexity to
solving a constrained optimization problem for the Shannon’s entropy function. We show
that the complexity can be expressed explicitly in terms of the root of a certain polynomial
of degree b, where b is the so-called blockade range of a Rydberg atom. Our results are
put in a relation with the model of irreversible deposition of k-mers on a one-dimensional
lattice.
Keywords: Dynamic lattice systems, equilibrium lattice systems, complexity function, configurational
entropy, jammed configuration, maximal packing, Rydberg atoms.
Math. Subj. Class. (2020): 82B20, 82C20, 05B40, 05A15, 05A16
*It is a pleasure for us to thank Jean-Marc Luck and Pavel Krapivsky for fruitful exchanges during the concomi-
tant elaboration of their preprint [43] and of the present work. T. Došlić gratefully acknowledges partial support
by the Slovenian ARIS via Program P1-0383, grant no. J1-3002, and by COST Action CA21126 NanoSpace.
Financial support of the Croatian Science Foundation (project IP-2022-10-2277) is gratefully acknowledged by
S. Šebek.
Lastly, we thank the anonymous referee for helpful comments that have led to improvements of the presentation
of the article.
†Corresponding author.
E-mail addresses: tomislav.doslic@grad.unizg.hr (Tomislav Došlić), mate.puljiz@fer.unizg.hr (Mate
Puljiz), stjepan.sebek@fer.unizg.hr (Stjepan Šebek), josip.zubrinic@fer.unizg.hr (Josip Žubrinić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 24 (2024) #P4.05
1 Introduction
Rydberg atom is a name given to an atom which has been excited into a high energy level
so that one of its electrons is able to travel much farther from the nucleus than usual (up
to 106 times more, see [22]). In physics community, Rydberg atoms have been intensely
studied and have become a testing ground for various quantum mechanical problems in
quantum information processing, quantum computation and quantum simulation [58]. See
[29] for a comprehensive description of the physics of Rydberg atoms and their remarkable
properties. Due to their large size, Rydberg atoms can exhibit very large electric dipole
moments which results in strong interactions between two close Rydberg atoms. This
causes a blockage effect that prohibits the excitation of an atom located close to an atom
that is already excited to a Rydberg state [3, 12, 34, 37, 54, 63]. The simplest setting for
studying Rydberg atoms and their blockage effect is on a finite one-dimensional lattice. In
this setting, each atom occupies one site and each two excited atoms are at least b ≥ 1
sites apart. The positive integer b is referred to as the blockade range of the model. We
will be interested in maximal (or jammed) configurations where no further atoms can be
excited. Note that in such a configuration each two excited atoms are at most 2b sites apart.
In physics literature, jammed states in similar deposition models have the interpretation of
metastable states at low enough temperature and/or high enough density, and are referred
to as valleys, pure states, quasi-states, and inherent structures [4, 5, 17, 33, 38, 50, 62].
The main question related to maximal configurations concerns the expected density of
the atoms excited to a Rydberg state. There are two natural ways to interpret this question.
One way to look at this problem is to consider the set of all the possible maximal configu-
rations and to sample one such configuration at random (which implies that all the maximal
configurations are equiprobable). This is referred to as the static (or equilibrium) model.
The static model is usually described by the so-called complexity function (also known as
configurational entropy), and the expected density of particles in a jammed configuration
converges to the argument of the maximum of the complexity function. This is exactly
the approach we take in this paper and our main result is the derivation of the mentioned
complexity function. Static model can be compared with the random sequential adsorp-
tion (RSA) model (also refered to as the dynamic model) where initially all atoms are in
the ground state, and are excited sequentially, at random, until a jammed configuration is
reached. The expected density of excited atoms with this underlying probability space is
called jamming limit. Assumption that the two models result in the same distribution of
maximal configurations has come to be known as Edwards’s flatness hypothesis (see [2]
for a recent review). However, there seems to be no a priori reason for the two models to
have similar properties. It is interesting to note that there is a long history to the question
of how similar the two models are (see e.g. the discussion in [8, page 681], or, in a more
subtle continuum context, how a similar confusion of different probability models led to
some extended discussion over a false “proof” [53]).
The dynamic version of the problem was already studied in literature. In [28, 42, 48] it
was found that the jamming limit is
ρb-Ryd∞ =
∫ 1
0
exp
−2 b∑
j=1
1− yj
j
 dy.
The jamming limit was also computed for an equivalent model of deposition of linear
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 3
polymers (k-mers) in [44, §7.1]
ρk-mer∞ = k
∫ ∞
0
exp
−u− 2 k−1∑
j=1
1− e−ju
j
 du = k ∫ 1
0
exp
−2 k−1∑
j=1
1− yj
j
 dy.
The equivalence of models is reflected in the fact that ρk-mer∞ = k · ρb-Ryd∞ for b = k − 1.
In the static model, it all comes down to counting the maximal configurations. It is
known that in similar models, the number of different maximal configurations with pre-
scribed density 0 ≤ ρ ≤ 1 tends to grow exponentially with the length L of configuration,
see [9, 10, 14, 13, 15, 16, 18, 23, 27, 35, 40, 45, 46, 49, 52, 55, 56, 59]. Denoting this
number by JL(ρ), it is common to describe it using the so-called complexity function (also
called configurational entropy) f(ρ) for which it holds that JL(ρ) ∼ eLf(ρ). It turns out
(see e.g. Figure 9) that the density ρb-Ryd? maximizing the complexity function is slightly
different than the expected density (jamming limit) of the dynamic model. This falsifies
the above mentioned Edward’s flatness hypothesis. Recall that ρb-Ryd? is the limit (as L
tends to infinity) of the most probable densities in the equilibrium models that assign equal
probabilities to all jammed configurations.
Our main goal is to compute the complexity f(ρ) of jammed configurations of Rydberg
atoms using direct combinatorial reasoning. The problem reduces to solving a constrained
optimization problem for the Shannon’s entropy function. We show that the complexity
function can be expressed explicitly in terms of the root of a certain polynomial of degree
b. This work has been carried out simultaneously with [43]. The authors there introduce a
novel method for determining the same complexity function. Their method is inspired by
the theory of renewal processes.
The described model of Rydberg atoms on a one-dimensional lattice is equivalent to
a number of other models already present in the literature. The case b = 1 is the famous
model introduced by Flory [26] describing the mechanism of vinyl polymerization. This is
in turn essentially equivalent to the Page-Rényi car parking problem [31, 51] (which is a
discrete version of the famous model introduced by Rényi in [57]) describing the jammed
configurations of cars of length 2. The equivalence is obtained by replacing each excited
atom with a car taking up both the atom’s and its right neighbor’s site. Clearly, this only
works for configurations not having an excited atom at the rightmost site. This means that
the total number of jammed configurations is actually different in these two models, but
only up to a constant factor, which does not affect the shape of the complexity function of
these models. In chemistry, this model appears in the context of the irreversible deposition
of dimers [26, 36], and in graph theory, the jammed configurations correspond to maximal
matchings in a path graph, see [21].
Similarly, the general case b > 1 corresponds to irreversible deposition of k-mers (k =
b + 1) in a linear polymer of length L. The equivalence (again, up to a constant factor)
is obtained by replacing each excited atom with a polymer taking up b + 1 consecutive
sites, starting from the atom’s position, see Figure 1. In this, and all the following figures,
bullets (•) represent Rydberg atoms (in the Rydberg model) or occupied sites (in the k-mer
deposition model), while empty bullets (◦) represent neutral atoms (in the Rydberg model)
or vacant sites (in the k-mer deposition model). Notice that the gaps between adjacent
k-mers in jammed configurations of this deposition model are of size at most k − 1. This
equivalence allows us to easily transfer our results on Rydberg atoms to the setting of k-
4 Ars Math. Contemp. 24 (2024) #P4.05
mer deposition model. The problem of irreversible deposition of k-mers was extensively
studied in the literature, see [1, 6, 24, 32, 42, 44, 61].
◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ • ◦ ◦ ◦
◦ • • • • ◦ • • • • ◦ ◦ ◦ • • • • • • • • • • • • ◦ ◦ ◦ • • • • • • • • ◦ ◦ • • • •
Figure 1: A jammed configuration of Rydberg atom model with blockade range b and the
corresponding jammed configuration of the k-mer deposition model when b = 3, k = 4.
In graph theory, the k-mer deposition model is equivalent to Pk-packings of a path
graph PL, and jammed configurations in the former correspond to maximal packings in the
latter. The maximal Pk-packings of PL were previously studied in [20].
Another equivalent formulation of the Rydberg atom model appeared recently in [19,
§3.2.1] where the authors of the present paper considered the settlement model consisting
of k-story buildings on a one-dimensional tract of land. The tract of land is oriented east-
west and each story of each building has to receive the sunlight from both east and west.
The rest of the paper is organized as follows. In Section 2 we calculate the asymp-
totics for the number of jammed configurations in the model of Rydberg atoms, which is
expressed in terms of the maximum of the Shannon’s entropy function over a certain finite
set determined by the constraints of the model. In Section 3 we use these results in order
to obtain the formula for the associated complexity function. We derive the expression
for the complexity f(ρ) which, for a chosen density ρ, depends explicitly on a positive
root of a certain polynomial whose degree coincides with the blockade range of the model.
Further on, in Section 4, we put our findings in relation with the model for the deposition
of k-mers on the linear polymer and draw conclusions from the obtained results. There,
we also provide some results on the qualitative properties of the maximizers of mentioned
complexity functions, for various blockade ranges b, and put them in comparison with their
counterparts in the theory of RSA. Finally, in Section 5 we recapitulate our findings and
indicate several possible directions of future research.
Notation
We write ML ∼ NL if the two positive sequences (ML)L and (NL)L have the same
growth, as L→∞, up to a sub-exponential factor, i.e. if
lim
L→∞
lnML − lnNL
L
= 0.
2 Jammed configurations of Rydberg atoms
As already stated in the introduction, the main goal of this paper is to compute the complex-
ity function f(ρ) of jammed configurations of Rydberg atoms. Crucial step towards obtain-
ing a complexity function of such models in general is to inspect the set of all jammed con-
figurations of a model. Each configuration is a binary 0/1 sequence which we sometimes
interpret as a sequence of empty/occupied sites or, in Rydberg model, as neutral/excited
atoms. The total number of all configurations of length L in the model is denoted by JL.
The total number of configurations of lengthL consisting ofN ones (occupied sites, excited
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 5
atoms) is denoted by JN,L. The density (saturation, coverage) of any such configuration
of length L with N ones is defined as N/L ∈ [0, 1].
In order to determine the complexity function, it is not enough to work only with JL.
We need to be more precise. We need to know the behavior of the number of different
jammed configurations of length L, where precisely N atoms are excited to the Rydberg
state. The main result of this section (see Lemma 2.4) provides asymptotics of the quantity
JN,L for Rydberg atom model.
Let us first consider several concrete examples of jammed configurations of our model
to get a better feeling of their possible shapes. Figure 2 displays three different jammed
configurations in the chain of L = 16 atoms, where the blockade range b is equal to two,
i.e. each two excited atoms are at least two sites apart. Since Rydberg atoms in a jammed
• ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ •
◦ • ◦ ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ • ◦ ◦ •
• ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ • ◦
Figure 2: Three jammed configurations in the chain of L = 16 atoms with blockade range
b = 2. The number of Rydberg atoms in these configurations is N = 6, 5, 4 (from top to
bottom).
configuration are separated by clusters of empty sites whose length is at least b (so that the
constraint imposed by the blockage effect is satisfied), and at most 2b (since we can excite
another atom in the middle of an empty range of size 2b + 1, hence such a configuration
would not be jammed), it is easy to see that it holds⌈
L
2b+ 1
⌉
≤ N ≤
⌈
L
b+ 1
⌉
, (2.1)
where N is the number of excited atoms, L is the length of the configuration, and b is the
blockade range. In the particular case of L = 16 and b = 2, this implies that 4 ≤ N ≤ 6.
Hence, Figure 2 shows one jammed configuration for each possible value ofN . Notice that
relation (2.1) implies that
1
2b+ 1
− 1
L
<
N
L
≤ 1
b+ 1
, (2.2)
and this in turn implies that in the limit, as L → ∞, the density ρ = N/L, of Rydberg
atoms in jammed configurations, lies within the bounds
1
2b+ 1
≤ ρ ≤ 1
b+ 1
. (2.3)
As a first result in the direction of better understanding the double sequence JN,L for
Rydberg atom model, we provide the bivariate generating function for this sequence in the
general case of blockade range b ≥ 1.
Lemma 2.1. The bivariate generating function of the sequence JN,L associated with
jammed configurations of Rydberg atoms, when the blockade range is equal to b, is given
by
Fb(x, y) =
∑
L
∑
N
JN,Lx
NyL =
(1− y)2 + xy − xyb+1 − xyb+2 + xy2b+2
(1− y)(1− y − xyb+1 + xy2b+2) .
6 Ars Math. Contemp. 24 (2024) #P4.05
Proof. As already mentioned, configurations of Rydberg atoms can be represented as 0/1
sequences. Due to the fact that we can determine whether the blockage effect has been
taken into account, and whether the configuration represented with such a sequence is
jammed, just by inspecting finite size patches of a given sequence, we can apply the so-
called transfer matrix method (see [60, §4.7] or [25, §V], and also [47, §2–4]). This is a
well known method for counting words of a regular language. Since Rydberg atoms in a
jammed configuration are separated with at least b, and at most 2b neutral atoms, every
jammed configuration will be composed of blocks that start with a Rydberg atom and then
have a cluster of neutral atoms of length between b and 2b. Such blocks are displayed in
Figure 3.
• ◦ · · · ◦︸ ︷︷ ︸
b atoms
• ◦ · · · ◦︸ ︷︷ ︸
b+1 atoms
· · · • ◦ · · · ◦︸ ︷︷ ︸
2b atoms
Figure 3: Building blocks of jammed configurations of Rydberg atoms with blockade range
b.
These building blocks are encoded with the polynomial
pb(x, y) = xy
b+1 + xyb+2 + · · ·+ xy2b+1.
Now we only need to take care of the beginning and the end of jammed configurations.
Notice that in front of the first block we can have some neutral atoms. More precisely,
the number of neutral atoms that can appear at the left end of the jammed configuration is
between 0 and b. These starting blocks are encoded with the polynomial
sb(x, y) = 1 + y + y
2 + · · ·+ yb.
Similarly, after the last block from the set of blocks shown in Figure 3 (if there are any, i.e.
if we want to have more than just one atom in the Rydberg state), we need to have a block
that again starts with a Rydberg atom, and then has a cluster of neutral atoms of length
between 0 and b. These ending blocks are encoded with the polynomial
eb(x, y) = xy + xy
2 + · · ·+ xyb+1.
Notice that each of the blocks shown in Figure 3 can be glued to any other block listed
in this figure. This implies that we do not even need to work with powers of the transfer
matrix, but we can directly take powers of the polynomial pb(x, y) in order to obtain the
desired bivariate generating function. A simple calculation gives
Fb(x, y) = 1 +
∞∑
n=0
sb(x, y) · pb(x, y)n · eb(x, y)
= 1 +
sb(x, y) · eb(x, y)
1− pb(x, y)
=
(1− y)2 + xy − xyb+1 − xyb+2 + xy2b+2
(1− y)(1− y − xyb+1 + xy2b+2) .
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 7
Remark 2.2. By using the same technique, we can easily compute the bivariate generating
function enumerating the number of jammed configurations of prescribed length, and with
some fixed number of occupied sites, in the k-mer deposition model. The building blocks
here are composed of a cluster of k consecutive sites occupied by a single k-mer, followed
by a cluster of empty sites of length between 0 and k − 1 (see Figure 4). These building
• · · · •︸ ︷︷ ︸
k sites
• · · · •︸ ︷︷ ︸
k sites
◦ • · · · •︸ ︷︷ ︸
k sites
◦ ◦ · · · • · · · •︸ ︷︷ ︸
k sites
◦ · · · ◦︸ ︷︷ ︸
k−1 sites
Figure 4: Building blocks of jammed configurations of k-mer deposition model.
blocks are encoded with a polynomial
pk(x, y) = x
kyk + xkyk+1 + · · ·+ xky2k−1,
where x is again a formal variable associated with the number of occupied sites, and y is
a formal variable associated with the length of a configuration. Similarly as in the case of
the Rydberg atom model, at the left end of a jammed configuration, we can have a cluster
of vacant sites of length between 0 and k − 1. These starting blocks are encoded with the
polynomial
sk(x, y) = 1 + y + y
2 + · · ·+ yk−1.
It is clear that we can end a jammed configuration with any of the building blocks shown
in Figure 4, so we can set ek(x, y) = 1. Using again the fact that each of the blocks from
Figure 4 can be glued to any other block listed in that figure, we can work directly with
powers of the polynomial pk(x, y) to obtain
Fk(x, y) =
∞∑
n=0
ak(x, y) · pk(x, y)n =
ak(x, y)
1− pk(x, y)
=
1− yk
1− y − xkyk + xky2k . (2.4)
Notice that we are not adding 1 to the bivariate generating function in (2.4). The reason
is that starting with a cluster of 0 vacant sites and setting n = 0 already counts the empty
configuration.
The sequence JN,L has already been studied in the literature, but in the context of
maximal Pk-packings of a path graph PL (see [20]). The bivariate generating function
enumerating the total number of maximal k-packings in PL, with exactly N copies of Pk,
is given in [20, Corollary 2.4], and the only difference between that bivariate generating
function and the one we obtained in (2.4), is that x is not raised to power k. The reason is
that the author in [20] is interested in the number of copies of Pk (i.e. the number of de-
posited k-mers) in jammed configurations, and we are interested in the total number of sites
occupied by those deposited k-mers. The bivariate generating function from (2.4) is also
obtained in [43, formula (5.3)], where authors use a novel approach inspired by the theory
of renewal processes. Using the same technique, they also obtain the bivariate generating
function which coincides with the one we obtained in Lemma 2.1, which enumerates the
total number of jammed configurations of length L of Rydberg atoms with blockade range
b, with precisely N excited atoms (see [43, formula (6.5)]).
It is easy to see from the bivariate generating function from Lemma 2.1 that, for b = 2,
J16 = 96 (i.e. there are 96 jammed configurations in the chain of L = 16 atoms, when
8 Ars Math. Contemp. 24 (2024) #P4.05
the blockade range is b = 2). Out of those 96 jammed configurations, 45 of them have 4
Rydberg atoms (J4,16 = 45), 50 of them have 5 Rydberg atoms (J5,16 = 50), and only
one has 6 Rydberg atoms (J6,16 = 1). This particular one is exactly the first jammed
configuration shown in Figure 2.
We could now proceed like the authors in [43] and use the bivariate generating function
developed in Lemma 2.1 to obtain the complexity function of jammed configurations of
Rydberg atoms by means of the Legendre transform. However, we will use a direct combi-
natorial argument. To this end, we introduce a slightly different way of counting jammed
configurations in the Rydberg model with blockade range b, than the one introduced in
Lemma 2.1. Denote with B the block of b + 1 adjacent atoms where only the first one is
excited to the Rydberg state (see Figure 5). Using again the fact that each two Rydberg
B = • ◦ ◦ · · · ◦︸ ︷︷ ︸
b atoms
Figure 5: Block consisting of b+1 adjacent atoms where only the first one is excited to the
Rydberg state.
atoms have at least b and at most 2b neutral atoms separating them, it is clear that every
jammed configuration consists of blocks B separated by clusters of neutral atoms of length
0 ≤ a ≤ b (see Figure 6). Denote by Ma the number of gaps with a neutral atoms. The
◦ · · · ◦︸ ︷︷ ︸
a1 atoms
B ◦ · · · ◦︸ ︷︷ ︸
a2 atoms
B ◦ · · · ◦︸ ︷︷ ︸
a3 atoms
B · · ·B ◦ · · · ◦︸ ︷︷ ︸
aN atoms
B
Figure 6: The shape of jammed configurations in Rydberg model with blockade range b and
exactly N Rydberg atoms, ending with a block B (displayed in Figure 5). Gaps between
blocks B, and in front of the first block B, consist of neutral atoms and are of length
0 ≤ ai ≤ b.
total number of jammed configurations of the shape shown in Figure 6, with L atoms in
total, out of which precisely N atoms are excited to the Rydberg state, is given as(
N
M0,M1, . . . ,Mb
)
=
N !∏
0≤a≤bMa!
, (2.5)
with Ma satisfying
b∑
a=0
Ma = N, (2.6)
b∑
a=0
aMa = L− (b+ 1)N. (2.7)
The constraint (2.6) expresses that the total number of gaps is N . Notice that we have
N blocks B (since we want to have precisely N Rydberg atoms), and that gaps of size
0 ≤ a ≤ b can be added in front of the first block B, and between each two blocks B.
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 9
The constraint (2.7) implies that the total number of neutral atoms is L − N . Clearly
we need L − N neutral atoms in addition to N Rydberg atoms to have a configuration
of length L. Equation (2.5) accounts for the jammed configurations ending precisely on
B. There are also jammed configurations where the last block B is truncated, and there
are only 0 ≤ c < b neutral atoms after the last atom excited to the Rydberg state. The
contribution of such jammed configurations to the value of JN,L is comparable to (2.5),
but since complexity function ignores sub-exponential factors, it suffices to determine the
asymptotics of the sum
JN,L ∼
∑
(M0,M1,...,Mb)∈RN,L
(
N
M0,M1, . . . ,Mb
)
, (2.8)
where
RN,L = {(M0,M1, . . . ,Mb) ∈ Nb+10 :M0 +M1 + · · ·+Mb = N and
M1 + 2M2 + · · ·+ bMb = L− (b+ 1)N}. (2.9)
We write H for the Shannon’s entropy function given as
H(p0, p1, . . . , pb) = −
b∑
i=0
pi ln pi, (2.10)
where pi ≥ 0, for 0 ≤ i ≤ b, and p0 + p1 + · · ·+ pb = 1.
Remark 2.3. In case pi = 0 for some i, we set 0 · ln 0 = 0.
The following lemma is the key result of this section, and it constitutes a crucial step in
computing the complexity function of our model as it provides the asymptotics of JN,L in
terms of the maximum of the entropy function.
Lemma 2.4.
JN,L ∼ exp
(
L · max
(M0,M1,...,Mb)∈RN,L
N
L
·H
(
M0
N
,
M1
N
, . . . ,
Mb
N
))
, as L→∞.
where the set RN,L is defined in (2.9), and the function H is defined in (2.10).
Proof. Note that
max
(M0,M1,...,Mb)∈RN,L
(
N
M0,M1, . . . ,Mb
)
≤
∑
(M0,M1,...,Mb)∈RN,L
(
N
M0,M1, . . . ,Mb
)
≤ |RN,L| max
(M0,M1,...,Mb)∈RN,L
(
N
M0,M1, . . . ,Mb
)
.
As the number |RN,L| of terms in the sum is at most (N + 1)b+1 ≤ (L+ 1)b+1, which is
polynomial in L, the sum, asymptotically, grows as its largest term. It is, therefore, enough
to determine the asymptotics of
JN,L ∼ max
(M0,M1,...,Mb)∈RN,L
(
N
M0,M1, . . . ,Mb
)
, as L (and N)→∞.
10 Ars Math. Contemp. 24 (2024) #P4.05
By following the proof of Lemma 2.2 in [11] we can conclude that(
N + b
b
)−1
NN
M0
M0M1
M1 · · ·MbMb
≤
(
N
M0,M1, . . . ,Mb
)
≤ N
N
M0
M0M1
M1 · · ·MbMb
.
Note that in case any Ma is zero, the expression 00 is to be interpreted as 1. Since
(
N+b
b
)
is of polynomial growth, we get(
N
M0,M1, . . . ,Mb
)
∼ N
N
M0
M0M1
M1 · · ·MbMb
=
(
N
M0
)M0 ( N
M1
)M1
· · ·
(
N
Mb
)Mb
,
(2.11)
as N →∞. Note that(
N
M0
)M0 ( N
M1
)M1
· · ·
(
N
Mb
)Mb
= exp
(
N ·H
(
M0
N
,
M1
N
, . . . ,
Mb
N
))
.
Hence
JN,L ∼ max
(M0,M1,...,Mb)∈RN,L
exp
(
N ·H
(
M0
N
,
M1
N
, . . . ,
Mb
N
))
, as L→∞,
and consequentially
JN,L ∼ exp
(
L · max
(M0,M1,...,Mb)∈RN,L
N
L
·H
(
M0
N
,
M1
N
, . . . ,
Mb
N
))
, as L→∞,
which is exactly what we wanted to prove.
Remark 2.5. One could obtain the asymptotics in (2.11) from Stirling’s approximation
N ! ∼ (N/e)N , as N →∞, where sub-exponential factors are ignored.
3 Complexity function of jammed configurations of Rydberg atoms
In this section we compute the complexity function, sometimes referred to as configura-
tional entropy, of jammed configurations of Rydberg atoms. We first recall the definition
of complexity function of a certain model.
Definition 3.1. For a fixed density ρ ∈ [0, 1], let JbρLc,L denote the number of configu-
rations of length L with density bρLc /L ≈ ρ. The complexity function f : [0, 1] → R is
then defined as
f(ρ) = lim
L→∞
ln JbρLc,L
L
, (3.1)
for each ρ ∈ [0, 1] for which this limit exists.
Remark 3.2. If the limit above does not exist for a certain ρ, one can still define (upper)
complexity at that point by replacing lim in the definition with lim sup. And if there are no
configurations with a certain density ρ, we still write f(ρ) = 0.
Remark 3.3. This definition implies that the number of configurations with the density
bρLc /L ≈ ρ grows as eLf(ρ) for large L.
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 11
The guiding idea behind introducing the complexity function is to describe what portion
of the total number of configurations take up configurations with a particular density. The
problem is that, as L grows to infinity, the actual proportions tend to the delta distribution
concentrated on the ‘most probable’ density ρ?.
As an example, the distribution of densities (the sum of digits divided by the length) of
binary sequences of length L is a symmetric binomial distribution re-scaled to the interval
[0, 1]. The limiting distribution is then the delta distribution δ0.5 which is, essentially, the
consequence of the law of large numbers.
This convergence to a delta distribution results from the fact that the number of config-
urations with a certain density grows exponentially with a rate that depends on the density.
For large L, the number of configurations with density having the largest rate overtakes, in
proportion, configurations having any other density. The complexity function then quan-
tifies the distribution of all configurations with respect to their densities in a more refined
way.
Another consequence of the fact that the number of configurations having density with
the largest rate dominates, in proportion, any other density is that the total number of all
configurations grows at the same exponential rate as the number of configurations having
this ‘most probable density’. To be precise, if ρ? denotes the density at which the com-
plexity function f attains its maximum and if JL is the total number of all configurations
of length L, then JL ∼ eLf(ρ?) for large L.
Remark 3.4. In Lemma 2.1 we derived the generating function for the sequence JN,L
within the Rydberg atom model. Plugging x = 1 into this generating function gives the
generating function for JL, the total number of configurations of length L in Rydberg atom
model
Fb(1, y) =
(1− y)2 + y − yb+1 − yb+2 + y2b+2
(1− y)(1− y − yb+1 + y2b+2)
=
1 + y(1 + y + · · ·+ yb)(1 + y + · · ·+ yb−1)
1− yb+1(1 + y + · · ·+ yb) .
From here, we can infer the asymptotics of JL for large L by inspecting the roots of the
polynomial 1− yb+1(1+ y+ · · ·+ yb) in the denominator. More precisely, if yb is the root
with the smallest modulus, then the logarithm of wb = |yb|−1 gives the exponential growth
rate of the sequence JL
JL ∼ wLb = eL lnwb .
The discussion in the previous paragraph now implies the relation f(ρb-Ryd? ) = lnwb.
The following theorem is the main result of this paper and provides an elegant expres-
sion for the complexity function of jammed configurations of Rydberg atoms f(ρ) in terms
of a root of a certain polynomial.
Theorem 3.5. The complexity function of jammed configurations of Rydberg atoms with
blockade range b ∈ N is given as
f(ρ) =
{
ρ
[
− ln 1−z
1−zb+1 −
(
1
ρ − (b+ 1)
)
ln z
]
, if 12b+1 < ρ ≤ 1b+1 ,
0, otherwise,
12 Ars Math. Contemp. 24 (2024) #P4.05
where z ≥ 0 is a real root of the polynomial
p(z) =
b∑
i=0
(
i+ b+ 1− 1
ρ
)
zi (3.2)
for which the expression f(ρ) is the largest.
Remark 3.6. When 12b+1 < ρ <
1
b+1 the leading coefficient of the polynomial p(z) given
in (3.2) is positive, while the constant term is negative. This guaranties the existence of
at least one positive real root z > 0. If ρ = 1b+1 , then z = 0 is the root of p(z) and the
formula gives f( 1b+1 ) = 0.
Remark 3.7. Since (3.2) is a polynomial of degree b, it is possible to find its roots explicitly
for b ≤ 4 and numerically for b > 4. The explicit expression for the complexity in case
b = 1 is
f1-Ryd(ρ) = ρ ln ρ− (1− 2ρ) ln(1− 2ρ)− (3ρ− 1) ln(3ρ− 1),
and for b = 2
f2-Ryd(ρ) = (3ρ− 1) ln
√
−44ρ2 + 24ρ− 3− 4ρ+ 1
10ρ− 2 −
ρ ln
−350ρ3 + (25ρ2 − 10ρ+ 1)
√
−44ρ2 + 24ρ− 3 + 215ρ2 − 44ρ+ 3
ρ2
√
−44ρ2 + 24ρ− 3− 134ρ3 + 57ρ2 − 6ρ
.
In the case b = 1, the function f1-Ryd(ρ) recovers the result from [44, formula (7.20)] and
[41, §VII]. The graphs of the complexity function of jammed configurations of Rydberg
atoms with blockade range 1 ≤ b ≤ 10 are given in Figure 7. In that figure we also see
that, for each b, the maximum of the complexity function matches lnwb, the growth rate of
all jammed configurations. This was already discussed in Remark 3.4.
Proof of Theorem 3.5. Recall that in (2.2) we showed that
1
2b+ 1
− 1
L
<
N
L
≤ 1
b+ 1
,
and therefore, there are no jammed configurations with densities ρ > 1b+1 nor with densities
ρ < 12b+1 , for sufficiently large L. Thus, f(ρ) = 0 when ρ >
1
b+1 or ρ <
1
2b+1 . In case
ρ = 12b+1 , it is not hard to see that the number of configurations Jb L2b+1c,L is
Jb L2b+1c,L =
{
1, if (2b+ 1) | L,
0, otherwise.
This implies f( 12b+1 ) = 0 by the definition of complexity.
In the remainder, we fix 12b+1 < ρ ≤ 1b+1 . By Lemma 2.4, and by using the definition
of the complexity function (3.1), we have
f(ρ) = lim
L→∞
max
(M0,M1,...,Mb)∈RN,L
N
L
·H
(
M0
N
,
M1
N
, . . . ,
Mb
N
)
, (3.3)
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 13
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
0
ρ
0
0.05
0.1
0.3
lnw1 = lnw2
lnw3
lnw4
lnw5
lnw6
lnw7
lnw8
lnw9
lnw10
f
(ρ
)
b = 1
b = 2
b = 3
b = 4
b = 5
b = 6
b = 7
b = 8
b = 9
b = 10
Figure 7: The complexity function of jammed configurations of Rydberg atoms with block-
ade range 1 ≤ b ≤ 10.
where N = bρLc, provided that this limit exists. By rewriting (M0,M1, . . . ,Mb) ∈ RN,L
as
M0
N
≥ 0, M1
N
≥ 0, . . . , Mb
N
≥ 0
M0
N
+
M1
N
+ · · ·+ Mb
N
= 1
M1
N
+ 2
M2
N
+ · · ·+ bMb
N
=
L
N
− (b+ 1)
and denoting pi = MiN ∈ 1bρLcZ, the complexity (3.3) can be written as
f(ρ) = lim
L→∞
max
(p0,p1,...,pb)∈ 1bρLcRbρLc,L
ρ̂H (p0, p1, . . . , pb) , (3.4)
where ρ̂ = ρ̂(L) = NL =
bρLc
L . We claim that this limit exists and is equal to the maximum
of the constrained optimization problem
max
p0,p1,...,pb≥0
p0+p1+···+pb=1
p1+2p2+···+bpb= 1ρ−(b+1)
ρH (p0, p1, . . . , pb) , (3.5)
where pi ∈ R are no longer required to be fractions.
14 Ars Math. Contemp. 24 (2024) #P4.05
We argue as follows. Denote by (p∗0, p
∗
1, . . . , p
∗
b) the point at which the maximum in
(3.5) is attained. For each L ∈ N, let (p0(L), p1(L), . . . , pb(L)) be the point at which
maximum in (3.4) is attained. Clearly,
ρ̂H (p0(L), p1(L), . . . , pb(L)) ≤ ρ̂H(p∗0, p∗1, . . . , p∗b) ≤ ρH(p∗0, p∗1, . . . , p∗b).
The first inequality follows by substituting ρ̂ for ρ in (3.5) and the fact that one is now
optimizing over a larger set. The second inequality follows from ρ̂ ≤ ρ. Note that the right
hand side no longer depends on L, and thus
lim sup
L→∞
ρ̂H (p0(L), p1(L), . . . , pb(L)) ≤ ρH(p∗0, p∗1, . . . , p∗b).
Next, for each L ∈ N, we consider the point (t0(L), t1(L), . . . , tb(L)) ∈ 1bρLcRbρLc,L,
which is closest to the to the optimizer (p∗0, p
∗
1, . . . , p
∗
b). Note that, due to the density
argument, (t0(L), t1(L), . . . , tb(L)) → (p∗0, p∗1, . . . , p∗b) as L → ∞. This, along with the
continuity of H and the fact that ρ̂→ ρ implies the lower bound
ρH(p∗0, p
∗
1, . . . , p
∗
b) = lim
L→∞
ρ̂H(t0(L), t1(L), . . . , tb(L)) ≤
≤ lim inf
L→∞
ρ̂H (p0(L), p1(L), . . . , pb(L)) .
Putting everything together completes the argument that the limit
f(ρ) = lim
L→∞
ρ̂H (p0(L), p1(L), . . . , pb(L))
exists and that the complexity function is
f(ρ) = ρH(p∗0, p
∗
1, . . . , p
∗
b) = max
p0,p1,...,pb≥0
p0+p1+···+pb=1
p1+2p2+···+bpb= 1ρ−(b+1)
ρ ·H (p0, p1, . . . , pb) .
In order to obtain the expression for complexity f(ρ), it only remains to solve the con-
strained optimization problem (3.5). We define the Lagrangian function
L(p0, . . . , pb;λ, µ) = ρ ·H(p0, p1, . . . , pb)− λ(p0 + p1 + · · ·+ pb − 1)
− µ(p1 + 2p2 · · ·+ bpb −
1
ρ
+ (b+ 1)),
and find the stationary point by solving the system
−ρ(ln pi + 1)− λ− µi = 0, for i = 0, 1, . . . , b;
p0 + p1 + · · ·+ pb = 1;
p1 + 2p2 · · ·+ bpb =
1
ρ
− (b+ 1).
(3.6)
By multiplying i-th of the first (b+ 1) equations by pi and adding them together we get
−ρ
b∑
i=0
(pi ln pi + pi)− λ
b∑
i=0
pi − µ
b∑
i=0
ipi = 0,
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 15
and from here we obtain the expression for complexity in terms of the Lagrange multipliers
λ and µ which solve the system (3.6)
f(ρ) = ρH(p0, p1, . . . , pb) = ρ+ λ+ µ
(
1
ρ
− (b+ 1)
)
. (3.7)
Subtracting successive equations in (3.6) we get
−ρ(ln pi − ln pi−1)− µ = 0,
or equivalently
pi
pi−1
= e−µ/ρ.
Therefore pi = p0e−µi/ρ, for i = 1, . . . , b. From the very first equation in (3.6) we get
p0 = e
−λ/ρ−1,
and the whole system (3.6) now reduces to just two equations
e−λ/ρ−1
b∑
i=0
e−µi/ρ = 1; (3.8)
e−λ/ρ−1
b∑
i=0
ie−µi/ρ =
1
ρ
− (b+ 1). (3.9)
Setting z = e−µ/ρ, and eliminating e−λ/ρ−1 from equations (3.8) and (3.9), gives a single
polynomial equation of degree b
bzb + (b− 1)zb−1 + · · ·+ 2z2 + z =
[
1
ρ
− (b+ 1)
]
(zb + zb−1 + · · ·+ z + 1), (3.10)
which can be written as p(z) = 0 where p(z) is given in (3.2).
Now, in order to obtain the complexity, all we need is, for a fixed 12b+1 < ρ <
1
b+1 ,
to find a real root z > 0 of the polynomial p(z) for which the expression (3.7) is the
largest. The case ρ = 1b+1 , which gives z = 0, has to be treated separately. From relation
z = e−µ/ρ and equation (3.8) we have
µ = −ρ ln z;
λ = −ρ
(
1 + ln
1− z
1− zb+1
)
.
(3.11)
Plugging (3.11) into (3.7), gives the complexity expressed in terms of the root of p(z)
f(ρ) = ρ
[
− ln 1− z
1− zb+1 −
(
1
ρ
− (b+ 1)
)
ln z
]
.
Lastly, in case ρ = 1b+1 , already from the last two equations in (3.6) we can con-
clude p1 = p2 = · · · = pb = 0 and p0 = 1. This immediately gives f(ρ) = 0 as
H(1, 0, 0, . . . , 0) = 0, completing the proof.
16 Ars Math. Contemp. 24 (2024) #P4.05
Remark 3.8. Using the standard summation formulas, we can rewrite (3.10) as
bzb+2 − (b+ 1)zb+1 + z
(1− z)2 =
[
1
ρ
− (b+ 1)
]
1− zb+1
1− z , (3.12)
or equivalently[
(2b+ 1)− 1
ρ
]
zb+2 −
[
(2b+ 2)− 1
ρ
]
zb+1 −
[
b− 1
ρ
]
z +
[
(b+ 1)− 1
ρ
]
= 0.
As discussed in the introduction, the complexity function is associated to equilibrium
(or static) models of a certain phenomena and ρ?, the point at which the complexity func-
tion attains its maximum, is interpreted as the expected and most probable density observed
in such a model. This value ρ? is sometimes called the equilibrium density of the model and
Theorem 3.9 below shows how to calculate it. A different (and perhaps more natural) way
to look at Rydberg atom model is dynamically, within the framework of random sequential
adsorption (RSA). Initially neutral atoms are sequentially and at random excited (obeying
the blockade range constraint) until the jammed configuration is reached. The expected
density of the reached jammed configuration (the jamming limit) in this dynamical version
of the model, denoted by ρb-Ryd∞ , was computed in [42, §IV]
ρb-Ryd∞ =
∫ 1
0
exp
−2 b∑
j=1
1− yj
j
 dy.
It is interesting to compare ρb-Ryd? and ρ
b-Ryd
∞ for different blockade ranges b. Even though
0 20 40 60 80 100
b
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
ρb-Ryd∞
ρb-Ryd?
Figure 8: Comparison of ρb-Ryd? and ρ
b-Ryd
∞ for 1 ≤ b ≤ 99.
they are not the same, they seem to match quite nicely, see Figure 8. Additionally, as one
would expect, they both tend to zero for large b. One can see their differences more clearly
in Figure 9. This violation of Edwards flatness hypothesis is even more pronounced when
one inspects the asymptotics of the two sequences more closely. In Figure 10 we see the
graph of quantities b · ρb-Ryd? and b · ρb-Ryd∞ .
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 17
1
2
1
3 ρ
1-Ryd
?
ρ1-Ryd∞
ρ
0.00
0.05
0.10
0.15
0.20
0.25
f1-Ryd(ρ)
1
6
1
11 ρ
5-Ryd
?
ρ5-Ryd∞
ρ
0.00
0.05
0.10
0.15
0.20
f5-Ryd(ρ)
1
21
1
41 ρ
20-Ryd
?
ρ20-Ryd∞
ρ
0.00
0.02
0.04
0.06
0.08
0.10
f20-Ryd(ρ)
1
51
1
101 ρ
50-Ryd
?
ρ50-Ryd∞
ρ
0.00
0.01
0.02
0.03
0.04
0.05
f50-Ryd(ρ)
Figure 9: Complexity function of Rydberg atom model with blockade range b, for b ∈
{1, 5, 20, 50}. Also plotted in each graph are the equilibrium density ρb-Ryd? and the jam-
ming density ρb-Ryd∞ .
0 20 40 60 80 100
b
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
b · ρb-Ryd∞
b · ρb-Ryd?
Rényi’s parking constant
Figure 10: Comparison of b · ρb-Ryd? and b · ρb-Ryd∞ for 1 ≤ b ≤ 99.
It can be shown that these two sequences approach different constants as b grows large
lim
b→∞
b · ρb-Ryd∞ =
∫ ∞
0
exp
[
−2
∫ y
0
1− e−x
x
dx
]
dy = 0.7475979202 . . .
lim
b→∞
b · ρb-Ryd? = 1.
(3.13)
18 Ars Math. Contemp. 24 (2024) #P4.05
The constant appearing in the first limit is known as Rényi’s parking constant [57]. Both of
these two limits are easier to understand in the context of irreversible deposition of k-mers.
We deal with the k-mer deposition model in the following section where we revisit those
limits.
The calculation below, showing how to obtain the first limit in (3.13), and which we
provide for completeness, appears in [32]. First note
b∑
j=1
1− yj
j
=
b∑
j=1
∫ 1
y
tj−1 dt =
∫ 1
y
b∑
j=1
tj−1 dt =
∫ 1
y
1− tb
1− t dt
=
[
x = b(1− t)
dx = −b dt
]
=
∫ b(1−y)
0
1− (1− xb )b
x
dx,
and therefore
b · ρb-Ryd∞ = b
∫ 1
0
exp
−2 b∑
j=1
1− yj
j
 dy
= b
∫ 1
0
exp
[
−2
∫ b(1−y)
0
1− (1− xb )b
x
dx
]
dy
=
[
ỹ = b(1− y)
dỹ = −b dy
]
=
∫ b
0
exp
[
−2
∫ ỹ
0
1− (1− xb )b
x
dx
]
dỹ.
The dominated convergence theorem now implies
lim
b→∞
b · ρb-Ryd∞ =
∫ ∞
0
exp
[
−2
∫ y
0
1− e−x
x
dx
]
dy = 0.7475979202 . . .
Before we calculate the second limit in (3.13), we give a characterization of the value
ρb-Ryd? in terms of a root of a certain polynomial. Compare this with the same results
obtained by Došlić [20, discussion after Theorem 2.10] and Krapivsky–Luck [43, (3.4),
(3.14) and (6.6)].
Theorem 3.9. The value ρb-Ryd? , at which the complexity of the Rydberg atom model with
blockade range b, given in Theorem 3.5, attains its maximum, can be calculated as
ρb-Ryd? =
(1− z)(1− zb+1)
1 + b− bz − 2zb+1 − 2bzb+1 + zb+2 + 2bzb+2 , (3.14)
where z is the unique root of the polynomial
z2b+1 + · · ·+ zb+2 + zb+1 − 1,
on the interval 0 < z < 1.
Proof. We seek to find the density 12b+1 < ρ
b-Ryd
? <
1
b+1 at which the complexity f =
f b-Ryd in Theorem 3.5 attains its maximum. Again, we employ the Lagrangian function
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 19
method by setting
L(ρ, z;λ) = ρ
[
− ln 1− z
1− zb+1 −
(
1
ρ
− (b+ 1)
)
ln z
]
− λ
b∑
i=0
(
i+ b+ 1− 1
ρ
)
zi
= ρ ln
1− zb+1
1− z − (1− ρ(b+ 1)) ln z − λ
b∑
i=0
(
i+ b+ 1− 1
ρ
)
zi.
The stationary points of this function solve the following system
ln
1− zb+1
1− z + (b+ 1) ln z −
λ
ρ2
· 1− z
b+1
1− z = 0
−ρ(b+ 1)z
b
1− zb+1 +
ρ
1− z −
(1− ρ(b+ 1))
z
− λ
b∑
i=1
i
(
i+ b+ 1− 1
ρ
)
zi−1 = 0
b∑
i=0
(
i+ b+ 1− 1
ρ
)
zi = 0.
Using standard summation formulas, as in (3.12), it is possible to express ρ from the third
equation as
ρ =
(1− z)(1− zb+1)
1 + b− bz − 2zb+1 − 2bzb+1 + zb+2 + 2bzb+2 .
Plugging this into the second equation gives
0 = λ
b∑
i=1
i
(
i+ b+ 1− 1
ρ
)
zi−1.
From here, we conclude λ = 0. Finally, from the first equation we get
λ = ρ2
1− z
1− zb+1 ln
zb+1(1− zb+1)
1− z
and, combining this with λ = 0, gives
ln
zb+1(1− zb+1)
1− z = 0,
or
zb+1(1− zb+1) = 1− z.
We know from Theorem 3.5 that z 6= 1, so we can rewrite this equation as
z2b+1 + · · ·+ zb+2 + zb+1 − 1 = 0.
Clearly, there is a unique 0 < z < 1 solving this equation, and the corresponding
ρb-Ryd? =
(1− z)(1− zb+1)
1 + b− bz − 2zb+1 − 2bzb+1 + zb+2 + 2bzb+2
is the density at which the complexity in the Rydberg atom model with blockade range b is
the largest.
20 Ars Math. Contemp. 24 (2024) #P4.05
The previous theorem can be used to give a proof of the second limit in (3.13).
Corollary 3.10.
lim
b→∞
b · ρb-Ryd? = 1.
Proof. Since 0 < z = z(b) < 1 solves the equation
zb+1(1− zb+1)
1− z = z
2b+1 + · · ·+ zb+2 + zb+1 = 1 (3.15)
it follows
bz2b+1 < 1 < bzb+1
and therefore
lim
b→∞
z2b+1 = 0.
Multiplying by z and taking square root we also get
lim
b→∞
zb+1 = 0.
Finally, letting b→∞ in the identity zb+1(1− zb+1) = 1− z, gives
lim
b→∞
z = 1.
Note that
b · ρb-Ryd? =
b(1− z)(1− zb+1)
1 + b(1− z)[1− 2zb+1]− 2zb+1 + zb+2
so in order to get limb→∞ b · ρb-Ryd? = 1, it suffices to show limb→∞ b(1− z) =∞. To see
this, note that from (3.15) it follows
(b+ 1) ln z = ln(1− z)− ln(1− zb+1)
and hence
lim
b→∞
(b+ 1)(1− z) = lim
b→∞
1− z
ln z
·
[
ln(1− z)− ln(1− zb+1)
]
= −1 · [−∞− 0] = +∞
which completes the argument.
4 Complexity function of jammed configurations for irreversible de-
position of k-mers
It is easy to see that the Rydberg atom model with blockade range b is, up to scaling all
densities by a factor b+1, equivalent to the irreversible deposition of k-mers model where
k = b+1. As an immediate consequence of Theorem 3.5 we get the complexity of jammed
configurations for irreversible deposition of k-mers.
Corollary 4.1. For k ∈ N, k > 1, the complexity function of jammed configurations for
irreversible deposition of k-mers is
f(ρ) =
{
ρ
k
[
− ln 1−z
1−zk −
(
k
ρ − k
)
ln z
]
, if k2k−1 < ρ ≤ 1,
0, otherwise,
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 21
where z ≥ 0 is a real root of the polynomial
k−1∑
i=0
(
i+ k − k
ρ
)
zi
for which the expression f(ρ) is the largest.
2
3
3
5
4
7
5
9
1
2
1
ρ
0
0.05
0.1
0.3
lnw1 = lnw2
lnw3
lnw4
lnw5
lnw6
lnw7
lnw8
lnw9
lnw10
f
(ρ
)
k = 2
k = 3
k = 4
k = 5
k = 6
k = 7
k = 8
k = 9
k = 10
k = 11
Figure 11: The complexity function of jammed configurations for irreversible deposition
of k-mers, for 2 ≤ k ≤ 11.
Figure 11 shows the complexity function for all 2 ≤ k ≤ 11. Note that the support of
the complexity function is now contained in the interval [1/2, 1]. In Figure 12 we compare
the equilibrium density ρk-mer? and the jamming density ρ
k-mer
∞ , for 2 ≤ k ≤ 100. In this
model it is even more obvious that the Edwards hypothesis is violated. The limits of these
two sequences as k grows large are
lim
k→∞
ρk-mer∞ =
∫ ∞
0
exp
[
−2
∫ y
0
1− e−x
x
dx
]
dy = 0.7475979202 . . .
lim
k→∞
ρk-mer? = 1.
(4.1)
Note that these limits are equivalent to those in (3.13). The convergence of jamming limits
of k-mer deposition models (as k grows to infinity) to the Rényi’s parking constant is
discussed in [44, §7.1] (see also [28, 32]).
Clearly, the second limit from (4.1) follows from Corollary 3.10 as ρk-mer? = k · ρb-Ryd?
for b = k − 1. Below, we provide a direct alternative proof of this fact.
22 Ars Math. Contemp. 24 (2024) #P4.05
Theorem 4.2.
lim
k→∞
ρk-mer? = 1.
Proof. The quantity we are interested in, ρk-mer? , is equivalent to the quantity called the
efficiency ε(k) in the context of packing Pk into Pn. It was shown in [20] that the efficiency
is determined by the smallest singularity wk of the generating function Fk(1, y), i.e., by
the smallest zero of its denominator. Hence we start by setting x = 1 into the rightmost
expression in (2.4),
Fk(1, y) =
1− yk
1− y − yk − y2k =
1−yk
1−y
1− yk 1−yk1−y
.
We rewrite its denominator as 1− qk(y), where qk(y) = qk 1−y
k
1−y , and denote the smallest
solution of equation qk(y) = 1 by wk. This equation has only one positive solution, since
qk(0) = 0, qk(1) = k > 1 for large k and q′k(y) > 0 for all y > 0. Moreover, the same
reasoning provides a better lower bound for wk, since qk( 12 ) = 2
(1−k)(1 − 2−k) < 1.
Hence 1/2 < wk < 1.
Consider now the expression
ε(k) = ρk-mer? =
k
wkq′k(x)
derived in [20]. First we rewrite q′k(wk) as
q′k(x) = x
k 1− xk
1− x
[
2k
x
− k
x(1− xk) +
1
1− x
]
.
After plugging in x = wk, the term outside the brackets becomes equal to one, and by
multiplying through by wk we arrive at
wkq
′
k(wk) =
(
2− 1
1− wkk
)
k +
wk
1− wk
.
We are seeking upper bounds to the right-hand side. The first term is bounded from above
by k, since the expression in parentheses cannot exceed one. It remains to bound the second
term. As mentioned before, wk is the only positive solution of the equation 1− qk(x) = 0.
We claim that, for a given (large) positive a, wk < 1 − ak for large enough k. So let us
suppose otherwise, that for a given a > 0, wk > 1− ak is valid for all k. It means that the
function 1− qk(x) has a positive value for x = 1− ak . By evaluating both sides, we obtain
that (
1− a
k
)k
−
(
1− a
k
)2k
<
a
k
is valid for all k. This is a contradiction, since the left-hand side has a positive limit,
e−a − e−2a > 0, while the right-hand side tends to zero as k tends to infinity. Hence,
wk < 1 − ak for large enough k. Now the second term can be bounded from above by ak ,
and the whole expression for wkq′k(wk) is bounded from above by
a+1
a k. Since a can be
taken arbitrarily large, it means that the reciprocal value of wkq′k(wk), which is equal to
our ρk-mer? , comes arbitrarily close to one, and our claim follows.
T. Došlić et al.: Complexity function of jammed configurations of Rydberg atoms 23
The convergence is quite slow, most likely logarithmic. We note another unusual thing
in Figure 12. The equilibrium density ρk-mer? attains the minimum value for k = 9. The
interpretation being that the polymers of length 9 pack the least efficiently of all polymers
assuming the equilibrium model. This phenomenon was previously observed in [20].
0 20 40 60 80 100
k
0.74
0.76
0.78
0.80
0.82
0.84
0.86
0.88
ρk-mer∞
ρk-mer?
Rényi’s parking constant
Figure 12: Comparison of ρk-mer? and ρ
k-mer
∞ for 2 ≤ k ≤ 100.
5 Conclusions
In this paper we have computed the complexity function (or configurational entropy) of
jammed configurations of Rydberg atoms with a given blockade range on a one-dimensional
lattice. We employed a purely combinatorial method which allowed us to compute the com-
plexity function by solving a constrained optimization problem. Along the way we have
explored and elucidated numerous connections between the considered problem and other
models, such as, e.g., the random sequential adsorption and packings of blocks of a given
length into one-dimensional lattices. In most cases, we have not followed those links very
far. We believe that many interesting results could be obtained by deeper investigations
of those connections. As an example, we mention here that explicit expressions for the
number of maximal packings of given size from reference [20] could be directly translated
into expressions for the number of jammed configurations of Rydberg atoms. By the same
reasoning one can show that the total number of all jammed configurations of N Rydberg
atoms with blockade range b on all one-dimensional lattices is given by (b+ 1)N+1.
The methods employed here could be easily adapted for other one-dimensional struc-
tures with low connectivity such as, e.g., cactus chains. Another class of promising struc-
tures could be various simple graphs decorated by addition of certain number of vertices of
degree one to each of their vertices.
Similar problems were considered under various guises also for finite portions of rect-
angular lattices, mostly for narrow strips of varying length. Among the best known prob-
lems of this type are the so-called unfriendly seating arrangements. See [7, 30] for their
history and some recent developments. To the same class belong the problems concerned
with privacy, such as the ones considered in [39]. All cited references were concerned with
one-dimensional lattices and/or narrow strips in the square grid, mostly with ladders. It
would be interesting to consider those problems in finite portions of the regular hexagonal
24 Ars Math. Contemp. 24 (2024) #P4.05
lattice.
Another interesting thing to do would be to study the behavior (and the difference) of
ρ∞ and ρ? for different lattices/substrates. In other words, to investigate the difference
between the jamming limit of dynamical models and the most probable densities in the
equilibrium models. A drastic example is presented by the expected density of independent
sets in stars: there are exactly two maximal independent sets in Sn = K1,n−1, one of them
with size 1 and the other with size n−1. If both of them are equally probable, the expected
size is n/2. Under dynamical model, however, the smaller one is much less probable than
the bigger one, and the expected size is 1n +
n−1
n (n − 1) = n − 2 + 2n . It would be
interesting to know more about such differences and to know for which classes of graphs
they are extremal.
Our final remark is that the jammed configurations of Rydberg atoms with a given
blockade range b are known as maximal b-independent sets in the language of graph theory.
It might be worth investigating to what extent can similar problems be formulated also in
terms of b-dominance in graphs.
ORCID iDs
Tomislav Došlić https://orcid.org/0000-0002-8326-513X
Mate Puljiz https://orcid.org/0000-0003-0912-8345
Stjepan Šebek https://orcid.org/0000-0002-1802-1542
Josip Žubrinić https://orcid.org/0009-0002-6522-9554
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