# Hochster–Eagon type theorem
for Serre’s $(S_{n})$ condition

Mitsuyasu Hashimoto Partially supported by JSPS KAKENHI Grant number 20K03538
and MEXT Promotion of Distinctive Joint Research Center Program
JPMXP0619217849.

###### Abstract

Let $(A,\mathfrak{m})\rightarrow(B,\mathfrak{n})$ be a pure homomorphism
between Noetherian commutative rings. If $B/\mathfrak{m}B$ is an Artinian
ring, then we have $\dim A=\dim B$ and $\mathop{\operator@font depth}\nolimits
A\geq\mathop{\operator@font depth}\nolimits B$. Using this version of
Hochster–Eagon theorem, we prove the following. Let $A\rightarrow B$ be a pure
homomorphism between Noetherian commutative rings. Assume that the fiber ring
$\kappa(\mathfrak{p})\otimes_{A}B$ is Artinian for each
$\mathfrak{p}\in\mathop{\operator@font Spec}\nolimits A$, and $B$ satisfies
Serre’s $(S_{n})$ condition. Then $A$ also satisfies Serre’s $(S_{n})$
condition. In particular, if a finite group $G$ acts on $B$ and the order
$|G|$ of $G$ is invertible in $B$, and if $B$ is Noetherian with the $(S_{n})$
condition, then the ring of invariants $A=B^{G}$ also satisfies the $(S_{n})$
condition.

0002020 Mathematics Subject Classification. Primary 13E05; Secondary 13A50.
Key Words and Phrases. pure homomorphism, Serre’s $(S_{n})$ condition,
Hochster–Eagon theorem

##  1 Introduction

We say that a homomorphism of commutative rings $f:A\rightarrow B$ is pure if
for any $A$-module $W$, the map $j_{W}:W\rightarrow B\otimes_{A}W$ given by
$j_{W}(w)=1\otimes w$ is injective. If $f:A\rightarrow B$ is pure, many good
ring-theoretic properties of $B$ are inherited by $A$. For example, if $B$ is
Noetherian (resp. a normal domain), then so is $A$ [HR, (6.15)]. The
Hochster–Eagon theorem [HE, Proposition 12], a striking result in this
direction appeared in 1971, states that if $f:A\rightarrow B$ is an integral
homomorphism between commutative rings, $B$ is Noetherian and Cohen–Macaulay,
and $A$ is a direct summand subring of $B$ through $f$ (that is, there is a
left inverse $g:B\rightarrow A$ as an $A$-linear map (that is, $gf=1_{A}$)),
then $A$ is also Noetherian and Cohen–Macaulay. As an application, Hochster
and Eagon proved that the non-modular invariant subring of a finite group
action on a Cohen–Macaulay ring is again Cohen–Macaulay [HE]. The assumption
of integral extension is necessary, as any finitely generated domain over a
field, Cohen–Macaulay or not, is a direct summand subring of a finitely
generated Cohen–Macaulay domain by Kawasaki’s arithmetic Macaulayfication
theorem [Kaw, Theorem 1.3]. Although Cohen–Macaulay property is not inherited
by a pure subring in genral, some important results in this direction are
known, see [Bou, HH, HR, Sch].

In this paper, we prove an analog of Hochster–Eagon theorem for Serre’s
$(S_{n})$ condition. We say that a Noetherian commutative ring $R$ satisfies
the condition $(S_{n})$ if $\mathop{\operator@font depth}\nolimits
R_{\mathfrak{p}}\geq\min(n,\dim R_{\mathfrak{p}})$ for any
$\mathfrak{p}\in\mathop{\operator@font Spec}\nolimits R$. This is equivalent
to say that for any $\mathfrak{p}\in\mathop{\operator@font Spec}\nolimits R$
such that $\mathop{\operator@font depth}\nolimits R_{\mathfrak{p}}<n$, we have
$R_{\mathfrak{p}}$ is Cohen–Macaulay. Our main theorem is the following.

######  Theorem 2.5.

Let $f:A\rightarrow B$ be a pure ring homomorphism. Let $B$ be a Noetherian
ring which satisfies Serre’s $(S_{n})$ condition. If the fiber ring
$B\otimes_{A}\kappa(\mathfrak{p})$ is zero-dimensional for any prime ideal
$\mathfrak{p}$ of $A$, then $A$ satisfies $(S_{n})$, too.

If $f:(A,\mathfrak{m})\rightarrow(B,\mathfrak{n})$ is a pure local
homomorphism such that $B/\mathfrak{m}B$ is Artinian, then we have that $\dim
A=\dim B$ and $\mathop{\operator@font depth}\nolimits
A\geq\mathop{\operator@font depth}\nolimits B$ (Proposition 2.3), and the
proof of Theorem 2.5 is reduced to this version of Hochster–Eagon theorem. As
a corollary, we have a result in non-modular invariant theory of finite
groups; if a finite group $G$ acts on a Noetherian ring $B$ and the order
$|G|$ of $G$ is invertible in $B$, the $(S_{n})$ property of $B$ is inherited
by $A$ for any $n\geq 1$ (Corollary 2.6).

##  2 The results

### L

et $R$ be a commutative ring and $f:M\rightarrow N$ be an $R$-linear map
between $R$-modules. We say that $f$ is pure or $R$-pure if for any $R$-module
$W$, the map $f\otimes 1_{W}:M\otimes_{R}W\rightarrow N\otimes_{R}W$ is
injective. Clearly, a pure linear map is injective. A ring homomorphism
between commutative rings $\varphi:A\rightarrow B$ is said to be pure if it is
so as an $A$-linear map. For each $A$-module $W$, we denote by
$j_{W}:W=A\otimes_{A}W\rightarrow B\otimes_{A}W$ the map $\varphi\otimes
1_{W}$. So $\varphi$ is pure if and only if $j_{W}$ is injective for any
$A$-module $W$. If $A$ is a subring of $B$ and the inclusion map
$A\hookrightarrow B$ is pure, then we say that $A$ is a pure subring of $B$.

### L

et $\varphi:A\rightarrow B$ be a pure ring homomorphism, and
$\mathfrak{p}\in\mathop{\operator@font Spec}\nolimits A$. Then
$\kappa(\mathfrak{p})=A\otimes_{A}\kappa(\mathfrak{p})\rightarrow
B\otimes_{A}\kappa(\mathfrak{p})$ is injective, and hence the fiber ring
$B\otimes_{A}\kappa(\mathfrak{p})$ is not zero, where
$\kappa(\mathfrak{p})=A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$ is the
residue field of $A_{\mathfrak{p}}$. It follows that
${}^{a}\varphi:\mathop{\operator@font Spec}\nolimits
B\rightarrow\mathop{\operator@font Spec}\nolimits A$ is surjective, since
$({}^{a}\varphi)^{-1}(\mathfrak{p})\cong\mathop{\operator@font
Spec}\nolimits(B\otimes_{A}\kappa(\mathfrak{p}))\neq\emptyset$ for each
$\mathfrak{p}$. In particular, a pure ring homomorphism between local rings is
a local homomorphism.

### L

et $\varphi:A\rightarrow B$ be a pure homomorphism. If $B$ is Noetherian, then
so is $A$ [HR, (6.15)].

######  Lemma 2.1.

Let $(A,\mathfrak{m})$ be a Noetherian local ring, and $f:A\rightarrow B$ be a
pure ring homomorphism. Then there exists some maximal ideal $\mathfrak{n}$ of
$B$ such that $\mathfrak{n}\cap A=\mathfrak{m}$, and $A\rightarrow
B_{\mathfrak{n}}$ is pure.

###### Proof.

Let $E=E_{A}(A/\mathfrak{m})$ be the injective hull of the residue field of
$A$. Then $j_{E}(1)\neq 0$ in $B\otimes_{A}E$ [HR, (6.11)], where $1$ is the
image of $1\in A/\mathfrak{m}$ in $E$. So there exists some maximal ideal
$\mathfrak{n}$ of $B$ such that $j_{E}(1)$ is still nonzero in
$B_{\mathfrak{n}}\otimes_{A}E$. Then $A\rightarrow B_{\mathfrak{n}}$ is pure
by [HR, (6.11)] again. As $\mathop{\operator@font Spec}\nolimits
B_{\mathfrak{n}}\rightarrow\mathop{\operator@font Spec}\nolimits A$ is
surjective, we must have $\mathfrak{n}\cap A=\mathfrak{m}$. ∎

######  Lemma 2.2.

If $A$ is a Noetherian ring and $f:A\rightarrow B$ is a pure ring
homomorphism, then $\dim A\leq\dim B$.

###### Proof.

Let
$\mathfrak{p}_{0}\supsetneq\mathfrak{p}_{1}\supsetneq\mathfrak{p}_{2}\supsetneq\cdots\supsetneq\mathfrak{p}_{d}$
be a chain of prime ideals of $A$. Note that $A_{\mathfrak{p}_{0}}\rightarrow
B_{\mathfrak{p}_{0}}$ is pure by [HR, (6.2)]. By Lemma 2.1, there exists some
prime ideal $P_{0}$ of $B$ lying over $\mathfrak{p}_{0}$ such that
$A_{\mathfrak{p}_{0}}\rightarrow(B_{\mathfrak{p}_{0}})_{P_{0}B_{\mathfrak{p}_{0}}}=B_{P_{0}}$
is pure. Using this argument to the pure ring homomorphism
$A_{\mathfrak{p}_{0}}\rightarrow B_{P_{0}}$, we know that there exists some
prime ideal $P_{1}\subset P_{0}$ such that $P_{1}\cap A=\mathfrak{p}_{1}$ and
$A_{\mathfrak{p}_{1}}\rightarrow B_{P_{1}}$ is pure. Continuing this, we can
take a chain of prime ideals

$P_{0}\supsetneq P_{1}\supsetneq P_{2}\supsetneq\cdots\supsetneq P_{d}$

such that $P_{i}\cap A=\mathfrak{p}_{i}$ and $A_{\mathfrak{p}_{i}}\rightarrow
B_{P_{i}}$ is pure. This proves the lemma. ∎

######  Proposition 2.3.

Let $f:(A,\mathfrak{m})\rightarrow(B,\mathfrak{n})$ be a pure local ring
homomorphism such that $B$ is Noetherian and the fiber ring $B/\mathfrak{m}B$
is Artinian. Then $\dim A=\dim B$ and $\mathop{\operator@font depth}\nolimits
A\geq\mathop{\operator@font depth}\nolimits B$.

###### Proof.

We have $\dim A\leq\dim B\leq\dim A+\dim B/\mathfrak{m}B=\dim A$ by Lemma 2.2
and the assumption that $\dim B/\mathfrak{m}B$ is Artinian.

Let $r=\mathop{\operator@font depth}\nolimits A$ and $s=\mathop{\operator@font
depth}\nolimits B$. We prove that $r\geq s$ by induction on $s$. If $s=0$,
then the assertion is trivial. Assume that $s>0$. If $r=0$, then there exists
some $x\in\mathfrak{m}$ such that $0:_{A}x=\mathfrak{m}$. As $f$ is an
injective map, $\mathfrak{m}B\subset 0:_{B}x\subset\mathfrak{n}$. As
$B/\mathfrak{m}B$ is Artinian, $B/(0:_{B}x)$ is Artinian and is nonzero. Hence

$\mathop{\operator@font Ass}\nolimits_{B}B\supset\mathop{\operator@font
Ass}\nolimits_{B}Bx=\mathop{\operator@font
Ass}\nolimits_{B}B/(0:_{B}x)=\\{\mathfrak{n}\\},$

and $s=0$. This is a contradiction. So $r>0$. For each
$P\in\mathop{\operator@font Ass}\nolimits B$, we have $P\cap
A\neq\mathfrak{m}$, since $\mathfrak{n}\cap A=\mathfrak{m}$,
$P\subsetneq\mathfrak{n}$, and $B/\mathfrak{m}B$ is zero-dimensional. By prime
avoidance, there exists some

$y\in\mathfrak{m}\setminus\left((\bigcup_{P\in\mathop{\operator@font
Ass}\nolimits B}(P\cap A))\cup(\bigcup_{\mathfrak{p}\in\mathop{\operator@font
Ass}\nolimits A}\mathfrak{p})\right).$

So $y$ is $A$-regular $B$-regular. By [HR, (6.2)], $A/yA\rightarrow B/yB$ is
pure, and $(B/yB)/\mathfrak{m}(B/yB)$ is Artinian. By induction assumption, we
have

$r-1=\mathop{\operator@font depth}\nolimits A/yA\geq\mathop{\operator@font
depth}\nolimits B/yB=s-1$

and hence $r\geq s$, as required. ∎

######  Theorem 2.4 (Hochster–Eagon [HE]).

Let $f:A\rightarrow B$ be a pure ring homomorphism. Let $B$ be Noetherian and
Cohen–Macaulay. If the fiber ring $B\otimes_{A}\kappa(\mathfrak{m})$ is zero-
dimensional for any maximal ideal $\mathfrak{m}$ of $A$, then $A$ is
Cohen–Macaulay.

###### Proof.

It suffices to show that $A_{\mathfrak{m}}$ is a Cohen–Macaulay local ring for
any maximal ideal $\mathfrak{m}$ of $A$. Replacing $f$ by
$f_{\mathfrak{m}}:A_{\mathfrak{m}}\rightarrow B_{\mathfrak{m}}$, we may assume
that $(A,\mathfrak{m})$ is local. Then we can find a maximal ideal
$\mathfrak{n}$ of $B$ lying over $\mathfrak{m}$ such that $A\rightarrow
B_{\mathfrak{n}}$ is pure. As a localization of a zero-dimensional ring is
still zero-dimensional, we may assume that $f$ is a local homomorphism between
local rings. By the Cohen–Macaulay property of $B$ and Proposition 2.3, we
have

$\mathop{\operator@font depth}\nolimits A\geq\mathop{\operator@font
depth}\nolimits B=\dim B=\dim A\geq\mathop{\operator@font depth}\nolimits A,$

and the assertion follows. ∎

######  Theorem 2.5.

Let $f:A\rightarrow B$ be a pure ring homomorphism. Let $B$ be a Noetherian
ring which satisfies Serre’s $(S_{n})$ condition. If the fiber ring
$B\otimes_{A}\kappa(\mathfrak{p})$ is zero-dimensional for any prime ideal
$\mathfrak{p}$ of $A$, then $A$ satisfies $(S_{n})$, too.

###### Proof.

Let $\mathfrak{p}$ be a prime ideal of $A$ such that $\mathop{\operator@font
depth}\nolimits A_{\mathfrak{p}}<n$. Then we can find a prime ideal $P$ of $B$
lying over $\mathfrak{p}$ such that the local homomorphism
$A_{\mathfrak{p}}\rightarrow B_{P}$ is pure. Note that the dimension of the
fiber ring $\kappa(\mathfrak{p})\otimes_{A_{\mathfrak{p}}}B_{P}$ is zero-
dimensional. Since $\mathop{\operator@font depth}\nolimits
B_{P}\leq\mathop{\operator@font depth}\nolimits A_{\mathfrak{p}}<n$ by
Proposition 2.3 and $B$ satisfies the $(S_{n})$ condition, we have that
$B_{P}$ is Cohen–Macaulay. By Theorem 2.4, we have that $A_{\mathfrak{p}}$ is
Cohen–Macaulay. This shows that $A$ satisfies the $(S_{n})$ condition. ∎

######  Corollary 2.6.

Let $G$ be a finite group acting on a Noetherian ring $B$. Assume that the
order $|G|$ of $G$ belongs to $B^{\times}$, the unit group of $B$. If $B$
satisfies Serre’s $(S_{n})$ condition, then so does the ring of invariants
$B^{G}$.

###### Proof.

For $b\in B$, the polynomial $f(t)=\prod_{g\in G}(t-gb)$ is monic and lies in
$B^{G}[t]$. As $f(b)=0$, the element $b$ is integral over $B^{G}$. This shows
that $B^{G}\hookrightarrow B$ is an integral extension, and all the fiber
rings are zero-dimensional.

Let $R={\mathbb{Z}}[|G|^{-1}]$, and define the Reynolds operator to be
$\rho=|G|^{-1}\sum_{g\in G}g\in RG$. It is a central idempotent of the group
algebra $RG$ of $G$ over $R$. As $RG$ acts on $B$, the $G$-linear map
$\rho:B\rightarrow B$ is defined. It is easy to see that $\rho(B)\subset
B^{G}$ and $\rho$ is the identity map on $B^{G}$. So $\rho(B)=B^{G}$, and
$\rho:B\rightarrow B^{G}$ is the left inverse of the inclusion
$j:B^{G}\rightarrow B$. Moreover,

$\rho(ab)=|G|^{-1}\sum_{g\in G}a(gb)=a(\rho b)$

for $a\in B^{G}$ and $b\in B$. Namely, $\rho$ is the left inverse of $j$ as a
$B^{G}$-linear map, and thus $B^{G}$ is a direct summand subring of $B$. It
follows that $B^{G}$ is a pure subring of $B$. The corollary follows from
Theorem 2.5. ∎

##  References

  * [Bou] J.-F. Boutot, Singularité rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65–68.
  * [HE] M. Hochster and J. A. Eagon, Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020–1058.
  * [HH] M. Hochster and C. Huneke, Applications of the existence of big Cohen–Macaulay algebras, Adv. Math. 113 (1995), 45–117.
  * [HR] M. Hochster and J. Roberts, Rings of invariants of reductive groups action on regular rings are Cohen–Macaulay, Adv. Math. 13 (1974), 115–175.
  * [Kaw] Takesi Kawasaki, On arithmetic Macaulayfication of Noetherian rings, Trans. Amer. Math. Soc. 354 (2002), 123–149.
  * [Sch] H. Schoutens, Pure subrings of regular rings are pseudo-rational, Trans. Amer. Math. Soc. 360 (2008), 609–627.

Mitsuyasu HASHIMOTO

Department of Mathematics

Osaka Metropolitan University

Sumiyoshi-ku, Osaka 558–8585, JAPAN

e-mail<EMAIL_ADDRESS>


# Long-Range Anomalous Decay of the Correlation in Jammed Packings

Paolo Rissone<EMAIL_ADDRESS>Small Biosystems Lab, Department of
Condensed Matter Physics, Carrer de Marti i Franques, 1, 11, 08028, Barcelona,
Spain  Eric I. Corwin<EMAIL_ADDRESS>Department of Physics and Materials
Science Institute, University of Oregon, Eugene, Oregon 97403, USA  Giorgio
Parisi<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza
Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy  Istituto Nazionale di
Fisica Nucleare, Sezione di Roma I, P.le A. Moro 5, 00185 Rome, Italy
Institute of Nanotechnology (NANOTEC) - CNR, Rome unit, P.le A. Moro 5, 00185
Rome, Italy

###### Abstract

We numerically study the structure of the interactions occurring in three-
dimensional systems of hard spheres at jamming, focusing on the large-scale
behavior. Given the fundamental role they play in the configuration of jammed
packings, we analyze the propagation through the system of the weak forces and
of the variation of the coordination number with respect to the isostaticity
condition $\Delta Z$. We show that these correlations can be successfully
probed by introducing a correlation function weighted on the density-density
fluctuations. The results of this analysis can be further improved by
introducing a representation of the system based on the contact points between
particles. In particular, we find evidence that the weak forces and the
$\Delta Z$ fluctuations support the hypothesis of randomly jammed packings of
spherical particles being hyperuniform by exhibiting an anomalous long-range
decay. Moreover, we find that the large-scale structure of the density-density
correlation exhibits a complex behavior due to the superimposition of two
exponentially damped oscillating signals propagating with linearly depending
frequencies.

Introduction. Amorphous packings of nearly incompressible particles, such as
marbles and pebbles, have been the object of an intense investigation during
the past decades as they represent a suitable benchmark for studying a broad
range of dense-packing and optimization problems [1, 2]. This rising interest
led to the development of many experiments [3, 4, 5] and simulations [6, 7, 8]
that made possible an extensive study of the features of these systems.
Moreover, this field appeared to be the perfect environment to apply the
theories of frustrated interactions [9]. In particular, the application of the
replica theory [10] led to the elaboration of an exact analytical solution
valid in the limit of high-dimensional packings [11, 12, 13].

We focus on athermal packings of frictionless hard spheres (HSs) compressed
until particles come into mechanical contact with their nearest neighbors. The
trapped spheres form a rigid network and cannot explore the surrounding
environment (ergodicity breaking). Under these conditions, the system enters a
phase of matter known as “jamming” [14, 15]. It has been hypothesized that
saturated jammed systems (no space to add another particle) are hyperuniform
[16], implying that their radial distribution function (RDF) tends to zero
from negative values and as a power law [17] – $g(r)-1\propto r^{-4}$. Even
though the tendency of the jammed packings to hyperuniformity has been
observed (with deviations from the postulated behavior) [18], such power-law
scaling of the pair correlation function has never been directly measured.

In this framework, we find evidence of hyperuniformity in the long-range
correlation of the forces exchanged between adjacent particles and the
deviation of the number of contacts per particle from the average value,
$\Delta Z=Z-\langle Z\rangle$. On the one hand, jammed packings exhibit a
unique force network [19], whose long-range fluctuations demand study. On the
other hand, it has been shown that $\Delta Z$ exhibits interesting features at
jamming [20, 21, 22] and that the fluctuations of the coordination number
$\sigma_{Z}^{2}$ for a fixed $\Delta Z$ are similar to those of density
hyperuniformity [23]. The research for a static observable exhibiting a
nontrivial behavior close to jamming is motivated by the existence of a
corresponding long-ranged dynamical response. It has been shown [24, 25, 20]
that a local perturbation to the position of a pair of adjacent particles,
i.e., breaking the contact between particles, produces a response propagating
through the system up to a maximum length, the “response length”, $\xi_{R}$
that diverges at jamming [26]. The main hindrance to this analysis is
represented by the strong statistical noise exhibited by the correlation
functions in the long range and superimposing to the (weak) signal of
interest. To overcome this problem, we define a suitable pair correlation
function to point out the long-range behavior of the observables by filtering
out the interfering signals. Moreover, we introduce a representation of the
interparticle network based on the contact points between particles instead of
their centers of mass. We show that the shift to a system of fictive particles
improves the resolution of the correlation function and is fundamental in
identifying the long-range features of the jammed packings.

System Setup. Given a system of $N$ randomly distributed monodisperse HSs of
diameter $\sigma$, let us introduce the interaction via the dimensionless
interaction potential

$U=\sum_{\langle
i,j\rangle}\left(1-\frac{|\mathbf{r}_{i}-\mathbf{r}_{j}|}{\sigma_{ij}}\right)\,,$
(1)

where $\mathbf{r}_{i,j}$ is the position of particles $i,j$ and $\sigma_{ij}$
is the distance between the centers of particles $i$ and $j$ when they are in
contact. Note that, for monodisperse HSs, $\sigma_{ij}=\sigma$ and that for
particles in kissing contact $r=|\mathbf{r}_{i}-\mathbf{r}_{j}|=\sigma$ so
that particles’ interaction potential is zero.

The system is controlled via the “packing fraction” $\phi$, defined as the
fraction of the system volume occupied by the spheres. The jammed phase is
reached when the packing fraction hits the critical value $\phi_{J}\approx
0.64$ in three-dimensional systems [27, 28, 29, 30]. By starting in the
overjammed region $(\phi\approx 2\phi_{J})$, $\phi$ is gradually decreased by
gently shrinking the particles’ diameter until the system reaches jamming. The
jamming point is approached by iteratively minimizing the potential energy
[Eq. (1)] using the “FIRE” algorithm [6] according to the protocol described
in [19] (Appendix B). The simulation ends when the targeted precision is
reached, i.e. when it is impossible to distinguish between a kissing contact
and a small overlap.

At the end of this protocol, the particles form a network of enduring contacts
that is stable only if the isostaticity condition holds [20, 31, 32, 33],
i.e., the average number of contacts per particle satisfies (in a first
approximation) $\langle Z\rangle\equiv Z_{iso}\approx 2d$, where $d$ is the
system dimension. As shown in [34], this property depends on the system
mechanical stability being ensured by the exact balancing between the affine
and the nonaffine (negative) components of the interparticle interactions over
the whole network. Within this picture, it is important to point out the
existence of “bucklers”, i.e., particles that still form part of the rigid
network but that are minimally constrained so that $Z=d+1$ [19]. A notable
exception to the isostaticity condition is represented by “rattlers”,
particles that are not part of the contact network and freely move inside
cages bounded by particles permanently in contact. The identification and the
exclusion of rattlers are fundamental for obtaining reliable results. By
adopting this method, we generated 96 critically jammed packings of
$N=16\,384$ particles in $d=3$.

Generalized RDF. Given a generic observable $\mathcal{O}$, we defined the
“generalized pair correlation function” as

$C^{s}_{\mathcal{O}}(r)=\frac{g^{s}_{\mathcal{O}}(r)}{g(r)}\,,$ (2)

where $r$ is the distance between particle pairs, $g(r)$ is the usual RDF [35]
and

$g^{s}_{\mathcal{O}}(r)=\frac{1}{C}\sum_{i,j}\delta(|\mathbf{r}_{i}-\mathbf{r}_{j}|-r)\mathcal{O}^{s}_{i}\mathcal{O}^{s}_{j}\,,$
(3)

$\mathbf{r}_{i}$ and $\mathbf{r}_{j}$ being the positions of particles $i$ and
$j$, $C$ the normalization factor and $s\in\Re$ a control parameter. Notice
that by choosing $s=0$ in Eq. (3), one gets the RDF (additional details can be
found in the Supplemental Material111See Supplemental Material for the
extensive description of the generalized RDF). We studied the two cases
$\mathcal{O}=f$ and $\mathcal{O}=\Delta Z\equiv Z-Z_{\rm iso}$, $f$ being the
force exchanged between particles and $\Delta Z$ the deviation of the number
of contacts per particle from isostaticity.

To study the long-range correlations with higher accuracy, we also introduced
the jammed packings’ representation with respect to the contact points between
particles instead of the centers of mass. To do this, let us consider the
densest packing in which HSs can arrange (see the inset in Fig. 1). Each
contact point between the HSs can be seen as the center of a fictive particle
with radius $\sigma^{\prime}=\sigma/2$. Thus, given a system of $N$ particles
and $Z_{\rm iso}=2d$ average contacts per particle, the new contacts-based
system will be formed by $N^{\prime}\leq 2dN=6N$ particles.

Contacts RDF. Switching from the centers-of-mass-based to the contact-points
description of the network, allowed us to study the radial distribution
function of the contact-centered model (called “contacts RDF” in what follows)
with a much higher resolution. Figure 1 shows the short-range behavior of the
contacts RDF (blue line) compared to the real spheres RDF (red dashed line).

Figure 1:  Contacts and original RDFs (in blue and red, respectively) in $d=3$
as a function of the distance (expressed in the respective diameter units
$r/\sigma^{\prime}$ and $r/\sigma$). Each peak (labeled with a different
letter) occurs in correspondence to a different configuration of the fictive
particles, as shown in the correspondent representations (which, however, do
not exhaust all the possibilities), along with the distance at which each
discontinuity appears (red lines). (Inset) Schematic representation of the
contacts-centered spheres model in $d=3$. Each fictive particle (dashed black
lines) results from the contact of two HSs and has a diameter
$\sigma^{\prime}=\sigma/2$, being $\sigma$ the diameter of the original HS.

The contacts RDF points out new features of the pair correlation function at
jamming, evidencing discontinuities that were much smoother [peaks
(e),(f),(g)] or completely absent [peak (b)] in the original system
description. The higher accuracy of the contacts RDF is further reflected by
the enhanced sharpness of peaks (c) and (d). The origin of each one of these
discontinuities can be easily addressed. The first peak ($r=\sigma^{\prime}$)
is due to a nearest-neighbor contact, while the others are determined by
different possible configurations of HSs forming a chain of contacts. Notably,
peaks (c) and (d) in Fig.1 correspond to “real” $\delta$ functions, i.e., to
mechanically rigid configurations of perfectly centrosymmetric particles with
respect to the transmission of forces. The local centrosymmetry ensures a zero
nonaffine component of the response and therefore the full mechanical
stability [37]. All the other peaks are the result of a wide range of possible
arrangements.

Figure 2:  Contacts density-density correlation $g(r/\sigma^{\prime})$
multiplied by the factor $e^{a\,r/\sigma^{\prime}}$ with $a\approx 0.3$. Two
sets of distinct oscillations given by Eq. (4) appear to be superimposed. The
first one, $f_{m}(r)$, propagating in the middle range
$[4:9]\,r/\sigma^{\prime}$ and the second, $f_{l}(r)$, in the long-range
region $[9:22]\,r/\sigma^{\prime}$. By fitting the resulting function (red
line) we found that $p_{l}\approx 7.5$ and $p_{m}\approx 3.8$. The distance is
expressed in diameter units.

Interestingly, as shown in Fig.2, the mid-range and long-range behavior of the
correlation function appears to be exponentially damped as $e^{-ar}$, with
$a\approx 0.3$. The enhancement of such behavior by considering $g(r)\times
e^{ar}$ pointed out a superposition of two oscillatory functions of type

$f_{i}(r)=c_{i}\,e^{a_{i}r}\cos(p_{i}r+\psi_{i}),$ (4)

where $c_{i},a_{i},p_{i},\psi_{i}$ are the function parameters and $i=m,l$
denotes the mid- and long-range regions, respectively. By performing the
resulting eight parameters fit222See Supplemental Material for a complete
description of the 8 parameters fit $g(r)=f_{m}(r)+f_{l}(r)$, we found
$p_{m}\approx 2p_{l}=7.514\pm 0.003$ and $a_{m}\approx 2a_{l}=0.70\pm 0.01$.
This result proves that because the contribution of the long-range
oscillations is small compared to the mid-range ones, the $f_{l}(r)$ can be
considered as $\mathcal{O}(2)$ correction to the $\mathcal{O}(1)$ mid-range
leading term $f_{m}(r)$.

Weak Forces Correlation. At the jamming point, each particle of the system
gets trapped in a fixed position by mechanical contact with its nearest
neighbors. The stability of the resulting configuration is ensured by the
balance of all the forces exchanged in these contact points, determining the
formation of a complex forces network spreading through the whole system [39,
40, 5]. Within this picture, it is possible to distinguish between “strong”
forces, which form a backbone crossing the whole system, and “weak” forces
spreading only in small subregions of the system confined by branches of the
main network (see Supplemental Material333See the Supplemental Material for a
detailed explanation on origin of the weak forces). Let us consider Eq. (3)
with $\mathcal{O}=f$. By choosing $s<0$ the resulting $g_{f}^{s}(r)$ will be
“weighted” on the weak forces so that the smaller the force, the bigger its
contribution to the correlation. Note that not any value of $s$ can be chosen.
In fact, as shown in [19], the force distribution can be described as function
$P(f)\propto f^{\theta}$, with $\theta\approx 0.4$, Thus, the average force
(raised to the power $s$) can be estimated as

$\langle f^{s}\rangle\sim\int dff^{s}P(f)=\int
dff^{s}f^{\theta}\propto\frac{1}{1+\theta+s}\,.$ (5)

which diverges unless $s>s_{\min}\equiv-1-\theta\approx-1.4$.

Figure 3:  (a) Weak force correlation function $C^{s=-1}_{f}(r)$ in $d=3$ and
enlargement of the long-range region in log-log scale (inset). The density-
density correlations propagating at long range have been filtered by
introducing the average over the oscillation period $T$, $\langle
C^{s}_{f}(r)\rangle_{T}$ (orange circles). The log-log scale in the inset
points out a long-range power-law behavior (red line) decaying with an
exponent $\gamma_{f}^{s=-1}=0.7\pm 0.3$ in the range [7:23]. (b) Weak forces
correlation function $C^{s=-1/2}_{f}(r)$ in $d=3$. The fluctuations analysis
points out a power-law decay with an exponent $\gamma_{f}^{s=-1/2}=4.1\pm 0.3$
in the range [7:23]. Notice that both insets show the modulus of the
correlation function. The distances are expressed in diameter units.

Figure 3 shows the weak forces correlation $C^{s}_{f}(r)$ for $s=-1$ and
$s=-1/2$. In both cases, the local density-density oscillations appear to be
not completely damped, propagating up to large length scales and masking any
power-law decay. Therefore, we introduced the correlation function averaged
over the period $T$ of the short-range oscillations $\langle
C^{s}_{f}(r)\rangle_{T}$ (orange circles), filtering out most of the
additional periodic component. The period $T$ is roughly equal to the particle
diameter but is tuned for each dataset. By fitting in the mid- and long-range
region $[7:23]\,r/\sigma^{\prime}$ and the resulting correlation functions to
the power law $f(r)=Ar^{-\gamma_{f}^{s}}+C$ for $s=(-1,-1/2)$ (red line in the
insets) with $C\approx 0$, we respectively found $\gamma_{f}^{s=-1}=0.7\pm
0.3$ and $\gamma_{f}^{s=-1/2}=4.1\pm 0.3$. These results prove that the
fluctuations of the weak force correlation function at large length scales
decay with the power law expected by the hyperuniformity theory, with the only
constraint being a fine tuning of the selected (weak) forces. In fact, for
$s=-1$ the fluctuations rapidly go to zero in the short range, whereas for
$s=-1/2$ they exhibit a long-ranged anomalous decay with the measured
nontrivial exponent.

Contacts Correlation. It has been proved that, at jamming, the coordination
number per particle – $Z$ – plays an essential role in determining the system
features [23]. The study of the fluctuations $\sigma_{\Delta Z}^{2}$ led to
the definition of a structural relaxation length $\xi_{\Delta Z}\propto\Delta
Z^{-\nu}$, where $\Delta Z=Z-2d$, $d$ is the system dimension and $\nu$ is a
nontrivial exponent. Here we point out that the $\Delta Z$ correlation
exhibits the expected anomalous behavior.

As previously discussed, changing to a contacts-based system representation
also implies changing from a system of $N$ particles to one of
$N^{\prime}=2dN$ fictive particles. In this framework, we defined the number
of contacts per (fictive) particle as

$Z^{\prime}_{l}=\frac{Z_{i}+Z_{j}}{2}\,,$ (6)

where $l=(1,\dots,N^{\prime})$ and $i,j=(1,\dots,N)$ with $i\neq j$. Thus, we
chose $\mathcal{O}=\Delta Z^{\prime}=Z^{\prime}-2d$ and $s=1$ Eqs. (3) and
(2). The resulting correlation function $C^{s=1}_{\Delta Z^{\prime}}(r)$
(shown in Fig.4) clearly exhibits a nontrivial behavior in the middle and
long-range superimposed onto some oscillations as a result of the incomplete
damping of the density-density correlations. As described above, this power-
low trend can be seen by fitting the period $T$ of the density-density
oscillations and eventually recomputing Eq. (3) by considering spherical
shells of thickness $T$ (orange circles in Fig.4). Analogously to the previous
case, we fit the resulting function according to the power law
$f(r)=Ar^{-\gamma_{\Delta Z^{\prime}}}+C$ in the range [4:23], finding a
nontrivial exponent $\gamma_{\Delta Z^{\prime}}=3.9\pm 0.2$.

Figure 4:  Contact correlation function $C^{s=1}_{\Delta Z^{\prime}}(r)$ in
$d=3$. The density-density oscillations at short range (blue line) have been
filtered by introducing the correlation averaged over the oscillation period
$T$, $\langle C_{\Delta Z^{\prime}}^{s=1}(r)\rangle$ (orange circles). (Inset)
Enlargement of the mid- and long-range regions of the modulus of the
$C^{s=1}_{\Delta Z^{\prime}}(r)$ (blue line) in log-log scale. The correlation
function exhibits a power-law decay (red line) propagating with a nontrivial
exponent $\gamma_{\Delta Z^{\prime}}=3.9\pm 0.2$. The distance is expressed in
diameter units.

Conclusion. We studied the mid- and large-scale spatial correlations in HS
packings at jamming by defining the correlation functions [Eqs. (3) and (2)]
for the weak forces $f$ and the variation of the coordination number per
particle $\Delta Z^{\prime}$. We found that the correlation functions exhibit
a long-range anomalous behavior, supporting the hyperuniformity hypothesis and
marking the transition to the jammed phase. In fact, both the weak forces and
the $\Delta Z^{\prime}$ correlation exhibit a power-law decay with nontrivial
exponents $\gamma^{s=-1}_{f}\approx\gamma_{\Delta Z^{\prime}}\approx 4$.
Moreover, we introduced a representation of the system based on the particles’
contact points instead of their centers of mass. This change pointed out new
features of the density-density correlation function, which appears to be a
sum of two different oscillating signals propagating in the middle and long-
range. A theoretical explanation for this two-terms form could be recovered by
writing integral equations of the hypernetted chain kind [42] for the
correlation function. However this problem has not been explored yet. It would
be of great interest to obtain such proof. Changing to the contacts-based
representation of the system also increased the sensitivity of the correlation
function at long range.

These results show that static observables other than the density can be used
to characterize the main features of jammed systems, marking (expected)
theoretical behaviors that remained hidden by studying the density
fluctuations of the pair correlation function. A further step in this analysis
would involve the study of bigger systems, i.e. systems with a larger number
of particles $N$, implying easier isolation of the power-law decay with
respect to the local oscillations. Moreover, it would be of great interest to
characterize these correlation functions at a packing fraction $\phi<\phi_{J}$
close to jamming and varying it up to the jamming point.

This work was supported by a Simons Foundation grant ($\\#454939$, EC;
$\\#454949$, GP).

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