Homological Percolation Dynamics

• Homological percolation is defined by the emergence of giant k-cycles that remain nontrivial in an ambient torus, extending classical connectivity concepts.
• The framework uses inclusion-induced maps on homology groups from random filtrations to identify macroscopic cycles.
• Models across continuum and lattice settings reveal critical thresholds linked to zeros of the expected Euler characteristic curve.

Searching arXiv for recent and foundational papers on homological percolation and closely related topological percolation models. Homological percolation is a higher-dimensional analogue of classical percolation in which the relevant macroscopic objects are not only connected components, but also homology classes that remain nontrivial in an ambient space. In the modern formulation developed for random subsets of the flat torus $T^d$, one studies a filtration $\{X_t\}$ and the inclusion-induced maps $H_k(X_t)\to H_k(T^d)$; the resulting nonzero image classes are the “giant” or “essential” $k$-cycles that define the percolative transition in degree $k$ (Bobrowski et al., 2020). Across continuum, lattice, and random-field models, the subject links percolation theory, algebraic topology, random geometric complexes, and topological data analysis, with a recurring empirical theme that the onset of giant cycles is closely related to zeros of the expected Euler characteristic curve (Bobrowski et al., 2019).

1. Conceptual framework and formal definitions

Classical percolation is fundamentally a $0$-dimensional phenomenon: the emergence of a giant connected component is an $H_0$ event. Homological percolation extends this viewpoint to higher dimensions by asking when a random space contains macroscopic cycles that represent ambient homology classes rather than merely local topological fluctuations (Bobrowski et al., 2020).

A standard setting is a random filtration $\{X_t\}$ of subsets of a compact ambient space $M$, with

$X_s \subset X_t \subset M \qquad (s<t).$

In the toroidal framework used repeatedly in the literature, the ambient space is the flat torus

$\{X_t\}$0

chosen because it is locally Euclidean, has no boundary, and has nontrivial homology in every degree $\{X_t\}$1, with

$\{X_t\}$2

This makes it meaningful to ask whether a random subspace already contains representatives of the ambient torus classes (Bobrowski et al., 2019).

For each parameter value $\{X_t\}$3, the inclusion $\{X_t\}$4 induces

$\{X_t\}$5

The image $\{X_t\}$6 consists of the homology classes in $\{X_t\}$7 that remain nontrivial in the ambient space. These are the “giant” cycles in the toroidal formulation. Two basic events are

$\{X_t\}$8

Accordingly, $\{X_t\}$9 is the event that at least one giant $H_k(X_t)\to H_k(T^d)$0-cycle exists, while $H_k(X_t)\to H_k(T^d)$1 is the event that all ambient $H_k(X_t)\to H_k(T^d)$2-dimensional torus classes are represented (Bobrowski et al., 2019). In the continuum Boolean model on $H_k(X_t)\to H_k(T^d)$3, the analogous notation is

$H_k(X_t)\to H_k(T^d)$4

with corresponding vacancy events $H_k(X_t)\to H_k(T^d)$5 for the complement (Bobrowski et al., 2020).

This distinction between ambiently essential cycles and arbitrary cycles is fundamental. The random set $H_k(X_t)\to H_k(T^d)$6 or $H_k(X_t)\to H_k(T^d)$7 may have many local holes or transient persistent-homology features, but homological percolation is specifically concerned with those classes whose image in ambient homology is nonzero. A long-lived persistent feature need not be giant in this sense, and conversely a giant class is defined by ambient essentiality rather than persistence length alone (Bobrowski et al., 2019).

In the finite-graph covering-space setting, the same idea is reformulated using covering maps. For a covering $H_k(X_t)\to H_k(T^d)$8, the first homology group associated with the cover is

$H_k(X_t)\to H_k(T^d)$9

where nontrivial cycles are those that fail to lift trivially to the cover. Here homological percolation refers to the appearance of open $k$0-cycles in Bernoulli edge percolation that are nontrivial in this quotient (Woolls et al., 2020). This suggests a common principle: the relevant cycles are those obstructed from being reduced to local or lift-trivial structure.

2. Toroidal continuum theory and giant $k$1-cycles

A rigorous continuum formulation was introduced for the Boolean occupied set

$k$2

where $k$3 is a Poisson process on $k$4, under the thermodynamic scaling

$k$5

In this setting, homological percolation means the emergence of giant $k$6-cycles in $k$7, that is, classes in $k$8 whose image in $k$9 is nonzero (Bobrowski et al., 2020).

The central theorem establishes threshold sequences

$k$0

with $k$1, and exponential decay below threshold as well as exponential convergence to certainty above threshold (Bobrowski et al., 2020). In particular, if $k$2, then giant $k$3-cycles are exponentially unlikely, while if $k$4, then all giant $k$5-cycles are exponentially likely.

The lowest nontrivial dimension is exact: $k$6 Thus giant $k$7-cycles appear exactly at the classical occupied percolation threshold (Bobrowski et al., 2020). At the opposite end,

$k$8

so giant $k$9-cycles are controlled from above by the vacancy threshold (Bobrowski et al., 2020). The threshold hierarchy is monotone: $0$0 which formalizes the principle that higher-dimensional macroscopic structures emerge no earlier than lower-dimensional ones (Bobrowski et al., 2020).

A crucial structural identity is the toroidal duality formula

$0$1

which yields the complementarity

$0$2

This occupied-vacant duality ties giant occupied $0$3-cycles to giant vacant $0$4-cycles and underlies the relationship between lower- and upper-dimensional thresholds (Bobrowski et al., 2020).

The same ambient-essential notion reappears in finite-volume toroidal lattice models. In plaquette and permutohedral models on $0$5, an $0$6-cycle is called giant if its class remains nontrivial in $0$7, and the sharp-threshold theory concerns the jump from $0$8 to $0$9 (Duncan et al., 2020). This places the torus at the center of the subject as the natural ambient space for defining macroscopic cycles.

3. Lattice models, duality, and exact threshold phenomena

Finite toroidal lattice models provide a rigorous parallel to continuum homological percolation. In plaquette percolation, the ambient complex is the cubical decomposition of

$H_0$0

and the random subcomplex is formed by taking the full $H_0$1-skeleton and then including each $H_0$2-face independently with probability $H_0$3. In the permutohedral model, the ambient torus carries the tiling induced by the Voronoi cells of $H_0$4, and each $H_0$5-dimensional permutohedron is included independently with probability $H_0$6 together with all its faces (Duncan et al., 2020).

For plaquettes, the relevant events are

$H_0$7

while for permutohedra, for each $H_0$8,

$H_0$9

These encode, respectively, the existence of at least one giant cycle and the spanning of all ambient homology in degree $\{X_t\}$0 (Duncan et al., 2020).

The topological backbone of the theory is duality. In the cubical model,

$\{X_t\}$1

and in the permutohedral model,

$\{X_t\}$2

These identities are the higher-dimensional analogue of planar primal-dual duality in ordinary percolation (Duncan et al., 2020).

Sharp-threshold theorems show that, for fixed median threshold functions $\{X_t\}$3, there is a transition from nonexistence of giant cycles to surjectivity onto ambient homology. For plaquettes,

$\{X_t\}$4

and an analogous statement holds for permutohedra (Duncan et al., 2020).

The middle-dimensional self-dual case yields exact thresholds. If $\{X_t\}$5, then for plaquettes,

$\{X_t\}$6

and for permutohedra,

$\{X_t\}$7

These are finite-volume higher-dimensional analogues of Kesten’s $\{X_t\}$8 theorems for bond percolation on the square lattice and site percolation on the triangular lattice (Duncan et al., 2020).

At the extremal homological dimensions, exact constants are again available. For plaquettes,

$\{X_t\}$9

while for permutohedra,

$M$0

This embeds homological percolation within the classical threshold landscape while preserving genuinely higher-dimensional content (Duncan et al., 2020).

4. Euler characteristic, persistent homology, and empirical threshold proxies

A major empirical development is the observed relation between homological-percolation thresholds and zeros of the expected Euler characteristic curve. For a filtration $M$1, the Euler characteristic is

$M$2

the Euler characteristic curve is

$M$3

and the expected Euler characteristic curve is

$M$4

In four different models on $M$5—site percolation on cubical and permutahedral lattices, the Poisson–Boolean model, and Gaussian random fields—the simulations strongly indicate that the zeros of $M$6 approximate the critical values for homological percolation (Bobrowski et al., 2019).

The common empirical observation is that, in all tested models and in dimensions $M$7, $M$8 has exactly $M$9 zeros,

$X_s \subset X_t \subset M \qquad (s<t).$0

and the giant-cycle thresholds satisfy

$X_s \subset X_t \subset M \qquad (s<t).$1

Because the limiting thresholds are not directly accessible in simulation, the paper uses the birth time of the first giant $X_s \subset X_t \subset M \qquad (s<t).$2-cycle as a finite-size proxy for $X_s \subset X_t \subset M \qquad (s<t).$3 (Bobrowski et al., 2019).

The discrete cubical model has

$X_s \subset X_t \subset M \qquad (s<t).$4

while the permutahedral model has

$X_s \subset X_t \subset M \qquad (s<t).$5

For the Poisson–Boolean model,

$X_s \subset X_t \subset M \qquad (s<t).$6

and for Gaussian random fields,

$X_s \subset X_t \subset M \qquad (s<t).$7

These formulas furnish analytically or numerically accessible threshold proxies in settings where direct critical-value analysis is difficult (Bobrowski et al., 2019).

The discrepancy

$X_s \subset X_t \subset M \qquad (s<t).$8

exhibits a sign pattern in dimensions $X_s \subset X_t \subset M \qquad (s<t).$9: $\{X_t\}$00 In symmetric even-dimensional models, the middle-dimensional threshold satisfies

$\{X_t\}$01

by symmetry and duality in the permutahedral case, and up to tiny numerical or discretization error in the zero-mean Gaussian case (Bobrowski et al., 2019).

Persistent homology supplies the operational language for distinguishing giant from small cycles. Giant cycles are persistent-homology features that are ambiently essential in the torus, while small cycles are born and die within the filtration but are trivial in the ambient space. In the simulations, giant classes are detected by checking which persistent-homology classes map to nontrivial classes in $\{X_t\}$02, and the first birth among them is taken as the finite-size percolation time (Bobrowski et al., 2019).

A related empirical observation appears in two-dimensional continuum percolation with disks. There, the union

$\{X_t\}$03

is analyzed via the alpha complex $\{X_t\}$04, with filtration parameter

$\{X_t\}$05

The longest-persisting $\{X_t\}$06 invariant is observed to be born at or near the percolation transition

$\{X_t\}$07

with the mean birth value moving closer to $\{X_t\}$08 as $\{X_t\}$09 increases (Speidel et al., 2018). This does not define giant torus-wrapping cycles, but it reinforces the broader idea that persistent topological signal localizes near percolative criticality.

5. Variants beyond the toroidal giant-cycle paradigm

Not all topological percolation models use the toroidal ambient-essential definition, but several closely related theories broaden the scope of homological percolation.

One codimension-one theory studies a random cubical set in $\{X_t\}$10 formed from open $\{X_t\}$11-faces. In this model, holes are bounded connected components of the complement, equivalently generators of $\{X_t\}$12 in finite windows, and a graph is constructed whose vertices are holes and whose edges encode shared boundary faces. The resulting hole percolation threshold satisfies

$\{X_t\}$13

implying

$\{X_t\}$14

If $\{X_t\}$15, then the number $\{X_t\}$16 of infinite hole clusters obeys

$\{X_t\}$17

and for $\{X_t\}$18,

$\{X_t\}$19

for any dual-lattice points $\{X_t\}$20 (Hiraoka et al., 2018). This is homological in a strict sense because the percolating objects are codimension-one homology generators, though it is not formulated through inclusion into a torus.

A different finite-graph variant defines homological triviality via coverings. For Bernoulli edge percolation on a sequence $\{X_t\}$21 covered by an infinite graph $\{X_t\}$22, the lower cycle erasure threshold is

$\{X_t\}$23

where

$\{X_t\}$24

The principal theorem states that

$\{X_t\}$25

while

$\{X_t\}$26

Here $\{X_t\}$27 is the homological distance, the size of the shortest nontrivial cycle on $\{X_t\}$28 (Woolls et al., 2020). This formulation identifies the growth rate of the shortest cover-detectable nontrivial cycle as the key parameter governing homological change.

Hyperbolic percolation yields another homological threshold proxy. For finite local quotients $\{X_t\}$29 of hyperbolic tilings, the induced first homology of the random open subgraph is

$\{X_t\}$30

and the rank difference function

$\{X_t\}$31

leads to the homological upper bound

$\{X_t\}$32

For $\{X_t\}$33, the explicit bound obtained is

$\{X_t\}$34

This approach does not define giant torus classes, but it uses the emergence of nontrivial first homology in finite approximants as a finite-volume signature of supercriticality (Delfosse et al., 2014).

A more recent continuum framework for random simplicial complexes proves a sharp phase transition for up-connectivity of $\{X_t\}$35-simplices through $\{X_t\}$36-simplices, with critical intensities

$\{X_t\}$37

and exponential decay below threshold together with a supercritical linear lower bound (Pabst, 18 Jun 2025). This is not homological percolation in the strict sense, since it concerns higher-order connectivity rather than homology groups, but it provides a rigorous higher-dimensional percolation framework adjacent to the homological theory.

6. Simplicial complexes, growth phenomena, and open problems

In growing simplicial complexes, homological percolation transitions have been studied through Betti numbers rather than ambient torus classes. A minimal model with growth and preferential attachment exhibits two successive transitions controlled by the first and second Betti numbers. The giant cluster size scales as

$\{X_t\}$38

the first Betti number as

$\{X_t\}$39

and the second Betti number as

$\{X_t\}$40

For $\{X_t\}$41, the first two critical values are reported as

$\{X_t\}$42

with essential singularities

$\{X_t\}$43

$\{X_t\}$44

$\{X_t\}$45

and

$\{X_t\}$46

These transitions are described as infinite order of the Berezinskii–Kosterlitz–Thouless type (Lee et al., 2020).

This line of work differs from the toroidal giant-cycle framework because the order parameters are the Betti numbers of the giant cluster rather than inclusion-induced images in ambient homology. It also exhibits delocalization: $\{X_t\}$47 continues to increase even after $\{X_t\}$48 appears, in contrast to the Kahle localization observed in static random simplicial complexes. The proposed mechanism is that delocalization arises when the merging rate of two-dimensional simplexes is less than the birth rate of isolated simplexes (Lee et al., 2020). This suggests that growth fundamentally alters topological criticality.

Across the subject, several open problems recur. In the continuum toroidal theory, the conjectured sharp picture is

$\{X_t\}$49

but only the case $\{X_t\}$50 is proved exactly (Bobrowski et al., 2020). In the Euler-characteristic program, the central relation

$\{X_t\}$51

is supported experimentally rather than theoretically, and the limiting behavior of

$\{X_t\}$52

remains unresolved except in symmetry-forced cases (Bobrowski et al., 2019). In finite toroidal lattice models, it remains open whether the threshold functions converge to true constants in all dimensions and homological degrees (Duncan et al., 2020). In covering-space graph models, the logarithmic regime suggests $\{X_t\}$53 in non-amenable settings, but this is supported by evidence rather than a full general theorem (Woolls et al., 2020).

Taken together, these results establish homological percolation as a family of topological phase transitions in which the macroscopic order parameter is no longer mere connectivity. Depending on the model, the percolating objects may be ambient-essential torus cycles, codimension-one holes, nontrivial cycles relative to a cover, or Betti-number-detectable structures in growing simplicial complexes. The unifying theme is the replacement of graph-theoretic largeness by largeness measured in homology.