Quasi-Independent Percolation

• Quasi-independent percolation is a phenomenon where multi-fluid percolation events on a honeycomb lattice, though originally coupled, become independent in the scaling limit.
• The approach leverages Boolean function analysis and Fourier–Walsh expansion to quantify influences and pivotality in percolation events.
• Numerical experiments and conjectures validate that as the lattice mesh refines, the joint probability of percolation events converges to the product of individual probabilities.

Quasi-independent percolation describes the emergent phenomenon whereby distinct percolation events, defined within a shared random environment on a lattice (notably, the honeycomb lattice with Potts coloring), become asymptotically independent as the mesh size of the lattice tends to zero. Despite inherent coupling at the level of configurations, critical connectivity events for multiple "fluids"—each defined by distinct subsets of cell colorings—exhibit vanishing correlation in the scaling limit. This phenomenon is rigorous within the context of three fluids on the honeycomb lattice, where the percolation events are constructed via a four-state Potts model at infinite temperature and analyzed through Boolean function and Fourier–Walsh techniques (Novikov, 2019).

1. Model Formulation: Multicolor Percolation and Event Definitions

Consider percolation on the regular honeycomb (hexagonal) lattice with mesh $\delta=1/n$. The domain is a fixed regular hexagon $P$ of side 1, centered at a cell $O$; $M_n$ denotes the set of all lattice cells entirely contained in $P$. Each cell is independently assigned a color from $\{0,1,2,3\}$ (four-state Potts coloring) with equal probability $1/4$. For each $k\in\{1,2,3\}$, "fluid $k$" occupies those cells whose color is either $0$ or $P$0. Fluid $P$1 percolates between two sets $P$2 if there exists a nearest-neighbor path through cells colored in $P$3 connecting $P$4 to $P$5.

The six sides of $P$6 are labeled $P$7 through $P$8 counter-clockwise. For $P$9, define the event

$O$0

The four-state Potts coloring can be equivalently recoded into three correlated $O$1 colorings, so that each $O$2 becomes a standard $O$3 percolation event in its respective coloring channel.

2. Asymptotic Quasi-Independence Theorem

The fundamental result is the quasi-independence theorem of Izyurov and Magazinov (Theorem 2.3), specifying that as the lattice becomes fine ($O$4, $O$5), the triple percolation events become mutually independent in the scaling limit. Explicitly,

$O$6

This establishes that, although the events are defined in terms of highly coupled structures (the same underlying coloring), the joint probability of simultaneous percolation converges to the product of the individual event probabilities as the mesh vanishes (Novikov, 2019).

3. Fourier–Walsh Analysis of Percolation Events

To analyze the dependencies among these percolation events, the approach reformulates each crossing event as a Boolean function $O$7. Each admits a unique Fourier–Walsh expansion:

$O$8

with Parseval’s identity yielding $O$9 and, for $M_n$0-valued $M_n$1, $M_n$2. The "influence" of coordinate $M_n$3 on $M_n$4 is $M_n$5, where $M_n$6 is pivotal for $M_n$7 on input $M_n$8 if flipping $M_n$9 flips the value $P$0.

The joint-triple probability for the percolation crossings is expressed as

$P$1

and $P$2. The difference of interest reduces to

$P$3

bounded in absolute value by the largest individual coordinate influence:

$P$4

4. Geometric Bounds and the Role of Pivotality

The key geometric observation states that, for a site $P$5 to be pivotal for $P$6, the logic of percolation implies dual open paths from that site to designated boundary segments. Thus,

$P$7

where $P$8 is the center cell and $P$9 is the boundary of $\{0,1,2,3\}$0. Kesten’s theorem for the honeycomb lattice at $\{0,1,2,3\}$1 ensures that the probability of an open path from the origin to macroscopic distance $\{0,1,2,3\}$2 vanishes as $\{0,1,2,3\}$3. This bound is pivotal to establishing the quasi-independence limit, as it leads all individual site influences to zero in the fine mesh regime (Novikov, 2019).

5. Conjectures and Empirical Investigations

The work sets forth several conjectures based on both analytic considerations and numerical experiments:

• Positive Triple-Correlation (Conjecture 2.2): For all $\{0,1,2,3\}$4,

$\{0,1,2,3\}$5

• Center-to-Boundary Correlation (Conjecture 2.3): For $\{0,1,2,3\}$6 being the event that fluid $\{0,1,2,3\}$7 percolates from the center $\{0,1,2,3\}$8 to the boundary,

$\{0,1,2,3\}$9

• Super-Multiplicativity in the Limit (Conjecture 2.4):

$1/4$0

• General Positive Triple-Correlation (Conjecture 2.5): For arbitrary pairs $1/4$1, the joint triple percolation probability is at least the product of the marginals.

Numerical experiments for $1/4$2 up to $1/4$3 (sample sizes up to $1/4$4) yield deviations of the normalized correlation ratio around $1/4$5, without systematic violation of positivity (Novikov, 2019).

Conjecture	Statement	Nature (Inequality/Limit)
2.2	$1/4$6	Inequality for all $1/4$7
2.3	$1/4$8	Inequality for all $1/4$9
2.4	$k\in\{1,2,3\}$0	Asymptotic strict lower bound
2.5	$k\in\{1,2,3\}$1 for general choices	Generalized inequality

6. Flexibility of Approach and Further Extensions

The Fourier–analytic proof combined with percolation arguments via Kesten’s theorem is robust under a variety of modifications to the lattice geometry and event specification. The underlying regular hexagon $k\in\{1,2,3\}$2 can be replaced with an arbitrary Jordan domain, and "designated sides" generalized to any choice of three disjoint boundary arcs. The analytic machinery adapts seamlessly, granting broad scope to the asymptotic quasi-independence phenomenon for multiclass percolation events.

Additional generalization to more than three fluids (up to four for the four-state Potts system), or to higher-state Potts models decomposed into Ising-like submodels, still yields asymptotic independence of their respective crossing events in the scaling limit. A plausible implication is that quasi-independence is a generic feature of multiclass percolation models based on well-mixed local randomness and hyperuniform domains (Novikov, 2019).

7. Outlook and Relations to Classical and Modern Percolation Theory

Quasi-independence of percolation events, as established for multiclass Potts systems, highlights the interplay between combinatorial structure, Fourier analytic methods, and geometric percolation estimates. The methodology draws on the machinery of O'Donnell’s "Analysis of Boolean Functions" for functional decomposition, and classical results of Grimmett and Kesten for the scaling analysis of crossing probabilities and influences. This line of research links the probabilistic decoupling of complex dependent systems in the fine mesh (scaling) limit to both statistical mechanics and theoretical computer science frameworks for event influence and critical phenomena.

The phenomenon affirms that, despite the strong microscopic correlations imposed by the shared underlying random field, macro-observable percolation events asymptotically exhibit factorization, reinforcing the foundational assumption of independence for large-scale events in critical percolation.