🔬 condensed matter

Complete local expansion of the availability function in random sequential adsorption of aligned squares at low density: Termination at fourth order

This paper derives and confirms through numerical simulations that the low-coverage expansion of the availability function for random sequential adsorption of aligned squares terminates exactly at the fourth order, as no more than four previously deposited squares can simultaneously overlap with a trial exclusion region.

Original authors: F. Tolea, M. Tolea

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: F. Tolea, M. Tolea

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to park identical square cars on a giant, empty, flat field. You have a strict rule: you can't park a car where it would overlap with any car already there. You park them one by one, in random spots, until you can't fit any more cars without crashing. This process is called Random Sequential Adsorption (RSA).

The paper by F. Tolea and M. Tolea is a mathematical investigation into the early stages of this parking game. Specifically, they want to know: "As we fill up the field, how much empty space is left for the next car?"

They call this remaining space the "availability function."

The Problem: It's Not Just Simple Subtraction

At first, you might think the math is easy. If one car blocks a certain area, then two cars should block twice that area, right?

No. It's more like a game of Tetris with invisible bubbles.

The Bubble: When you park a car, it doesn't just block the space the car occupies. It also creates a "forbidden zone" (an exclusion region) around it where no other car can park. For a square car, this forbidden zone is a larger square.
The Overlap: If you park two cars close together, their forbidden zones might overlap. If you just added the size of both zones, you would be double-counting that overlapping middle section. You have to subtract the overlap to get the true blocked area.
The Chain Reaction: But wait! If you have three cars, their zones might all overlap in one tiny spot. If you subtracted the overlaps of pairs, you might have accidentally removed that triple-overlap spot too many times. You have to add it back.

The authors use a method called Inclusion-Exclusion. Think of it like a group of friends trying to count how many people are in a room, but they keep miscounting because they are standing in groups:

Count everyone individually (First Order).
Realize you counted the people standing in pairs twice, so subtract the pairs (Second Order).
Realize you subtracted the people standing in groups of three too many times, so add them back (Third Order).
Realize you messed up the groups of four, so subtract them again (Fourth Order).

The Big Discovery: The "Magic Number" Four

The most exciting part of this paper is that for aligned squares (cars parked parallel to the axes), this counting game stops exactly at four.

The authors prove a geometric fact: It is physically impossible to arrange more than four non-overlapping parked cars such that their forbidden zones all overlap at the same single point.

You can have 1, 2, or 3 cars whose "no-parking bubbles" overlap.
You can even have 4 cars whose bubbles overlap.
But you cannot have 5 cars doing this.

Because of this geometric limit, the math for the "available space" doesn't need an infinite list of corrections. It stops after the fourth correction. The authors calculated the exact numbers for these first four steps.

The Result: A Perfect Formula

They derived a specific formula (a polynomial) that tells you exactly how much space is left for a new car, based on how much of the field is already covered ( $q$ ).

The formula looks like this:
$\text{Available Space} = 1 - 4q + 3.5q^2 - 0.88q^3 + 0.059q^4$

The $1$: The whole empty field.
The $-4q$ : The space taken by single cars.
The $+3.5q^2$ : The correction for when two cars' bubbles overlap.
The $-0.88q^3$ : The correction for when three bubbles overlap.
The $+0.059q^4$ : The final correction for when four bubbles overlap.

Why This Matters (According to the Paper)

The authors explain that this formula is exact for the "local" picture. It perfectly describes the geometry of how cars block each other in small groups.

However, they also point out a limitation. This formula predicts that you can't fit any more cars once the field is about 34.5% full. But in reality, if you keep playing the game, you can actually fill the field up to about 56% full before it's truly jammed.

Why the difference?
The formula only looks at local overlaps (neighbors bumping into neighbors). The real game involves global organization. As the field gets fuller, the cars arrange themselves in complex, long-range patterns that this simple local formula can't see. The "jamming" happens because of these long-range connections, not just because of immediate neighbors.

Summary

In simple terms, the authors solved a complex geometry puzzle:

They figured out exactly how much space is lost when you park square cars randomly.
They proved that for squares, you only need to count up to groups of four cars to get the perfect local answer.
They created a precise formula for this, which acts like a "perfect map" for the early stages of parking, but eventually runs out of steam because the real-world parking lot gets too crowded and complex for simple local rules to explain.

They also double-checked their math with computer simulations, and the numbers matched perfectly.

Technical Summary: Complete Local Expansion of the Availability Function in Random Sequential Adsorption of Aligned Squares

Problem Statement
The paper addresses the Random Sequential Adsorption (RSA) of aligned squares on a two-dimensional surface. Specifically, it focuses on the low-coverage regime to derive an exact, closed-form expansion of the availability function, $\alpha(q)$ . This function represents the fraction of positions on the plane where an additional square can be inserted without overlapping previously deposited squares, given a fractional surface coverage $q = \rho L^2$ (where $\rho$ is the number density and $L$ is the side length). While the one-dimensional RSA problem (Rényi's parking problem) is exactly solvable, higher-dimensional systems exhibit complex geometric correlations that make analytical solutions difficult. Previous work has provided numerical estimates for coefficients in the low-density expansion of $\alpha(q)$ for various shapes, but explicit closed-form expressions derived directly from geometric overlap configurations have been lacking beyond the leading order.

Methodology
The authors employ a systematic inclusion–exclusion principle applied to the geometry of exclusion zones.

Geometric Framework: Each deposited square of side $L$ creates an exclusion zone of side $2L$ (area $4L^2$ ) where the center (or reference corner) of a new square cannot be placed. The availability function is defined as $1$ minus the fraction of the plane covered by the union of these exclusion zones.
Cluster Expansion Logic: The expansion of $\alpha(q)$ $α (q)$ is constructed by correcting overcounting errors inherent in summing independent exclusion zones:
- First Order: Subtracts the area of individual exclusion zones.
- Second Order: Adds back the area where two exclusion zones overlap (correcting double subtraction).
- Third Order: Subtracts the area where three exclusion zones overlap (correcting over-addition).
- Fourth Order: Adds back the area where four exclusion zones overlap.
Termination Condition: A central geometric observation drives the methodology: for aligned squares, no more than four mutually non-overlapping deposited squares can simultaneously overlap the exclusion region of a trial square. Consequently, the inclusion–exclusion hierarchy for local overlaps terminates exactly at the fourth order.
Calculation: The coefficients are derived by explicitly enumerating and integrating over all geometrically admissible configurations of $n$ -body overlaps ( $n=1$ to $4$). The integrals are factorized into one-dimensional components due to the axis-aligned nature of the squares. Combinatorial symmetry factors ( $1/n!$ ) are applied to account for ordered vs. unordered clusters.
Validation: The analytical coefficients are confirmed via numerical simulations on a discrete lattice, where the coefficients correspond to normalized counts of compatible blocker clusters.

Key Contributions and Results
The primary contribution is the derivation of the complete, exact low-density polynomial for the availability function of aligned squares up to its maximal geometrically allowed order. The resulting expansion is:

$\alpha(q) = 1 - 4q + \frac{7}{2}q^2 - \frac{8}{9}q^3 + \frac{17}{288}q^4$

The specific coefficients are:

$a_1 = -4$ : Represents the single-square exclusion area ( $4L^2$ ).
$a_2 = 7/2$ : The first collective correction, representing the average overlap of two exclusion zones.
$a_3 = -8/9$ : The correction for triple overlaps.
$a_4 = 17/288$ : The final correction for quadruple overlaps.

The paper demonstrates that this polynomial is not an arbitrary truncation but the complete local overlap hierarchy. The authors calculate the root of this quartic polynomial, finding a saturation point of $q^* \approx 0.3455$ .

Significance and Claims
The paper claims significance in the following areas:

Exactness of Local Expansion: It provides the first explicit, closed-form expressions for the availability function coefficients up to the fourth order for aligned squares, derived directly from geometric overlap kernels rather than numerical fitting.
Definition of Local vs. Non-Local Physics: The work establishes a clear boundary between local geometric constraints and global jamming phenomena. The quartic polynomial accurately describes the system up to the point where local correlations are exhausted ( $q \approx 0.3455$ ). However, the actual RSA jamming limit for aligned squares is known to be approximately $0.5620$.
Limitations of Local Models: The discrepancy between the quartic root ($0.3455$) and the true jamming density ($0.5620$) demonstrates that the approach to jamming is controlled by nonlocal collective correlations generated by the irreversible deposition history, which cannot be captured by any finite-order local expansion.
Analytical Benchmark: The derived coefficients serve as rigorous analytical benchmarks for testing numerical RSA simulations and approximate theories of irreversible adsorption.

The authors emphasize that while the local expansion is exact up to fourth order, the failure of this polynomial to predict the true jamming limit highlights the intrinsic complexity of higher-dimensional RSA systems, where global history-dependent correlations dominate the saturation process.