Inverse Polynomial Scaling

• Inverse polynomial scaling is a mathematical phenomenon where a system's critical threshold inversely relates polynomially to its primary parameter, reflecting controlled growth.
• It demonstrates that sets or measures with polynomial growth are structurally supported by coset nilprogressions and approximate groups, ensuring precise algebraic characteristics.
• In statistical physics, this concept quantifies percolation thresholds, enabling asymptotic inversion methods to estimate system sizes and critical probabilities.

Inverse polynomial scaling refers to a set of mathematical phenomena in which a structural property or threshold in a combinatorial, probabilistic, or dynamical system relates polynomially to a primary system parameter, and its inverse law characterizes the dual relationship. This concept appears across disparate areas, notably in additive combinatorics—where it underpins the structure of sets or measures of controlled polynomial growth—and in statistical physics models such as anisotropic bootstrap percolation, where critical phenomena emerge via inverse relationships between system size and percolation thresholds.

1. Inverse Theorems for Sets and Measures with Polynomial Growth

Let $A$ be a finite non-empty subset of a group $G$. The polynomial growth condition is the existence of $d > 0$ such that

$|A^n| \leq n^d |A| \quad \text{for all large } n,$

where $A^n$ denotes the $n$-fold product set. Its measure-theoretic analogue considers a symmetric probability measure $\mu$ on $G$ and requires

$$\|\mu^{*n}\|_{\ell^2}^{-2} \leq n^d \|\mu\|_{\ell^2}^{-2},$$

with $\mu^{*n}$ the $n$-fold convolution. The inverse theorem, as established by Tao building on the work of Breuillard–Green–Tao, asserts that sets or measures satisfying such growth bounds are essentially supported on coset nilprogressions of rank and step $O(d)$. This imposes precise algebraic and combinatorial structure (Tao, 2015).

2. Structural Descriptions: Coset Nilprogressions and Approximate Groups

The critical structural construct is the coset nilprogression. For $v_1, \ldots, v_r \in G$ and positive integers $N_1, \ldots, N_r$, the nilprogression $P(v_1,\ldots,v_r; N_1,\ldots,N_r)$ comprises all group-words with each $v_i^{\pm1}$ appearing at most $N_i$ times, generating a nilpotent subgroup of class $s$. If $H \triangleleft G$ is a finite normal subgroup and $P$ a nilprogression in the quotient $N_G(H)/H$, the pullback $$HP = \{ h \tilde{p} : h \in H,\, \tilde{p} \in \text{preimage of } P \}$$ forms a coset nilprogression.

Approximate groups play a central role: $A$ is a $K$-approximate group if $1 \in A$, $A = A^{-1}$, and $A^2 \subset X A$ for some set $X$ of size $|X| \le K$. A is "controlled" by a coset nilprogression $HP$ if $HP \subset A^{O(1)}$, $|HP| \ll |A|$, and $A \subset X HP$ for some finite $X$.

3. Proof Sketch and Growth Estimation Method

Beginning with the polynomial growth hypothesis, a Balog–Szemerédi–Gowers argument produces an $O_d(1)$–approximate group comprising a large subset of $A^{O(1)}$. The Breuillard–Green–Tao inverse theorem then yields a coset nilprogression $HP$ of rank and step $O(d)$ containing (up to bounded index) this approximate group. Further combinatorial arguments show that $A \subset X HP^{O(1/n)} \subset X HP$, and for all $m \geq n$,

$|A^m| = \Theta_d(m^d |A|).$

In measure terms, $$\|\mu^{*m}\|_{\ell^2}^{-2} = \Theta_d(m^d \|\mu\|_{\ell^2}^{-2})$$. The entropy formulation takes the form

$\log|A^m| = \log|A| + f(\log m) + O_d(1)$

with $f$ a piecewise-linear map of at most $O(d)$ segments, slopes in $\{0, 1, ..., O(d)\}$.

4. Abelian Case and Littlewood-Offord Phenomena

When $G$ is abelian, a coset nilprogression is a coset of a generalized arithmetic progression, reducing inverse polynomial scaling to well-known additive combinatorics structures. The Nguyen–Vu inverse Littlewood–Offord theorem asserts that, under suitable anti-concentration bounds for sums of random signs of group elements, all but a small subset of summands must lie in a low-rank coset progression. A symmetrized non-abelian analogue generalizes this to settings where the group is non-commutative and the summands have large order (Tao, 2015).

5. Inverse Scaling in Anisotropic Bootstrap Percolation

In statistical mechanics, the (1,2)-rule anisotropic bootstrap percolation serves as a paradigmatic example of inverse polynomial scaling (Enter, 2014). The model involves a square region of size $N=L^2$ where each site is initially occupied with probability $p$. The system percolates if eventually all sites become occupied, governed by the update rule that an empty site becomes occupied if at least three of its six specific anisotropic neighbors are occupied.

A key estimate identifies the probability $P(p)$ that a specific rectangle can nucleate the percolating state:

$$P(p) \asymp \exp\left[-\frac{1}{6p}(\ln \tfrac{1}{p})^2 + \frac{1}{3}\ln \tfrac{1}{p} + O(1)\right].$$

The critical percolation probability $p_c(N)$, defined by $N P(p_c(N)) \approx 1$ as $N \to \infty$, inherits a sharp asymptotic:

$$p_c(N) = \frac{(\ln\ln N)^2}{6\,\ln N}\left[1 - \frac{\ln\ln N}{3\,\ln N} + o\left(\frac{\ln\ln N}{\ln N}\right)\right].$$

The inverse scaling relationship is observed in the critical size $N_c(p)$ needed for a fixed $p \ll 1$:

$$N_c(p) = \exp\left[\frac{1}{6p}(\ln \frac{1}{p})^2 - \frac{1}{3}\ln\frac{1}{p} + O(1)\right].$$

6. General Inversion Methodology and Broader Applicability

The inversion of asymptotic expansion is a robust general tool in probability theory and combinatorics: whenever the probability of a critical event satisfies

$P(p) = \exp\{-A g(p) + B h(p) + O(1)\}$

with $g(p) \gg h(p) \gg 1$ as $p \to 0$, critical thresholds $p_c(N)$ and $N_c(p)$ are obtained by inverting relations between $P(p)$ and the primary parameter. This method generalizes beyond polynomial scaling to cover stretched-exponential and other non-polynomial regimes in higher-dimensional percolation or models reducible to effective lower-dimensional slices (Enter, 2014).

The "inverse polynomial scaling" principle thus classifies the structure and threshold behavior in diverse settings. In additive and non-abelian combinatorics, it identifies coset nilprogressions as the unique structural support for such growth, while in percolation theory, it quantifies critical phenomena via asymptotic inversion, with exponents determined by the system's geometric or group-theoretic structure.