Local Character Expansion Results

• Local character expansion results are a set of methods that quantify asymptotic behavior and structural properties in graphs and group representations using local optimization and spectral data.
• They refine expansion bounds by incorporating local optimality constraints, effectively reducing the number of unlikely configurations in random regular graphs.
• The improved asymptotic bounds yield better constants, especially as the regular degree increases, enhancing our understanding of expander graph properties.

Local character expansion results comprise a family of techniques for quantifying the asymptotic behavior and structural properties of objects in mathematics, ranging from expanders in graph theory to characters of group representations, often by leveraging local optimization, local spectral data, or combinatorial conditions. These expansions provide sharper bounds, structural formulas, and connections to combinatorial or harmonic analytic invariants. The methodology and impact of local character expansions varies by context, but common unifying themes include exploiting local optimality constraints, local spectral information, and locally-defined invariants to extract global consequences.

1. Classical Union Bound and Expansion in Random Regular Graphs

The union-bound technique for showing that random $\Delta$-regular graphs are expanders was pioneered by Bollobás, calculating the probability that a vertex set $S$ has small edge expansion and summing over all possible subsets. For $G$ a random $\Delta$-regular graph, fixing $|S| = u \leq n/2$ and edge cut size $c=|\partial S|$, the configuration model gives the probability

$$P(\vec{s},\vec{s}') = \frac{|S|!}{s_0!s_1!\cdots s_\Delta!} \prod_{i=0}^{\Delta} \binom{\Delta}{i}^{s_i} \cdot \frac{|V\setminus S|!}{s'_0!s'_1!\cdots s'_\Delta!} \prod_{i=0}^{\Delta} \binom{\Delta}{i}^{s'_i} \cdot c! \cdot \frac{(\Delta|S|-c)!! (\Delta|V\setminus S|-c)!!}{(\Delta n)!!}$$

for specified out-degree vectors $\vec{s}$ and $\vec{s}'$.

Applying the union bound over $u$ and $c$, the sum

$$\sum_{u \leq n/2} \sum_{c \leq \phi u} {n \choose u} P(\vec{s},\vec{s}')$$

aims to be $o(1)$ as $n \to \infty$, ensuring high-probability expansion lower bounds.

However, this approach can be loose because it counts all sets $S$, not accounting for local improvements.

2. Local Optimality Constraints and Tightening the Expansion Bound

The central advance is the introduction of local optimality within the union-bound framework. A set $S$ is locally optimal if no single vertex swap between $S$ and $V\setminus S$ further decreases the cut size. For out-degree maxima $d = \max \{i : s_i > 0 \}$ and $d' = \max \{i : s'_i > 0 \}$, local optimality implies

$d + d' \leq \Delta+1$

and, with mild perturbations,

$d + d' \leq \Delta.$

This additional structure imposes strong constraints on viable "bad" sets, significantly reducing the effective number of configurations. Moreover, probability calculations under this constraint show that sets with small expansion and local optimality have substantially diminished likelihoods.

Probabilistic estimates are modified: typical out-degree distributions align with Binomial$(\Delta,p)$ for some $p$, but local optimality further forces concentration, shrinking the parameter regime where bad sets occur.

3. Improved Asymptotic Expansion Bound

Bollobás’ original expansion bound for random $\Delta$-regular graphs is

$\frac{\Delta}{2} - \Theta(\sqrt{\Delta}),$

where the implied constant in the $\Theta(\sqrt{\Delta})$ is $\sqrt{\ln 2} \approx 0.83$.

By imposing local optimality, the asymptotic analysis is sharpened. Section 5 establishes that, using recurrence relations for the out-degree vector (via formulas like

$$\sum_{i=0}^d \beta\gamma^i\binom{\Delta}{i} = 1, \qquad \sum_{i=1}^d \beta i \gamma^i \binom{\Delta}{i} = \frac{1-\eta}{2}\Delta,$$

comparing distribution tails), one finds that there exists $\alpha < 2\sqrt{\ln 2}$ such that, with high probability,

$$i(G) \geq \left(1 - \frac{\alpha}{\sqrt{\Delta}} \right) \frac{\Delta}{2}.$$

While the overall form remains $\frac{\Delta}{2} - \Theta(\sqrt{\Delta})$, $\alpha$ is strictly less than the prior constant, making the expansion bound strictly stronger for large $\Delta$.

4. Scaling Behavior and Large Degree Analysis

Crucially, the benefit of the local improvement method grows with $\Delta$. For moderate $\Delta$, the reduction in bad configurations is significant; for large $\Delta$, the probability penalty from the local optimality constraint becomes much more pronounced. Numerical results and theoretical estimates confirm that as $\Delta$ increases, the difference between this improved bound and Bollobás’ original bound grows.

In practical terms, the standard union bound used for expansion is typically wasteful because it does not restrict to locally optimal sets. After incorporating local optimality, the summation over configurations becomes much tighter and the estimation of expansion is more refined.

5. Methodological Implications and Extensions

The local improvement/expansion technique prompts several methodological extensions:

• By focusing on locally optimal configurations, one can prune unlikely events in other random graph or combinatorial contexts, potentially enabling sharper bounds for other graph parameters.
• Structural restrictions imposed by local optimality may help in analyzing related properties such as robustness to vertex perturbations, subgraph expansion, and percolation.

A plausible implication is that local character expansion principles could be systematically extended to optimize numerous random discrete models, provided locally optimal structures can be efficiently characterized.

6. Summary of Main Technical Advances

The improved local character expansion analysis for random regular graphs, as introduced, involves:

• Precise probability computations for configuration vectors encoding out-degree distributions
• Refined union bounds leveraging local optimality (no reduction of cut size under any single vertex exchange)
• A combinatorial lemma (Lemma 3.4) formalizing the constraint on maximal out-degrees
• Tracking of binomial-type probability mass functions under additional concentration imposed by local improvement
• Asymptotic improvements in the leading constant for the expansion lower bound
• Increased effect as the regular degree $\Delta$ becomes large

The result is a bound of the form

$$i(G) \geq \left(1-\frac{\alpha}{\sqrt{\Delta}}\right) \frac{\Delta}{2}$$

with $\alpha$ strictly less than previous bounds, applying not only to special cases but to the general random $\Delta$-regular graph model.

7. Significance for Expander Graph Theory

These local character expansion results contribute to fundamental understanding of expander graphs by showing that the structural rarity of locally optimal bad sets underpins expansion properties more strongly than previously realized. The approach has the potential to refine union-bound-based analyses throughout random discrete structures, and to inform the paper of expansion in high-degree and large-scale graph models.