Critical Erdős-Rényi Graph

• Critical Erdős-Rényi graphs are random graphs with p ≈ 1/n, marking the phase transition from sublinear to macroscopic component sizes.
• The analysis employs scaling laws where the largest component scales as n^(2/3) and the spanning tree diameter scales as n^(1/3), using methods like the Matrix-Tree Theorem.
• Advanced techniques such as cluster expansions and coupling arguments reveal universal behaviors and continuous interpolation between uniform and minimal spanning trees.

A critical Erdős-Rényi (ER) graph refers to the random graph $G(n,p)$ in the regime where the edge probability $p$ is tuned so that the global structure of the graph undergoes a phase transition—specifically, at or near the "critical window" where $p \sim 1/n$. This regime marks the threshold at which the largest component transitions from being sublinear to macroscopic in size. Critical ER graphs play a central role in probabilistic combinatorics, random graph theory, and the study of scaling limits of statistical physics models due to universal behaviors in component sizes, cycle structures, and associated observables such as the uniform or random spanning tree.

1. Definition and Scaling Regime

In the classical Erdős-Rényi model $G(n,p)$, a graph of $n$ vertices is formed by including each possible edge independently with probability $p \in [0,1]$. The critical regime is defined by $p = 1/n + \lambda n^{-4/3}$ for $\lambda \in \mathbb{R}$—the so-called "critical window." In this regime, several macroscopic features of the random graph, such as the size and diameter of the largest component, exhibit nontrivial scaling and limiting behaviors. At $p = c/n$ with $c<1$, all components are typically trees or unicyclic and sublinear in size; for $c>1$, a giant component of size $\Theta(n)$ emerges. The critical regime is characterized by the largest components having size and diameter scaling as $n^{2/3}$ and $n^{1/3}$, respectively.

2. Spanning Trees and Critical Behavior

A key observable in critical ER graphs is the structure and scaling of spanning trees, especially in the study of random spanning trees in random environment (RSTRE) and uniform spanning trees (UST). In the critical window, with high probability, the largest component $C_1$ is of size $\Theta(n^{2/3})$, and the diameter of the UST (formed, for example, on $C_1$ equipped with i.i.d. positive conductances) is typically $n^{1/3+o(1)}$, as established in scaling limit results for random graphs near criticality (Makowiec, 10 Jul 2025, Makowiec et al., 2023, Makowiec et al., 22 Oct 2024). The minimal spanning tree (MST) constructed from i.i.d. edge weights on $K_n$ also demonstrates $n^{1/3}$ scaling for its diameter in this regime, reflecting the fractal geometry at criticality.

3. Gibbs Measures, UST, and RSTRE Interpolation

Random spanning trees on a critical ER graph can be sampled according to various probability laws:

• Uniform Spanning Tree (UST): Each spanning tree has equal probability. On critical $G(n,p)$, the UST diameter is of order $n^{1/3}$ due to the geometry of the underlying critical component.
• Random spanning tree in random environment (RSTRE): The tree is sampled from the Gibbs measure:

$$P_\beta[T \mid \{w_e\}] = \frac{e^{-\beta H(T)}}{Z(\beta)}, \quad H(T) = \sum_{e\in T} w_e\,,$$

with $w_e$ i.i.d. edge weights and inverse temperature $\beta$. This interpolates between the UST at $\beta=0$ and the MST as $\beta\to\infty$ (Makowiec et al., 22 Oct 2024, Makowiec, 10 Jul 2025).

• In the critical ER regime (and more generally in the critical window of any sparse random graph), edge inclusion probabilities and global geometric functionals interpolate smoothly between these extremes as $\beta$ varies (Makowiec, 10 Jul 2025).

4. Key Observables: Diameter, Local and Global Metrics

The most studied observables on critical ER graphs in the context of random spanning trees are:

• Diameter: For UST or RSTRE at criticality (i.e., low disorder), the tree diameter scales as $n^{1/3+o(1)}$ (Makowiec et al., 22 Oct 2024, Makowiec, 10 Jul 2025, Makowiec et al., 2023). For larger $\beta$, as the model approaches the MST, the scaling remains $n^{1/3+o(1)}$.
• Component sizes: The largest component is of order $n^{2/3}$ vertices, with smaller components rapidly decaying in size.
• Edge-inclusion probabilities: At $\beta=0$, these are governed by effective resistances; at high $\beta$, edges in every MST have inclusion probability close to $1$, and other edges inclusion probability close to $0$ (Makowiec, 10 Jul 2025).

The summary table presents the dominant scaling regimes for the UST/random spanning tree diameter on the largest critical ER component:

Regime	Edge Probability $p$	UST/RSTRE Diameter
Subcritical	$p=c/n$, $c<1$	$O(\log n)$
Critical	$p = 1/n + O(n^{-4/3})$	$n^{1/3+o(1)}$
Supercritical	$p=c/n$, $c>1$	$O(\log n)$ (giant)

5. Analytical Techniques and Scaling Limits

The study of critical ER graphs leverages several advanced probabilistic and combinatorial tools:

• Matrix-Tree Theorem and Effective Resistance: Inclusion probabilities and degree moments in the UST rely on Laplacian pseudoinverses and electrical network interpretations (Sanmartín et al., 20 Sep 2024).
• Cluster Expansion and Large Deviations: RSTRE measures in the critical window require analysis of partition functions and interpolation between UST and MST regimes (Makowiec et al., 22 Oct 2024, Makowiec, 10 Jul 2025).
• Couplings and Stochastic Domination: Results on scaling of diameter and volume growth use couplings between the random spanning tree on the critical component and suitably chosen reference models (e.g., random 3-regular kernels, loop-erased random walks) (Makowiec et al., 22 Oct 2024).
• Critical Percolation Analogies: The local limit and scaling window geometry are tightly connected to continuum random tree (CRT) scaling limits and Aldous’ Brownian CRT.

6. Open Questions and Research Directions

Current research focuses on refining the scaling window exponents, determining limit laws for metric measure spaces arising from random spanning trees on critical ER graphs, and extending universality to other models:

• Scaling Limits: It is conjectured, and partially proven, that the UST rescaled on the largest critical ER component converges—in the Gromov–Hausdorff sense—to a universal "critical spanning tree" metric space, interpolating between the Brownian CRT and the MST continuum structure (Makowiec et al., 2023, Makowiec, 10 Jul 2025).
• Disorder Effects: The role of disorder in edge weights and the transition across $\beta$ in the RSTRE model is an active area, with open problems regarding the nature of the critical exponents $\gamma(\alpha)$ for diameter scaling in intermediate disorder regimes (Makowiec et al., 22 Oct 2024).
• Extensions: Generalization to configuration models, random regular graphs at criticality, and extension of scaling results for other observables (height, number of leaves, etc.) remain open (Makowiec, 10 Jul 2025).

7. Connections to Broader Random Graph Theory

Critical ER graphs, and the scaling behavior of random spanning trees therein, are prototypical in the sense that they illustrate universal phenomena at phase transitions in random structures. The techniques developed for their analysis—spectral methods, Laplacian-based concentration inequalities, cluster expansions—have found application in the study of random processes on networks, percolation, epidemic models, and high-dimensional combinatorics. The interplay between random environment models and critical connectivity properties exemplifies the interface of probability, statistical physics, and combinatorial optimization (Makowiec et al., 2023, Makowiec et al., 22 Oct 2024, Makowiec, 10 Jul 2025).