Random Regular Graphs

• Random regular graphs are graphs where every vertex has the same fixed degree, leading to sparse, homogeneous, and locally tree-like structures.
• They provide a canonical framework for asymptotic enumeration, sampling techniques, and transferring properties from Erdős–Rényi models to controlled regular settings.
• These graphs are pivotal in analyzing network dynamics, quantum transport, and optimization in disordered systems, with applications ranging from spin-glass models to percolation processes.

Searching arXiv for recent and foundational papers on random regular graphs to ground the article. A random regular graph is a random graph in which every vertex has the same degree. In the standard uniform model $G(n,d)$, one chooses uniformly from all simple $d$-regular graphs on the labeled vertex set $[n]=\{1,\dots,n\}$; equivalently, one may use the configuration model, in which each vertex receives $d$ half-edges, a uniform perfect matching of the $nd$ half-edges is sampled, and conditioning on simplicity yields the uniform law on simple $d$-regular graphs (Gao et al., 2015, Fernley, 13 Mar 2025). Because typical instances are sparse, homogeneous, and locally tree-like, random regular graphs form a canonical setting for asymptotic enumeration, exact and approximate sampling, extremal subgraph questions, interacting particle systems, and disordered quantum models (Tomasi et al., 2019, Concetti, 2017).

1. Ensemble, configuration model, and local weak limit

For a simple graph $G=(V,E)$ on $[n]$, $d$-regularity means

$\deg_G(v)=d \qquad \text{for all } v\in V.$

The state space $d$0 consists of all simple $d$1-regular graphs on $d$2, and the uniform random $d$3-regular graph is the uniform distribution on $d$4 (Gao et al., 2015). A broader notation $d$5 is used for a prescribed degree sequence $d$6; the regular case is $d$7 (Gao et al., 2019).

The configuration model realizes the same object before conditioning on simplicity. In the formulation used for the random $d$8-regular configuration model, one partitions $d$9 into $[n]=\{1,\dots,n\}$0 blocks of size $[n]=\{1,\dots,n\}$1, chooses a uniform perfect matching of the $[n]=\{1,\dots,n\}$2 half-edges, and interprets each matched pair as an edge between the corresponding vertices; the result is a random $[n]=\{1,\dots,n\}$3-regular multigraph on $[n]=\{1,\dots,n\}$4 (Fernley, 13 Mar 2025). This multigraph is simple with asymptotically positive probability, so high-probability statements proved in the configuration model transfer to the uniform simple model with positive limiting probability (Fernley, 13 Mar 2025).

A central structural fact is local convergence to the infinite $[n]=\{1,\dots,n\}$5-regular tree $[n]=\{1,\dots,n\}$6. For a uniformly chosen vertex $[n]=\{1,\dots,n\}$7, the radius-$[n]=\{1,\dots,n\}$8 ball $[n]=\{1,\dots,n\}$9 converges in distribution to the radius-$d$0 ball around the root in $d$1; the same remains valid for $d$2 with small fixed $d$3 (Fernley, 13 Mar 2025). This local weak limit is the basis for transferring tree-based recursion, cavity, and branching arguments to large finite random regular graphs.

2. Local geometry, expansion, and effective tree structure

Random regular graphs are locally tree-like in a stronger asymptotic sense than local weak convergence alone. For asymptotically almost every $d$4, all but $d$5 vertices have neighborhoods of radius $d$6 that are trees, and the diameter is $d$7 (Su, 2015). The same paper uses edge-expansion estimates of the form

$d$8

for asymptotically almost every $d$9-regular graph, together with a positive isoperimetric constant, to control spreading processes and metastability (Su, 2015).

For random regular graphs of connectivity $nd$0, the number of vertices at graph distance $nd$1 from a given vertex grows as

$nd$2

and the typical distance between two random vertices scales as

$nd$3

At distances $nd$4 much smaller than the diameter, such graphs look like regular trees with branching number $nd$5, while loops are typically long (Tomasi et al., 2019). This makes random regular graphs a precise, controllable setting for “infinite-dimensional” geometry without the boundary effects of the Bethe lattice: they are finite and homogeneous, and all vertices are equivalent (Tomasi et al., 2019).

In dense degree regimes, random regular graphs also fall into robust-expander classes. In particular, graphs on $nd$6 vertices with minimum degree at least $nd$7 are robust expanders, so a random $nd$8-regular graph with $nd$9 satisfies this condition with high probability (Granet et al., 2023). This robust-expansion viewpoint underlies asymptotic results on perfect matchings and Poisson overlap statistics.

3. Sampling, counting, and coupling with binomial random graphs

Algorithmic generation of uniform or nearly uniform random regular graphs has been a longstanding theme. Bayati, Kim, and Saberi gave a nearly-linear-time sequential algorithm for graphs with a prescribed degree sequence when

$d$0

producing almost uniform samples in time $d$1; for regular graphs, they showed that if

$d$2

their algorithm generates an asymptotically uniform $d$3-regular graph, improving the previous $d$4 range due to Kim and Vu [0702124].

A different line, based on switchings in the configuration model, yielded exact and approximate uniform samplers with explicit complexity guarantees. The sampler REG generates $d$5-regular graphs uniformly at random in expected time

$d$6

for $d$7, while the approximate sampler REG$d$8 runs in expected time

$d$9

and outputs a $G=(V,E)$0-regular graph whose distribution differs from uniform by $G=(V,E)$1 in total variation distance, again for $G=(V,E)$2 (Gao et al., 2015). The exact algorithm preserves uniformity by equalizing expected visit counts across strata of pairings, and the approximate version drops the rejection machinery while retaining asymptotic correctness (Gao et al., 2015).

Coupling random regular graphs to binomial random graphs provides a complementary transfer principle. Kim and Vu conjectured that if $G=(V,E)$3, then $G=(V,E)$4 should be asymptotically sandwichable between two Erdős–Rényi graphs with edge probabilities $G=(V,E)$5. Gao, Isaev, and McKay proved the conjecture with perfect containment on both sides for all

$G=(V,E)$6

and for smaller $G=(V,E)$7 obtained a weaker sandwich in which the upper parameter is approximately $G=(V,E)$8 (Gao et al., 2019). In the near-regular setting this becomes a general tool for transferring Hamiltonicity, universality, chromatic number, small subgraph counts, diameter, independence number, and edge-percolation thresholds from $G=(V,E)$9 to $[n]$0 (Gao et al., 2019).

4. Factors, perfect matchings, and sprinkling

Uniform perfect matchings in dense regular robust expanders exhibit strong edge-uniformity. If $[n]$1 is a $[n]$2-regular robust $[n]$3-expander on $[n]$4 vertices, $[n]$5 is a uniformly chosen perfect matching of $[n]$6, and $[n]$7, then

$[n]$8

Moreover, for any fixed matching $[n]$9, the random variable $d$0 is asymptotically Poisson with parameter $d$1, and for a spanning $d$2-regular subgraph $d$3 with $d$4, $d$5 is asymptotically Poisson with parameter $d$6 (Granet et al., 2023). Since random $d$7-regular graphs with $d$8 are robust expanders with high probability, these asymptotics describe the typical dense random-regular case as well (Granet et al., 2023).

A more structural decomposition question is whether the edge-disjoint union of two random regular graphs behaves like a random regular graph of the combined degree. The conjectural statement is that the distribution of the edge-disjoint union of two random regular graphs on the same vertex set is asymptotically equivalent to a random regular graph of degree $d$9, provided the combined degree grows with $\deg_G(v)=d \qquad \text{for all } v\in V.$0. This has been verified in several regimes: when $\deg_G(v)=d \qquad \text{for all } v\in V.$1, $\deg_G(v)=d \qquad \text{for all } v\in V.$2, and $\deg_G(v)=d \qquad \text{for all } v\in V.$3; when

$\deg_G(v)=d \qquad \text{for all } v\in V.$4

and in a dense–sparse regime specified by

$\deg_G(v)=d \qquad \text{for all } v\in V.$5

The same work also proves an asymptotic formula for the expected number of spanning regular subgraphs in a random regular graph (Isaev et al., 2023).

The reduction underlying these sprinkling results identifies concentration of

$\deg_G(v)=d \qquad \text{for all } v\in V.$6

—the number of spanning $\deg_G(v)=d \qquad \text{for all } v\in V.$7-regular subgraphs of a random $\deg_G(v)=d \qquad \text{for all } v\in V.$8-regular graph—as equivalent to the desired coupling between the union model and $\deg_G(v)=d \qquad \text{for all } v\in V.$9 (Isaev et al., 2023). This is a distinctly regular-graph analogue of the classical sprinkling method for $d$00. A plausible implication is that many multi-round exposure arguments in the Erdős–Rényi setting should have regular-graph counterparts whenever factor-count concentration is available.

5. Propagation processes, cover times, and finite-graph phase transitions

Bootstrap percolation on random regular graphs leads to an extremal optimization problem: the size of a minimal contagious set. For a $d$01-regular random graph with threshold $d$02, the cavity method yields several quantitative conjectures, notably

$d$03

meaning that 5-regular random graphs with threshold 3, and 6-regular random graphs with threshold 4, have contagious sets of asymptotic density $d$04 and $d$05, respectively (Guggiola et al., 2014). Equivalently, these are the minimal fractions of vertices that have to be removed from a 5-regular random graph to destroy its 3-core, and from a 6-regular random graph to destroy its 4-core (Guggiola et al., 2014).

For the contact process on a uniform random $d$06-regular graph with $d$07, initialized from full occupancy, the extinction time grows exponentially with the vertex number when $d$08, and the normalized extinction time converges in distribution to $d$09 for asymptotically almost every graph (Su, 2015). A more recent result identifies a finite-graph analogue of the second tree critical value $d$10. On the random $d$11-regular configuration model on $d$12, started from $d$13, let $d$14 be the time of the $d$15-th reinfection of vertex $d$16. Conditional on survival for $d$17 time, for every $d$18,

$d$19

whereas

$d$20

with $d$21 the Malthusian growth parameter on $d$22 (Fernley, 13 Mar 2025). This identifies the second transition as one between linear reinfections and reinfections following a long healthy period.

Random-walk exploration also acquires explicit asymptotics on random regular graphs. For the biased random walk that always prefers previously unvisited edges, on a random $d$23-regular graph $d$24 with fixed odd $d$25,

$d$26

with high probability, for vertex and edge cover times respectively (Johansson, 2018). In parallel, the random-cluster Glauber dynamics on random $d$27-regular graphs mixes in $d$28 for every $d$29, $d$30, and $d$31, with high probability over the graph; via Ullrich’s comparison, this yields $d$32 mixing for the Swendsen–Wang dynamics of the $d$33-state Potts model in the same tree-uniqueness regime (Blanca et al., 2020).

These results collectively show that random regular graphs are not merely static test cases. They support metastability, dynamic phase transitions, optimal-time MCMC in uniqueness regimes, and exploration laws with explicit constants, all while retaining enough homogeneity for rigorous asymptotics.

6. Quantum transport, Anderson localization, and spin glasses

Random regular graphs are a standard substrate for disordered quantum systems because they combine local tree-likeness with global finiteness. In the Anderson model on an RRG of connectivity $d$34, the Hamiltonian is

$d$35

and the graph serves as a proxy for the many-body localization problem on a Fock-space-like graph (Tomasi et al., 2019). For this connectivity, one study quotes an Anderson transition at

$d$36

while another uses

$d$37

for the same model (Tomasi et al., 2019, Bera et al., 2018).

The dynamical observable

$d$38

—the probability distribution of graph distance from an initial site after time $d$39, projected to a microcanonical window near the band center—reveals four space-time regimes on the RRG. In the delocalized phase, the wave-front position obeys a subdiffusive law

$d$40

and the return probability satisfies

$d$41

with the Thouless time scaling as

$d$42

The same work argues that $d$43 over a substantial intermediate-disorder range, and interprets the resulting subdiffusion as relevant to many-body localization on Fock-space-like graphs (Tomasi et al., 2019).

A complementary study of the return probability

$d$44

finds evidence for two distinct phases on the random regular graph: a fully ergodic phase, in which $d$45 decays polynomially with oscillations, and a nonergodic phase, in which the decay is stretched exponential (Bera et al., 2018). Comparing the mean $d$46 to the typical value $d$47 differentiates an ergodic phase from a nonergodic one, and the paper uses this diagnostic—after benchmarking on PLRBM and Rosenzweig–Porter ensembles—to argue for a nonergodic extended regime in the Anderson model on the RRG (Bera et al., 2018).

Random regular graphs also support sparse mean-field spin-glass formulations. For the Ising spin glass on a $d$48-regular random graph, the local tree-like property motivates cavity/Bethe recursions, but low-temperature analysis requires replica symmetry breaking. A recent construction extends the full RSB scheme to the Ising spin glass on a random regular graph by introducing a martingale approach that replaces the Parisi–Mézard tower of distributions with a well-defined stochastic-process formulation, together with order parameters and self-consistency equations for the free energy (Concetti, 2017). This suggests that random regular graphs occupy a distinctive position between fully connected SK theory and finite-dimensional disorder: sparse enough to preserve locality, yet mean-field enough to sustain full RSB structures.

Taken together, these developments portray random regular graphs as a unifying substrate for sparse mean-field phenomena. Their local convergence to regular trees supports cavity, branching, and uniqueness arguments; their global homogeneity suppresses boundary artifacts; and their combinatorial rigidity is strong enough to admit exact asymptotics for sampling, factorization, metastability, and quantum transport.