Operator $K$-theoretic analysis of random adjacency matrices

TL;DR

The paper analyzes the K-theoretic properties of random graph C*-algebras generated from diverse random graph models, showing that K1 is a.a.s. trivial and that stable isomorphism classes are captured by Cuntz polygons, with exact isomorphism to actual Cuntz algebras governed by cyclicity of K0. It develops a Fraïssé framework for Cuntz polygons, yielding a universal limit graph G that encodes asymptotic K-theoretic behavior, and uses random matrix theory (via M_E = A_E^t - I) to connect graph structure with K0, K1, and unit classes. The work pairs rigorous asymptotic results with extensive empirical K-data, validating theory and providing estimates where theory is incomplete, including explicit prime-factor formulas for Sylow subgroups and probabilistic predictions for flow equivalence of edge shifts. Collectively, the results bridge random graph theory, operator algebras, and symbolic dynamics, offering a probabilistic lens on the Elliott classification landscape for finite graph algebras and a universal Fraïssé limit that guides K-theoretic analyses.

Abstract

We appeal to results from combinatorial random matrix theory to deduce that various random graph $\mathrm{C}^*$-algebras are asymptotically almost surely Kirchberg algebras with trivial $K_1$. This in particular implies that, with high probability, the stable isomorphism classes of such algebras are exhausted by variations of Cuntz algebras that we term 'Cuntz polygons'. These probabilistically generic algebras can be assembled into a Fraïssé class whose limit structure $\mathbb{G}$ is consequently relevant to any $K$-theoretic analysis of finite graph $\mathrm{C}^*$-algebras. We also use computer simulations to experimentally verify the behaviour predicted by theory and to estimate the asymptotic probabilities of obtaining stable isomorphism classes represented by actual Cuntz algebras. These probabilities depend on the frequencies with which the Sylow $p$-subgroups of $K_0$ are cyclic and in some cases can be computed from existing theory. For random symmetric $r$-regular multigraphs, current theory can describe these frequencies for finite sets of odd primes $p$ not dividing $r-1$. A novel aspect of the collected data is the observation of new heuristics outside of this case, leading to a conjecture for the asymptotic probability of these graphs yielding $\mathrm{C}^*$-algebras stably isomorphic to Cuntz algebras. For other models of random multigraphs including Bernoulli (di)graphs, the data also allow us to estimate and heuristically explain the (surprisingly high) asymptotic probabilities of exact isomorphism to a Cuntz algebra. Recognising the role played by Cuntz--Krieger algebras in the theory of symbolic dynamics, we also collect supplemental data to estimate (and in some cases, actually compute) the asymptotic probability of a random subshift of finite type being flow equivalent to a full shift.

Figures (11)

• Figure 1: The graph $E_{(m_1,m_2,m_3,m_4,m_5)}$ of the Cuntz pentagon $\mathcal{P}_{(m_1,m_2,m_3,m_4,m_5)}$.
• Figure 2: Sample generated graph $\widehat{\mathbb{G}}_{20,3}$
• Figure 3: Sample generated graph $\mathbb{D}_{11,1/2}$
• Figure 4: Frequency distribution for $10^5$ observations of $K_0(\mathrm{C}^*(\mathbb{D}_{100,1/4}))_{2}$
• Figure 5: Frequency distribution for $10^5$ observations of $K_0(\mathrm{C}^*(\mathbb{E}_{100,1/2}))_{2}$

Theorems & Definitions (28)

• Proposition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Definition 2.5
• Theorem 2.6
• Remark 2.7
• Remark 2.8
• Theorem 3.2