Habitat connectivity follows percolation dynamics: below a critical threshold (~59.3%), ecosystems fragment into isolated patches; above it, landscape-spanning connectivity emerges nonlinearly.
Synchronization is a fundamental collective phenomenon observed across nature and art, from firefly flash coordination to power-grid frequency locking to ballet corps moving in unison. We model a ballet ensemble as a system of spatially-embedded Kuramoto coupled oscillators and study the phase transition from incoherence to synchrony as a function of coupling strength K.
When multiple autonomous agents must coordinate on a shared action—choosing the same meeting point, communication protocol, or trading strategy—each agent's prior belief about which action is "correct" shapes the outcome. We study how the degree of prior disagreement affects coordination in a pure coordination game with N agents and K actions.
Compute Ramsey numbers R(K_{s,t}, K_{s,t}) for s,t ≤ 8 via SAT solving and ILP. The growth rate exhibits a sharp transition: for t/s < 0.
Simulate random Clifford circuits (up to 256 qubits) with variable measurement rate p (fraction of qubits measured per layer). Entanglement entropy S_A of half-chain: at p=0 (no measurement), S_A scales as volume law S_A ∝ L.
Grokking—sudden generalization long after memorization—is difficult to predict. We identify a precursor: the Gradient Acceleration Index (GAI), the second derivative of gradient norm w.
Habitat connectivity follows percolation dynamics: below a critical threshold (~59.3%), ecosystems fragment into isolated patches; above it, landscape-spanning connectivity emerges nonlinearly.