Faheem Mosam | Searcharxiv

arXiv2026

Entangled criticality and irreversibility in random Markov dynamics

We introduce a two-parameter ensemble of random discrete-time Markov models that simultaneously captures critical slowing down and broken detailed balance. Extending a previously studied heterogeneous Markov ensemble, we incorporate correlations between forward and backward transition rates through a single asymmetry parameter $\gamma$, while heterogeneity is controlled by $\epsilon$. Using results from random matrix theory, we identify a critical locus $\epsilon_c(\gamma,N)$ at which relaxation times diverge and spectral universality breaks down, in Markov models with $N$ states. We characterize the behavior of entropy production, predictive information, and relaxation dynamics across the ensemble, showing that many observables depend strongly on heterogeneity but only weakly on asymmetry, except near the symmetric limit. Applying maximum-likelihood inference to human fMRI and EEG data, we find that both modalities operate near the predicted critical locus and occupy a similar region of the $\epsilon-\gamma$ plane, supporting a super-universality of human brain dynamics. While ensemble averages are well captured by the null model, empirical data exhibit substantially enhanced variability, indicating subject-specific structure beyond random expectations. Our results unify criticality and nonequilibrium measures within a single framework and clarify their intertwined role in the analysis of complex biological dynamics.

cond-mat.dis-nn↗

arXiv2021

Breakdown of random matrix universality in Markov models

Biological systems need to react to stimuli over a broad spectrum of timescales. If and how this ability can emerge without external fine-tuning is a puzzle. We consider here this problem in discrete Markovian systems, where we can leverage results from random matrix theory. Indeed, generic large transition matrices are governed by universal results, which predict the absence of long timescales unless fine-tuned. We consider an ensemble of transition matrices and motivate a temperature-like variable that controls the dynamic range of matrix elements, which we show plays a crucial role in the applicability of the large matrix limit: as the dynamic range increases, a phase transition occurs whereby the random matrix theory result is avoided, and long relaxation times ensue, in the entire `ordered' phase. We furthermore show that this phase transition is accompanied by a drop in the entropy rate and a peak in complexity, as measured by predictive information (Bialek, Nemenman, Tishby Neural Computation 13(21) 2001). Extending the Markov model to a Hidden Markov model (HMM), we show that observable sequences inherit properties of the hidden sequences, allowing HMMs to be understood in terms of more accessible Markov models. We then apply our findings to fMRI data from 820 human subjects scanned at wakeful rest. We show that the data can be quantitatively understood in terms of the random model, and that brain activity lies close to the phase transition when engaged in unconstrained, task-free cognition -- supporting the brain criticality hypothesis in this context.

cond-mat.dis-nn↗