This paper provides a comprehensive review and synthesis of phase transitions and criticality in random linear systems. Such systems, characterized by matrices with stochastic entries, are ubiquitous across statistical physics, machine learning, signal processing, and control theory. We examine how their macroscopic properties undergo abrupt qualitative changes as underlying parameters vary, analogous to physical phase transitions. Criticality, the state at these phase boundaries, exhibits unique statistical properties like scaling laws and universal behavior, often defying characterization by conventional perturbation theories. The paper reviews foundational concepts from Random Matrix Theory and statistical mechanics of disordered systems, connecting them to modern high-dimensional data analysis and artificial intelligence. We discuss theoretical frameworks, including the replica and cavity methods, and numerical approaches employed to uncover these transitions. Furthermore, we highlight the practical implications of understanding criticality for designing robust algorithms, predicting system performance limits, and comprehending emergent complexity. Our aim is to offer an interdisciplinary synthesis, bridging classical insights with cutting-edge developments, to provide a unified perspective on these profound phenomena across diverse research fields.
This paper provides a comprehensive review and synthesis of phase transitions and criticality in random linear systems. Such systems, characterized by matrices with stochastic entries, are ubiquitous across statistical physics, machine learning, signal processing, and control theory. We examine how their macroscopic properties undergo abrupt qualitative changes as underlying parameters vary, analogous to physical phase transitions. Criticality, the state at these phase boundaries, exhibits unique statistical properties like scaling laws and universal behavior, often defying characterization by conventional perturbation theories. The paper reviews foundational concepts from Random Matrix Theory and statistical mechanics of disordered systems, connecting them to modern high-dimensional data analysis and artificial intelligence. We discuss theoretical frameworks, including the replica and cavity methods, and numerical approaches employed to uncover these transitions. Furthermore, we highlight the practical implications of understanding criticality for designing robust algorithms, predicting system performance limits, and comprehending emergent complexity. Our aim is to offer an interdisciplinary synthesis, bridging classical insights with cutting-edge developments, to provide a unified perspective on these profound phenomena across diverse research fields.
Phase transitions represent fundamental transformations of matter, characterized by abrupt changes in macroscopic properties and often associated with singular behavior at critical points. While equilibrium aspects of phase transitions, particularly universal critical exponents and scaling laws, are well-understood, the non-equilibrium processes occurring near criticality, especially the production of entropy, remain a rich area of investigation. This paper explores the intricate relationship between critical entropy production and the emergence of scaling laws in systems undergoing phase transitions. We delve into how the rate of entropy production diverges or exhibits anomalous scaling near critical points, reflecting the system's attempts to dissipate energy and respond to external perturbations as it approaches a state of macroscopic change. By integrating concepts from non-equilibrium thermodynamics, statistical mechanics, and renormalization group theory, we aim to elucidate the underlying mechanisms that govern entropy generation in these critical regimes. We discuss theoretical frameworks that predict universal scaling behaviors for entropy production rates, analogous to those observed for equilibrium order parameters and susceptibilities. The implications of these scaling laws extend beyond theoretical physics, offering insights into complex systems, far-from-equilibrium dynamics, and the efficiency of energy conversion processes in critical environments. Our analysis highlights the critical entropy production as a unique signature of non-equilibrium dynamics during phase transitions, providing a deeper understanding of the interplay between fluctuations, dissipation, and universality near points of instability.
Non-Hermitian disordered systems have emerged as a central arena in modern physics, with ramifications spanning condensed matter, quantum, statistical, and high energy contexts. The same principles also underlie phenomena beyond physics, such as network science, complex systems, and biophysics, where dissipation, nonreciprocity, and stochasticity are ubiquitous. Here, we review the physics and mathematics of non-Hermitian disordered systems, with particular emphasis on non-Hermitian random matrix theory. We begin by presenting the 38-fold symmetry classification of non-Hermitian systems, contrasting it with the 10-fold way for Hermitian systems. After introducing the classic Ginibre ensembles of non-Hermitian random matrices, we survey various diagnostics for complex-spectral statistics and distinct universality classes realized by symmetry. As a key application to physics, we discuss how non-Hermitian random matrix theory characterizes chaos and integrability in open quantum systems. We then turn to the criticality due to the interplay of disorder and non-Hermiticity, including Anderson transitions in the Hatano-Nelson model and its higher-dimensional extensions. We also discuss the effective field theory description of non-Hermitian disordered systems in terms of nonlinear sigma models.
Clearly, in nature, but also in technological applications, complex systems built in an entirely ordered and regular fashion are the exception rather than the rule. In this thesis we explore how critical phenomena are influenced by quenched spatial randomness. Specifically, we consider physical systems undergoing a continuous phase transition in the presence of topological disorder, where the underlying structure, on which the system evolves, is given by a non-regular, discrete lattice. We therefore endeavour to achieve a thorough understanding of the interplay between collective dynamics and quenched randomness. According to the intriguing concept of universality, certain laws emerge from collectively behaving many-body systems at criticality, almost regardless of the precise microscopic realization of interactions in those systems. As a consequence, vastly different phenomena show striking similarities at their respective phase transitions. In this dissertation we pursue the question of whether the universal properties of critical phenomena are preserved when the system is subjected to topological perturbations. For this purpose, we perform numerical simulations of several prototypical systems of statistical physics which show a continuous phase transition. In particular, the equilibrium spin-1/2 Ising model and its generalizations represent -- among other applications -- fairly natural approaches to model magnetism in solids, whereas the non-equilibrium contact process serves as a toy model for percolation in porous media and epidemic spreading. Finally, the Manna sandpile model is strongly related to the concept of self-organized criticality, where a complex dynamic system reaches a critical state without fine-tuning of external variables. Our results reveal that the prevailing understanding of the influence of topological randomness on critical phenomena is insufficient. In particular, by considering very specific and newly developed lattice structures, we are able to show that -- contrary to the popular opinion -- spatial correlations in the number of interacting neighbours are not a key measure for predicting whether disorder ultimately alters the behaviour of a given critical system.
Critical phenomena, characterized by abrupt changes in macroscopic properties near phase transition points, exhibit universal scaling behavior independent of microscopic details. This paper explores the hypothesis that entropy serves as a fundamental universal scaling law governing these phenomena. We review the theoretical foundations of critical phenomena, including Landau theory and the renormalization group, highlighting the role of critical exponents and scaling functions. We then propose a framework where the configurational entropy, entanglement entropy, and other entropic measures exhibit universal scaling, linking them directly to the underlying critical exponents. Our methodology involves analyzing how entropy behaves under coarse-graining transformations near critical points, identifying a potential entropic fixed point in the renormalization group flow. We present theoretical arguments suggesting that the information content, as quantified by various entropy definitions, reflects the diverging correlation length and the emergent scale invariance at criticality. The discussion extends to how different entropic forms, such as Rényi and Tsallis entropies, might provide a more generalized framework for understanding critical behavior in complex and non-extensive systems. By demonstrating entropy's role as a universal scaling law, this work offers a unifying perspective on critical phenomena, deepening our understanding of phase transitions and the emergence of macroscopic order from microscopic fluctuations.
Critical phenomena, characterized by abrupt changes in the macroscopic properties of a system, exhibit a profound degree of universality, meaning that vastly different physical systems can display identical behavior near their critical points. This universality is typically understood through the lens of scaling laws and the Renormalization Group theory, which classify critical points into distinct universality classes. This paper proposes and explores the "Entropic Law of Critical Universality," an overarching principle positing that not only do thermodynamic quantities scale universally, but the fundamental information-theoretic measures of a system, particularly entropy in its various forms (thermodynamic entropy, entanglement entropy, Renyi entropy), also conform to universal scaling behaviors that uniquely characterize these universality classes. We review how entanglement entropy exhibits universal logarithmic corrections at quantum critical points, linked to the central charge of underlying conformal field theories. We also discuss how specific entropy behaves non-monotonically near quantum chromodynamics critical points and how entropic measures can identify phase transitions and their classes in various condensed matter systems, including graphene and spin models. The paper synthesizes theoretical frameworks and empirical evidence to demonstrate that entropy serves as a powerful and often more direct probe of the universal properties underlying critical phenomena, offering a comprehensive perspective on the shared information landscape of diverse critical systems.
The study of quantum criticality has historically been confined to Hermitian systems, where closed-system approximations neglect environmental interactions. However, a significant portion of quantum phenomena occurs in open systems, necessitating a departure from this conventional framework. This paper explores non-Hermitian quantum criticality in open many-body systems, delving into the unique phases and phase transitions that emerge due to dissipation, gain, and other non-conservative interactions. We establish a comprehensive framework that integrates concepts from biorthogonal quantum mechanics, Liouvillian dynamics, and tensor network methods to characterize critical phenomena, including dynamical quantum phase transitions and the emergence of non-Hermitian quantum many-body scars. Through an extensive literature review, we highlight how the interplay of non-Hermiticity with many-body correlations leads to novel critical behaviors, distinct from their Hermitian counterparts, and in some cases, surprisingly resilient or even enhanced. We discuss the role of exceptional points as hallmarks of non-Hermitian systems, their influence on energy spectra, and their implications for criticality and topological invariants. Furthermore, we examine the methodologies for defining observables and statistical ensembles in these non-Hermitian settings, emphasizing the distinctions and occasional controversies that arise. The findings provide critical insights into the fundamental principles governing quantum criticality in dissipative environments and pave new avenues for engineering and understanding open quantum technologies, from quantum sensing to robust quantum information processing.