Replica Symmetry Breaking Research Articles

Counting and hardness-of-finding fixed points in cellular automata on random graphs

Abstract We study the fixed points of outer-totalistic cellular automata on sparse random regular graphs. These can be seen as constraint satisfaction problems, where each variable must adhere to the same local constraint, which depends solely on its state and the total number of its neighbors in each possible state. Examples of this setting include classical problems such as independent sets or assortative/dissasortative partitions. We analyse the existence and number of fixed points in the large system limit using the cavity method, under both the replica symmetric (RS) and one-step replica symmetry breaking (1RSB) assumption. This method allows us to characterize the structure of the space of solutions, in particular, if the solutions are clustered and whether the clusters contain frozen variables. This last property is conjectured to be linked to the typical algorithmic hardness of the problem. We bring experimental evidence for this claim by studying the performance of the belief-propagation reinforcement algorithm, a message-passing-based solver for these constraint satisfaction problems.

Exploring statistical complexity and enabling speckle-free imaging with Dye-Kaolinite colloidal random lasers

This article extends our previous investigation into random lasers (RLs), focusing on the incorporation of Rhodamine 6G (R6G) dye in methanol with Kaolinite nanoclay scatterers as a disordered active medium for colloidal RLs. Building upon the prior exploration, which specifically focused on understanding the performance dependence related to the concentration of gain medium and scatterers, this report takes a closer look. It conducts a thorough statistical analysis of the incoherent dye-Kaolinite random lasing spectra, particularly in the context of replica symmetry breaking (RSB) and Lévy statistics. This study highlights that RSB is observed around the threshold energy and can serve as an indicator of the threshold. Gaussian distribution prevailed below, near, and far above the lasing threshold estimated from the levy (α) exponent, suggesting the absence of Lévy flight behavior. Additionally, Pearson correlation coefficients (PCC) were used to quantify the correlation between intensity fluctuations at different wavelengths, with the results visually represented by heatmaps. Furthermore, the random lasing output is utilized to achieve speckle-free imaging.

Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models

Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models

Small field chaos in spin glasses: Universal predictions from the ultrametric tree and comparison with numerical simulations

Replica symmetry breaking (RSB) for spin glasses predicts that the equilibrium configuration at two different magnetic fields are maximally decorrelated. We show that this theory presents quantitative predictions for this chaotic behavior under the application of a vanishing external magnetic field, in the crossover region where the field intensity scales proportionally to [Formula: see text], being N the system size. We show that RSB theory provides universal predictions for chaotic behavior: They depend only on the zero-field overlap probability function [Formula: see text] and are independent of other system features. In the infinite volume limit, each spin-glass sample is characterized by an infinite number of states that have a tree-like structure. We generate the corresponding probability distribution through efficient sampling using a representation based on the Bolthausen-Sznitman coalescent. Using solely [Formula: see text] as input we can analytically compute the statistics of the states in the region of vanishing magnetic field. In this way, we can compute the overlap probability distribution in the presence of a small vanishing field and the increase of chaoticity when increasing the field. To test our computations, we have simulated the Bethe lattice spin glass and the 4D Edwards-Anderson model, finding in both cases excellent agreement with the universal predictions.

A supersymmetric SYK model with a curious low energy behavior

We consider N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2, 4 supersymmetric SYK models that have a peculiar low energy behavior, with the entropy going like S = S0 + (constant)Ta, where a ≠ 1. The large N equations for these models are a generalization of equations that have been previously studied as an unjustified truncation of the planar diagrams describing the BFSS matrix quantum mechanics or other related matrix models. Here we reanalyze these equations in order to better understand the low energy physics of these models. We find that the scalar fields develop large expectation values which explore the low energy valleys in the potential. The low energy physics is dominated by quadratic fluctuations around these values. These models were previously conjectured to have a spin glass phase. We did not find any evidence for this phase by using the usual diagnostics, such as searching for replica symmetry breaking solutions.

Experimental evidence on the feasibility of employing a replica symmetry breaking classification random laser.

Replica symmetry breaking (RSB) has been introduced in a random laser to investigate the interactions between disorder and fluctuations. In this work, the dynamic difference between four non-energy transfer and Förster resonance energy transfer (FRET)-assisted random laser systems is investigated based on RSB. It is found that FRET is one of the key factors influencing RSB, and it is demonstrated that RSB in a random laser is not robust. This dynamic difference can be attributed to the different disorders induced by the gain mechanism in different random laser systems. This provides experimental evidence and theoretical support for the classification feasibility of RL with different emission mechanisms employing RSB.

Replica symmetry breaking in 1D Rayleigh scattering system: theory and validations

Spin glass theory, as a paradigm for describing disordered magnetic systems, constitutes a prominent subject of study within statistical physics. Replica symmetry breaking (RSB), as one of the pivotal concepts for the understanding of spin glass theory, means that under identical conditions, disordered systems can yield distinct states with nontrivial correlations. Random fiber laser (RFL) based on Rayleigh scattering (RS) is a complex disordered system, owing to the disorder and stochasticity of RS. In this work, for the first time, a precise theoretical model is elaborated for studying the photonic phase transition via the platform of RS-based RFL, in which we clearly reveal that, apart from the pump power, the photon phase variation in RFL is also an analogy to the temperature term in spin-glass phase transition, leading to a novel insight into the intrinsic mechanisms of photonic phase transition. In addition, based on this model and real-time high-fidelity detection spectral evolution, we theoretically predict and experimentally observe the mode-asymmetric characteristics of photonic phase transition in RS-based RFL. This finding contributes to a deeper understanding of the photonic RSB regime and the dynamics of RS-based RFL.

Exact Numerical Solution of the Fully Connected Classical and Quantum Heisenberg Spin Glass.

We present the mean field solution of the quantum and classical Heisenberg spin glasses, using the combination of a high precision numerical solution of the Parisi full replica symmetry breaking equations and a continuous time quantum MonteCarlo algorithm. We characterize the spin glass order and its low-energy excitations down to zero temperature. The Heisenberg spin glass has a rougher energy landscape than its Ising analog, and exhibits a very slow temperature evolution of its dynamical properties. We extend our analysis to the doped, metallic Heisenberg spin glass, which displays unexpectedly slow spin dynamics, reflecting the proximity to the melting quantum critical point and its associated Sachdev-Ye-Kitaev Planckian dynamics.

Observation of the photonic Hall effect and photonic magnetoresistance in random lasers

Modulation of scattering in random lasers (RLs) by magnetic fields has attracted much attention due to its rich physical insights. We fabricate magnetic gain polymer optical fiber to generate RLs. From macroscopic experimental phenomena, with the increase of the magnetic field strength, the magnetic transverse photocurrent exists in disordered multiple scattering of RLs and the emission intensity of RLs decreases, which is the experimental observation of photonic Hall effect (PHE) and photonic magnetoresistance (PMR) in RLs. At the microscopic level, based on the field dependence theory of magnetic disorder in scattered nanoparticles and the replica symmetry breaking theory, the magnetic-induced transverse diffusion of photons reduces the scattering disorder, and then decreases the intensity fluctuation disorder of RLs. Our work establishes a connection between the above two effects and RLs, visualizes the influence of magnetic field on RL scattering at the microscopic level, which is crucial for the design of RLs.

Observation and all-optical manipulation of replica symmetry breaking dynamics in a multi-Stokes-involved Brillouin random fiber laser photonic system

In this paper, we propose and demonstrate an all-optical control of RSB transition in a multi-wavelength Brillouin random fiber laser (MWBRFL). Multi-order Stokes light components can be subsequently generated by increasing the power of the Erbium-doped fiber amplifier (EDFA) inside the MWBRFL, providing additional disorder as well as multiple Stokes-involved interplay. It essentially allows diversified laser mode landscapes with adjustable average mode lifetime and random mode density of the 1st order Stokes, which benefits the switching between replica symmetry breaking (RSB) and replica symmetry (RS) states in an optically controlled manner. Results show that the average mode lifetime of the 1st order Stokes component gradually decreases from 250.0 ms to 1.2 ms as high orders from the 2nd to the 5th of Stokes components are activated. Meanwhile, the order parameter q of the 1st order Stokes random lasing emission presents distinct statistical distributions within the selective sub-window under various EDFA optical powers. Consequently, all-optical dynamical control of the 1st Stokes random laser mode landscapes with adjustable average mode lifetime turns out to be attainable, facilitating the RSB transition under an appropriate observation time window. These findings open a new avenue for exploring the underlying physical mechanisms behind the occurrence of the RSB phenomenon in photonic complex systems.

Effect of constraint relaxation on the minimum vertex cover problem in random graphs.

A statistical-mechanical study of the effect of constraint relaxation on the minimum vertex cover problem in Erdős-Rényi random graphs is presented. Using a penalty-method formulation for constraint relaxation, typical properties of solutions, including infeasible solutions that violate the constraints, are analyzed by means of the replica method and cavity method. The problem involves a competition between reducing the number of vertices to be covered and satisfying the edge constraints. The analysis under the replica-symmetric (RS) ansatz clarifies that the competition leads to degeneracies in the vertex and edge states, which determine the quantitative properties of the system, such as the cover and penalty ratios. A precise analysis of these effects improves the accuracy of RS approximation for the minimum cover ratio in the replica symmetry-breaking (RSB) region. Furthermore, the analysis based on the RS cavity method indicates that the RS/RSB boundary of the ground states with respect to the mean degree of the graphs is expanded, and the critical temperature is lowered by constraint relaxation.

Replica symmetry breaking in supervised and unsupervised Hebbian networks

Abstract Hebbian neural networks with multi-node interactions, often called Dense Associative Memories, have recently attracted considerable interest in the statistical mechanics community, as they have been shown to outperform their pairwise counterparts in a number of features, including resilience against adversarial attacks, pattern retrieval with extremely weak signals and supra-linear storage capacities. However, their analysis has so far been carried out within a replica-symmetric theory. In this manuscript, we relax the assumption of replica symmetry and analyse these systems at one step of replica-symmetry breaking, focusing on two different prescriptions for the interactions that we will refer to as supervised and unsupervised learning. We derive the phase diagram of the model using two different approaches, namely Parisi's hierarchical ansatz for the relationship between different replicas within the replica approach, and the so-called telescope ansatz within Guerra's interpolation method: our results show that replica-symmetry breaking does not alter the threshold for learning and slightly increases the maximal storage capacity. Further, we also derive analytically the instability line of the replica-symmetric theory, using a generalization of the De Almeida and Thouless approach.

Observation of Replica Symmetry Breaking in Standard Mode-Locked Fiber Laser.

We report the first experimental demonstration of the replica symmetry breaking (RSB) phenomenon in a fiber laser system supporting standard mode-locking (SML) regime. Though theoretically predicted, this photonic glassy phase remained experimentally undisclosed so far. We employ an ytterbium-based mode-locked fiber laser with a very rich phase diagram. Two phase transitions are observed separating three different regimes: cw, quasi-mode-locking (QML), and SML. The regimes are intrinsically related to the distinct dynamics of intensity fluctuations in the laser spectra. We set the connection between the RSB glassy phase with frustrated modes and onset of L-shaped intensity distributions in the QML regime, which impact directly the replica overlap measure.

Griffiths-Type Theorems for Short-Range Spin Glass Models

We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap R1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{1,2}$$\\end{document} implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}. In Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_2$$\\end{document} invariant models such as the standard Edwards–Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards–Anderson order parameter. This generalization of Griffiths’ theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model. There is a 25-minute video that explains the main results of the present work:https://youtu.be/BF3hJiY1xvI

One-Step Replica Symmetry Breaking of Random Regular NAE-SAT II

Continuing our earlier work in Nam et al. (One-step replica symmetry breaking of random regular NAE-SAT I, arXiv:2011.14270, 2020), we study the random regular k-nae-sat model in the condensation regime. In Nam et al. (2020), the (1rsb) properties of the model were established with positive probability. In this paper, we improve the result to probability arbitrarily close to one. To do so, we introduce a new framework which is the synthesis of two approaches: the small subgraph conditioning and a variance decomposition technique using Doob martingales and discrete Fourier analysis. The main challenge is a delicate integration of the two methods to overcome the difficulty arising from applying the moment method to an unbounded state space.

Ising sSon Frustrated Bronze‐Mean Hexagonal Quasicrystal

AbstractWe investigate the Ising model on the Bronze‐mean hexagonal quasicrystal (BMH QC), an aperiodic tiling with geometric frustration. Our extensive Monte Carlo simulations explore the model's rich phase diagram, revealing six distinct phases with diverse magnetic properties and degrees of frustration. We uncover exotic spin glass phases, signaled by the replica symmetry breaking and slow relaxation dynamics. We shed light on the intriguing magnetic properties of frustrated quasicrystals and open new avenues for studying exotic phases in condensed matter physics.

Replica-Symmetry Breaking Transitions in the Large Deviations of the Ground-State of a Spherical Spin-Glass

We derive, within the replica formalism, a generalisation of the Crisanti–Sommers formula to describe the large deviation function (LDF) L(e)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{L}(e)$$\\end{document} for the speed-N atypical fluctuations of the intensive ground-state energy e of a generic spherical spin-glass in the presence of a random external magnetic field of variance Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. We then analyse our exact formula for the LDF in much detail for the Replica symmetric, single step Replica Symmetry Breaking (1-RSB) and Full Replica Symmetry Breaking (FRSB) situations. Our main qualitative conclusion is that the level of RSB governing the LDF may be different from that for the typical ground-state. We find that while the deepest ground-states are always controlled by a LDF of replica symmetric form, beyond a finite threshold e≥et\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e\\ge e_{t}$$\\end{document} a replica-symmetry breaking starts to be operative. These findings resolve the puzzling discrepancy between our earlier replica calculations for the p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=2$$\\end{document} spherical spin-glass (Fyodorov and Le Doussal in J Stat Phys 154:466, 2014) and the rigorous results by Dembo and Zeitouni (J Stat Phys 159:1306, 2015) which we are able to reproduce invoking an 1-RSB pattern. Finally at an even larger critical energy ec≥et\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e_{c}\\ge e_{t}$$\\end{document}, acting as a “wall”, the LDF diverges logarithmically, which we interpret as a change in the large deviation speed from N to a faster growth. In addition, we show that in the limit Γ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma \\rightarrow 0$$\\end{document} the LDF takes non-trivial scaling forms (i) L(e)∼G((e-ec)/Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{L}(e) \\sim G((e-e_c)/\\Gamma )$$\\end{document} in the vicinity of the wall (ii) L(e)∼ΓηνF((e-etyp)/Γν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{L}(e) \\sim \\Gamma ^{\\eta \ u } F((e-e_{\ extrm{typ}})/\\Gamma ^{\ u })$$\\end{document} in the vicinity of the typical energy, characterised by two new exponents η≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta \\ge 1$$\\end{document} and ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} characterising universality classes. Via matching the latter allows us to formulate several conjectures concerning the regime of typical fluctuations, identified as e-etyp∼N-1/η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e-e_{\ extrm{typ}} \\sim N^{-1/\\eta }$$\\end{document} and Γ∼N-1/(ην)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma \\sim N^{-1/(\\eta \ u )}$$\\end{document}.

Statistical mechanics of the maximum-average submatrix problem

We study the maximum-average submatrix problem, wherein given an N × N matrix J, one needs to find the k × k submatrix with the largest average number of entries. We investigate the problem for random matrices J, whose entries are i.i.d. random variables, by mapping it to a variant of the Sherrington–Kirkpatrick spin-glass model at fixed magnetisation. We analytically characterise the phase diagram of the model as a function of the submatrix average and the size of the submatrix k in the limit N→∞ . We consider submatrices of size k=mN with 0<m<1 . We find a rich phase diagram, including dynamical, static one-step replica symmetry breaking (1-RSB) and full-step RSB. In the limit of m → 0, we find a simpler phase diagram featuring a frozen 1-RSB phase, where the Gibbs measure comprises exponentially many pure states each with zero entropy. We discover an interesting phenomenon, reminiscent of the phenomenology of the binary perceptron: there are efficient algorithms that provably work in the frozen 1-RSB phase.

Statistical mechanics of inference in epidemic spreading.

We investigate the information-theoretical limits of inference tasks in epidemic spreading on graphs in the thermodynamic limit. The typical inference tasks consist in computing observables of the posterior distribution of the epidemic model given observations taken from a ground-truth (sometimes called planted) random trajectory. We can identify two main sources of quenched disorder: the graph ensemble and the planted trajectory. The epidemic dynamics however induces nontrivial long-range correlations among individuals' states on the latter. This results in nonlocal correlated quenched disorder which unfortunately is typically hard to handle. To overcome this difficulty, we divide the dynamical process into two sets of variables: a set of stochastic independent variables (representing transmission delays), plus a set of correlated variables (the infection times) that depend deterministically on the first. Treating the former as quenched variables and the latter as dynamic ones, computing disorder average becomes feasible by means of the replica-symmetric cavity method. We give theoretical predictions on the posterior probability distribution of the trajectory of each individual, conditioned to observations on the state of individuals at given times, focusing on the susceptible infectious (SI) model. In the Bayes-optimal condition, i.e., when true dynamic parameters are known, the inference task is expected to fall in the replica-symmetric regime. We indeed provide predictions for the information theoretic limits of various inference tasks, in form of phase diagrams. We also identify a region, in the Bayes-optimal setting, with strong hints of replica-symmetry breaking. When true parameters are unknown, we show how a maximum-likelihood procedure is able to recover them with mostly unaffected performance.

Phase transition in von Neumann entropy from replica symmetry breaking

We study the entanglement transition in monitored Brownian SYK chains in the large-N limit. Without measurement the steady state n-th Rényi entropy is obtained by summing over a class of solutions, and is found to saturate to the Page value in the n → 1 limit. In the presence of measurements, the analytical continuation n → 1 is performed using the cyclic symmetric solution. The result shows that as the monitoring rate increases, a continuous von Neumann entanglement entropy transition from volume-law to area-law occurs at the point of replica symmetry unbreaking.