Phase transitions (PTs) are generally classified into second-order and first-order transitions, each exhibiting different intrinsic properties. For instance, a first-order transition exhibits latent heat and hysteresis when a control parameter is increased and then decreased across a transition point, whereas a second-order transition does not. Recently, hybrid percolation transitions (HPTs) are issued in diverse complex systems, in which the features of first-order and second-order PTs occur at the same transition point. Thus, the question whether hysteresis appears in an HPT arises. Herein, we investigate this fundamental question with a so-called restricted Erdős--Rényi random network model, in which a cluster fragmentation process is additionally proposed. The hysteresis curve of the order parameter was obtained. Depending on when the reverse process is initiated, the shapes of hysteresis curves change, and the critical behavior of the HPT is conserved throughout the forward and reverse processes.
In this paper we consider various models of hydraulic hysteresis in invasive mercury porosimetry. For simulating the hydraulic hysteresis is used isotropic site percolation on three-dimensional square lattices with $(1, d)$-neighborhood. The relationship between the percolation model parameters and invasive porosimetry data is studied phenomenologically. The implementation of the percolation model is based on libraries SPSL and SECP, released under license GNU GPL-3 using the free programming language R.
We report the critical point for site percolation for the "explosive" type for 2D square lattices using Monte Carlo simulations and compare it to the classical well known percolation. We use similar algorithms as have been recently reported for bond percolation and networks. We calculate the "explosive" site percolation threshold as $p_c=0.695$ and we find evidence that "explosive" site percolation surprisingly may belong to a different universality class than bond percolation on lattices, providing that the transitions (a) are continuous and (b) obey the conventional finite size scaling forms…
We experimentally investigate the critical behavior of a phase transition between two topologically different turbulent states of electrohydrodynamic convection in nematic liquid crystals. The statistical properties of the observed spatiotemporal intermittency regimes are carefully determined, yielding a complete set of static critical exponents in full agreement with those defining the directed percolation class in (2+1) dimensions. This constitutes the first clear and comprehensive experimental evidence of an absorbing phase transition in this prominent non-equilibrium universality class.
We study hysteresis in the random-field Ising model with an asymmetric distribution of quenched fields, in the limit of low disorder in two and three dimensions. We relate the spin flip process to bootstrap percolation, and show that the characteristic length for self-averaging $L^*$ increases as $exp(exp (J/Δ))$ in 2d, and as $exp(exp(exp(J/Δ)))$ in 3d, for disorder strength $Δ$ much less than the exchange coupling J. For system size $1 << L < L^*$, the coercive field $h_{coer}$ varies as $2J - Δ\ln \ln L$ for the square lattice, and as $2J - Δ\ln \ln \ln L$ on the cubic lattice. Its limiting…
The concept of midpoint percolation has recently been applied to characterize the double percolation transitions in negatively curved structures. Regular $d$-dimensional hypercubic lattices are in the present work investigated using the same concept. Specifically, the site-percolation transitions at the critical thresholds are investigated for dimensions up to $d=10$ by means of the Leath algorithm. It is shown that the explicit inclusion of the boundaries provides a straightforward way to obtain critical indices, both for the bulk and surface parts. At and above the critical dimension $d=6$, …
A minimal model for self-organized critical percolation on directed graphs with activating and de-activating links is studied. Unlike classical self-organized criticality, the variables that determine criticality are separated from the dynamical variables of the system and evolve on a slower timescale, resulting in robust criticality. While activity of nodes percolates across the network, the network self-organizes through local adjustment of links according to the criterion that a link's adjacent nodes' average activities become similar. As a result, the network self-organizes to the percolat…
We introduce a stochastic sandpile model where finite drive and dissipation are coupled to the activity field. The absorbing phase transition here, as expected, belongs to the directed percolation (DP) universality class. We focus on the small drive and dissipation limit, i.e. the so-called self-organised critical (SOC) regime and show that the system exhibits a crossover from ordinary DP scaling to a dissipation-controlled scaling which is independent of the underlying dynamics or spatial dimension. The new scaling regime continues all the way to the zero bulk drive limit suggesting that the …
We extend the Achlioptas model for the delay of criticality in the percolation problem. Instead of having a completely random connectivity pattern, we generalize the idea of the two-site probe in the Achlioptas model for connecting smaller clusters, by introducing two models: the first one by allowing any number k of probe sites to be investigated, k being a parameter, and the second one independent of any specific number of probe sites, but with a probabilistic character which depends on the size of the resulting clusters. We find numerically the complete spectrum of critical points and our r…
This is a comprehensive report on the phase transition between two turbulent states of electroconvection in nematic liquid crystals, which was recently found by the authors to be in the directed percolation (DP) universality class [K. A. Takeuchi et al., Phys. Rev. Lett. 99, 234503 (2007)]. We further investigate both static and dynamic critical behavior of this phase transition, measuring a total of 12 critical exponents, 5 scaling functions, and 8 scaling relations, all in full agreement with those characterizing the DP class in 2+1 dimensions. Developing an experimental technique to create …
We derive the critical nearest-neighbor connectivity $g_n$ as $3/4$, $3(7-9p_c^{tri})/[4(5-4p_c^{tri})]$, and $3(2+7p_c^{tri})/[4(5-p_c^{tri})]$ for bond percolation on the square, honeycomb and triangular lattice respectively, where $p_c^{tri}=2\sin(π/18)$ is the percolation threshold for the triangular lattice; and confirm these values via Monte Carlo simulations. On the square lattice, we also numerically determine the critical next-nearest-neighbor connectivity as $g_{nn}=0.687\;500\;0(2)$, which confirms a conjecture by Mitra and Nienhuis in J. Stat. Mech. P10006 (2004), implying the exac…
This Letter studies the critical point as well as the discontinuity of a class of explosive site percolation in Erdös and Rényi (ER) random network. The class of the percolation is implemented by introducing a best-of-m rule. Two major results are found: i). For any specific $m$, the critical percolation point scales with the average degree of the network while its exponent associated with $m$ is bounded by -1 and $\sim-0.5$. ii). Discontinuous percolation could occur on sparse networks if and only if $m$ approaches infinite. These results not only generalize some conclusions of ordinary perco…
The random Lorentz gas (RLG) is a minimal model for transport in disordered media. Despite the broad relevance of the model, theoretical grasp over its properties remains weak. For instance, the scaling with dimension d of its localization transition at the void percolation threshold is not well controlled analytically nor computationally. A recent study [Biroli et al. Phys. Rev. E L030104 (2021)] of the caging behavior of the RLG motivated by the mean-field theory of glasses has uncovered physical inconsistencies in that scaling that heighten the need for guidance. Here, we first extend analy…
We perform Monte Carlo simulations to determine the average excluded area $<A_{ex}>$ of randomly oriented squares, randomly oriented widthless sticks and aligned squares in two dimensions. We find significant differences between our results for randomly oriented squares and previous analytical results for the same. The sources of these differences are explained. Using our results for $<A_{ex}>$ and Monte Carlo simulation results for the percolation threshold, we estimate the mean number of connections per object $B_c$ at the percolation threshold for squares in 2-D. We study systems of squares…
Using three-dimensional representation of confluent cell layers, we map the amorphous solid to fluid phase transition in active cell layers onto the two-dimensional (2D) site percolation universality class. Importantly, we unify two distinct, predominant, pathways associated with this transition; namely (i) cell-cell adhesion and (ii) active traction forces. For each pathway, we independently vary the corresponding control parameter and focus on the emergent mechanical stress patterns as the monolayer transitions from a glassy- to a fluid-like state. Through finite-size scaling analyses, our r…
We present a pedagogical introduction to self-organized criticality (SOC), unraveling its connections with nonequilibrium phase transitions. There are several paths from a conventional critical point to SOC. They begin with an absorbing-state phase transition (directed percolation is a familiar example), and impose supervision or driving on the system; two commonly used methods are extremal dynamics, and driving at a rate approaching zero. We illustrate this in sandpiles, where SOC is a consequence of slow driving in a system exhibiting an absorbing-state phase transition with a conserved dens…
We propose a comprehensive theoretical description of hysteresis in capillary condensation of gases in mesoporous disordered materials. Applying mean-field density functional theory to a coarse-grained lattice-gas model, we show that the morphology of the hysteresis loops is influenced by out-of-equilibrium transitions that are different on filling and on draining. In particular, desorption may be associated to a depinning process and be percolation-like without explicit pore-blocking effects.
We consider the directed percolation process as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. The model is in a critical state when the activation probability is adjusted at some precise value p_c. Criticality is lost as soon as the probability to activate sites at the first attempt, p1, is changed. We show here that criticality can be restored by "compensating" the change in p1 by an appropriate change of the second time activation probability p2 in the opposite direction. At compensation, we observe that the bulk exponents of the process coincid…
Despite original claims of a first-order transition in the product rule model proposed by Achlioptas et al. [Science 323, 1453 (2009)], recent studies indicate that this percolation model, in fact, displays a continuous transition. The distinctive scaling properties of the model at criticality, however, strongly suggest that it should belong to a different universality class than ordinary percolation. Here we introduce a generalization of the product rule that reveals the effect of non-locality on the critical behavior of the percolation process. Precisely, pairs of unoccupied bonds are chosen…
We present a unified theory of fracture in disordered brittle media that reconciles apparently conflicting results reported in the literature. Our renormalization group based approach yields a phase diagram in which the percolation fixed point, expected for infinite disorder, is unstable for finite disorder and flows to a zero-disorder nucleation-type fixed point, thus showing that fracture has mixed first order and continuous character. In a region of intermediate disorder and finite system sizes, we predict a crossover with mean-field avalanche scaling. We discuss intriguing connections to o…
Extraction of interaction networks from multi-variate time-series is one of the topics of broad interest in complex systems. Although this method has a wide range of applications, most of the previous analyses have focused on the pairwise relations. Here we establish the potential of such a method to elicit aggregated behavior of the system by making a connection with the concepts from percolation theory. We study the dynamical interaction networks of a financial market extracted from the correlation network of indices, and build a weighted network. In correspondence with the percolation model…
We study a two-dimensional kinetic Ising model with Swendsen-Wang dynamics, replacing the usual percolation on top of Ising clusters by explosive percolation. The model exhibits a reversible first-order phase transition with hysteresis. Surprisingly, at the transition flanks the global bond density seems to be equal to the percolation thresholds.
The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality
University of Science and Technology of China（中国科学技术大学）；Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室）；CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China（上海量子科学研究与工程中心及中国科学院量子信息与量子物理学卓越创新中心）；Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China（中国科学院微观尺度物理科学国家研究中心及物理科学学院）
The critical behaviour of the Ising model in the absence of an external magnetic field can be specified either through spontaneous symmetry breaking (thermal criticality) or through cluster percolation (geometric criticality). We extend this to finite external fields for the case of the Potts' model, showing that a geometric analysis leads to the same first order/second order structure as found in thermodynamic studies. We calculate the Kertész line, separating percolating and non-percolating regimes, both analytically and numerically for the Potts model in presence of an external magnetic fie…
We study the site percolation under Achlioptas process (AP) with a product rule in a $2-dimensional$ (2D) square lattice. From the measurement of the cluster size distribution, $P_s$, we find that $P_s$ has a very robust power-law regime followed by a stable hump near the transition threshold. Based on the careful analysis on the $P_s$ distribution, we show that the transition should be discontinuous. The existence of the hysteresis loop in order parameter also verifies that the transition is discontinuous in 2D. Moreover we also show that the transition nature from the product rule is not the…
We present study of finite-size scaling and universality of crossing probabilities for the $q$-state Potts model. Crossing probabilities of the Potts model are similar ones in percolation problem. We numerically investigated scaling of $π_{s}$ - the probability of a system to percolate only in one direction for two-dimensional site percolation, the Ising model, and the q-state Potts model for $q=3,4,5,6,8,10$. We found the thermal scaling index $y= \frac{1}ν$ for $q<4$. In contrast, $y \ne \frac{1}ν$ for $q=4$.
Experimental results for covalent glasses have highlighted the existence of a new self-organized phase due to the tendency of glass networks to minimize internal stress. Recently, we have shown that an equilibrated self-organized two-dimensional lattice-based model also possesses an intermediate phase in which a percolating rigid cluster exists with a probability between zero and one, depending on the average coordination of the network. In this paper, we study the properties of this intermediate phase in more detail. We find that microscopic perturbations, such as the addition or removal of a…
Percolation Criticality of Amorphous-Amorphous Transitions in Compressed Glasses
Laboratoire Charles Coulomb (L2C), CNRS - Université Montpellier（夏尔·库隆实验室，法国国家科学研究中心-蒙彼利埃大学）；Departamento de Física y Matemáticas, Universidad Iberoamericana（物理与数学系，伊比利亚美洲大学）；Centro de Física, Instituto Venezolano de Investigaciones Científicas（物理中心，委内瑞拉科学研究研究所）；Departamento de Física, Universidad Simón Bolívar（物理系，西蒙·玻利瓦尔大学）
The question we address here is of whether phenomena of collective bankruptcies are related to self-organized criticality. In order to answer it we propose a simple model of banking networks based on the random directed percolation. We study effects of one bank failure on the nucleation of contagion phase in a financial market. We recognize the power law distribution of contagion sizes in 3d- and 4d-networks as an indicator of SOC behavior. The SOC dynamics was not detected in 2d-lattices. The difference between 2d- and 3d- or 4d-systems is explained due to the percolation theory.
We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small-world (the diameter grows either algebraically or logarithmically with the net size), assortative or disassortative (a measure of the tendency of like-degree nodes to be connected to one another), or possess various degrees of clustering. The percolation phase transition can be analyzed exactly in all these cases, due to the self-similar structure of the hierarchical nets. We find different types of criticality, illustrating the crucial effect of …
The aim of the paper is to present numerical results supporting the presence of conformal invariance in three dimensional statistical mechanics models at criticality and to elucidate the geometric aspects of universality. As a case study we study three dimensional percolation at criticality in bounded domains. Both on discrete and continuous models of critical percolation, we test by numerical experiments the invariance of quantities in finite domains under conformal transformations focusing on crossing probabilities. Our results show clear evidence of the onset of conformal invariance in fini…
Critical phenomena of a second-order percolation transition are known to be independent of cluster merging or pruning process. However, those of a hybrid percolation transition (HPT), mixed properties of both first-order and second-order transitions, depend on the processes. The HPT induced by cluster merging is more intrigue and little understood than the other. Here, we construct a theoretical framework using the so-called restricted percolation model. In this model, clusters are ranked by size and partitioned into small- and large-cluster sets. As the cluster rankings are updated by cluster…
A vast class of disordered conducting-insulating compounds close to the percolation threshold is characterized by nonuniversal values of transport critical exponents. The lack of universality implies that critical indexes may depend on material properties such as the particular microstructure or the nature of the constituents, and that in principle they can be influenced by suitable applied perturbations leading to important informations about the origin of nonuniversality. Here we show that in RuO$_2$-glass composites the nonuniversal exponent can be modulated by an applied mechanical strain,…
We present a numerical study on an interacting monomer-dimer model with nearest neighbor repulsion on a square lattice, which possesses two symmetric absorbing states. The model is observed to exhibit two nearby continuous transitions: the $Z_2$ symmetry-breaking order-disorder transition and the absorbing transition with directed percolation criticality. We find that the symmetry-breaking transition shows a non-Ising critical behavior, and that the absorbing phase becomes critical, in the sense that the critical decay of the dimer density observed at the absorbing transition persists even wit…
Entanglement transitions in quantum dynamics present a novel class of phase transitions in non-equilibrium systems. When a many-body quantum system undergoes unitary evolution interspersed with monitored random measurements, the steady-state can exhibit a phase transition between volume and area-law entanglement. There is a correspondence between measurement-induced transitions in non-unitary quantum circuits in $d$ spatial dimensions and classical statistical mechanical models in $d+1$ dimensions. In certain limits these models map to percolation, but there is analytical and numerical evidenc…
The critical exponent beta =0.17(1) for the three-dimensional random-field Ising model (RFIM) order parameter upon zero-field cooling (ZFC) has been determined using extinction-free magnetic x-ray scattering techniques for Fe(0.85)Zn(0.15)F2. This result is consistent with other exponents determined for the RFIM in that Rushbrooke scaling is satisfied. Nevertheless, there is poor agreement with equilibrium computer simulations, and the ZFC results do not agree with field-cooling (FC) results. We present details of hysteresis in Bragg scattering amplitudes and line shapes that help elucidate th…
We develop a percolation model motivated by recent experimental studies of gels with active network remodeling by molecular motors. This remodeling was found to lead to a critical state reminiscent of random percolation (RP), but with a cluster distribution inconsistent with RP. Our model not only can account for these experiments, but also exhibits an unusual type of mixed phase transition: We find that the transition is characterized by signatures of criticality, but with a discontinuity in the order parameter.
Environmental heterogeneity can drive genetic heterogeneity in expanding populations; mutant strains may emerge that trade overall growth rate for an improved ability to survive in patches that are hostile to the wild type. This evolutionary dynamic is of practical importance when seeking to prevent the emergence of damaging traits. We show that a sub-critical slow-spreading mutant can attain dominance even when the density of patches is below their percolation threshold and predict this transition using geometrical arguments. This work demonstrates a phenomenon of ''assisted percolation'', wh…
We propose a kind of Ginzburg--Landau equation with quenched randomness. There is a pinning--depinning transition in the system when the external magnetic force is changed. The transition is self-organized when the external magnetic field is slowly changed under the demagnetizing effect. The total magnetization increases stepwise and the probability distribution of the increase in the total magnetization approximately obeys a power law. A hysteresis loop is obtained when the external magnetic field is changed reciprocally. In our model, the coercivity in the magnetization curve is expressed as…
Spreading processes on networks are ubiquitous in both human-made and natural systems. Understanding their behavior is of broad interest; from the control of epidemics to understanding brain dynamics. While in some cases there exists a clear separation of time scales between the propagation of a single spreading cascade and the initiation of the next -- such that spreading can be modelled as directed percolation or a branching process -- there are also processes for which this is not the case, such as zoonotic diseases or spiking cascades in neural networks. For a large class of relevant netwo…
The Wilson-Cowan model constitutes a paradigmatic approach to understanding the collective dynamics of networks of excitatory and inhibitory units. It has been profusely used in the literature to analyze the possible phases of neural networks at a mean-field level, e.g., assuming large fully-connected networks. Moreover, its stochastic counterpart allows one to study fluctuation-induced phenomena, such as avalanches. Here, we revisit the stochastic Wilson-Cowan model paying special attention to the possible phase transitions between quiescent and active phases. We unveil eight possible types o…
The probability distributions of the order parameter for two models in the directed percolation universality class were evaluated. Monte Carlo simulations have been performed for the one-dimensional generalized contact process and the Domany-Kinzel cellular automaton. In both cases, the density of active sites was chosen as the order parameter. The criticality of those models was obtained by solely using the corresponding probability distribution function. It has been shown that the present method, which has been successfully employed in treating equilibrium systems, is indeed also useful in t…
We investigate variations of the well-known Achlioptas percolation problem, which uses the method of probing sites when building up a lattice system, or probing links when building a network, ultimately resulting in the delay of the appearance of the critical behavior. In the first variation we use two-dimensional lattices, and we apply reverse rules of the Achlioptas model, thus resulting in a speed-up rather than delay of criticality. In a second variation we apply an attractive (and repulsive) rule when building up the lattice, so that newly added sites are either attracted or repelled by t…
We investigate the critical behavior of the entanglement transition induced by projective measurements in (Haar) random unitary quantum circuits. Using a replica approach, we map the calculation of the entanglement entropies in such circuits onto a two-dimensional statistical mechanics model. In this language, the area- to volume-law entanglement transition can be interpreted as an ordering transition in the statistical mechanics model. We derive the general scaling properties of the entanglement entropies and mutual information near the transition using conformal invariance. We analyze in det…
Percolation and criticality of systems with competing interactions on Bethe lattices: limitations and potential strengths of cluster schemes
Sorbonne Université（索邦大学）；Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Duke University（杜克大学）；Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）
It has recently been discovered that random quantum circuits provide an avenue to realize rich entanglement phase diagrams, which are hidden to standard expectation values of operators. Here we study (2+1)D random circuits with random Clifford unitary gates and measurements designed to stabilize trivial area law and topologically ordered phases. With competing single qubit Pauli-Z and toric code stabilizer measurements, in addition to random Clifford unitaries, we find a phase diagram involving a tricritical point that maps to (2+1)D percolation, a possibly stable critical phase, topologically…
Multifractal scaling of critical wave functions at a disorder-driven (Anderson) localization transition is modified near boundaries of a sample. Here this effect is studied for the example of the spin quantum Hall plateau transition using the supersymmetry technique for disorder averaging. Upon mapping of the spin quantum Hall transition to the classical percolation problem with reflecting boundaries, a number of multifractal exponents governing wave function scaling near a boundary are obtained exactly. Moreover, additional exact boundary scaling exponents of the localization problem are extr…
Non-equilibrium critical phenomena generally exist in many dynamic systems, like chemical reactions and some driven-dissipative {reactive} particle systems. Here, by using computer simulation and theoretical analysis, we demonstrate the crucial role of the activation barrier on the criticality of dynamic phase transitions in a minimal reactive hard-sphere model. We find that at zero thermal noise, with increasing the activation barrier, the type of transition changes from a continuous conserved directed percolation into a discontinuous dynamic transition by crossing a \emph{tricritical} point.…
A wide variety of methods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrapping clusters exist on the torus. We apply this approach to percolation in continuum models, finding overlaps between objects with real-valued positions and orientations. In particular, we find precise v…
Probabilistic cellular automata are prototypes of non equilibrium critical phenomena. This class of models includes among others the directed percolation problem (Domany Kinzel model) and the dynamical Ising model. The critical properties of these models are usually obtained by fine-tuning one or more control parameters, as for instance the temperature. We present a method for the parallel evolution of the model for all the values of the control parameter, although its implementation is in general limited to a fixed number of values. This algorithm facilitates the sketching of phase diagrams a…
We investigate the dynamics of two-dimensional quantum spin systems under the combined effect of random unitary gates and local projective measurements. When considering steady states, a measurement-induced transition occurs between two distinct dynamical phases, one characterized by a volume-law scaling of entanglement entropy, the other by an area-law. Employing stabilizer states and Clifford random unitary gates, we numerically investigate square lattices of linear dimension up to $L=48$ for two distinct measurement protocols. For both protocols, we observe a transition point where the domi…
We present a random-matrix realization of a two-dimensional percolation model with the occupation probability $p$. We find that the behavior of the model is governed by the two first extreme eigenvalues. While the second extreme eigenvalue resides on the moving edge of the semicircle bulk distribution with an additional semicircle functionality on $p$, the first extreme exhibits a disjoint isolated Gaussian statistics which is responsible for the emergence of a rich finite-size scaling and criticality. Our extensive numerical simulations along with analytical arguments unravel the power-law di…
