General form of the probability distribution function (PDF) of the rescaled fractal dimension characterizing spatial structures of local order parameters in finite but large systems near the critical point is derived. To obtain the PDF, we assume that (i) sample-to-sample fluctuations of global order parameters are described by the generalized Gumbel distribution function, which is known to be valid for many of long-range correlated systems such as critical systems and (ii) spatial structures of local order parameters are always fractal at the critical point. From the obtained PDF, it is elucidated that fluctuations of fractal dimensions do not exhibit a universal character in contrast to the case of fluctuating global order parameters. In addition, we perform numerical calculations of the PDF for two-dimensional percolation systems near the critical concentration and confirm that the analytically calculated PDF well describes numerical results. We also discuss the influence of non-fractal critical clusters neglected under the assumption (ii).
General form of the probability distribution function (PDF) of the rescaled fractal dimension characterizing spatial structures of local order parameters in finite but large systems near the critical point is derived. To obtain the PDF, we assume that (i) sample-to-sample fluctuations of global order parameters are described by the generalized Gumbel distribution function, which is known to be valid for many of long-range correlated systems such as critical systems and (ii) spatial structures of local order parameters are always fractal at the critical point. From the obtained PDF, it is elucidated that fluctuations of fractal dimensions do not exhibit a universal character in contrast to the case of fluctuating global order parameters. In addition, we perform numerical calculations of the PDF for two-dimensional percolation systems near the critical concentration and confirm that the analytically calculated PDF well describes numerical results. We also discuss the influence of non-fractal critical clusters neglected under the assumption (ii).
Abstract Understanding phase transitions requires not only identifying order parameters but also characterizing how their correlations behave across scales. By quantifying how fluctuations at distinct spatial or temporal points are related, correlation functions reveal the structural organization of complex systems. In this work, we reexamine the theoretical foundations of these correlations in systems undergoing second-order phase transitions, with emphasis on the Ising model extended to non-integer spatial dimensions. We revisit the hypotesis that, at criticality, the equilibrium dynamics become effectively confined to the fractal edge of spin clusters and redo the analysis using fractional calculus. Within this framework, the fractal dimension that governs the correlations in that subspace is directly related to Fisher’s exponent η η , which quantifies the singular behavior of the correlation function near criticality. Importantly, this correlation fractal dimension is distinct from the fractal dimension associated with the order parameter. The fractional approach allows us to directly compute the correlation fractal dimension and to establish an explicit geometrical relation connecting the two fractal dimensions. Moreover, the formulation naturally extends to non-integer spatial dimensions, remaining valid below the upper critical dimension and yielding the correct value of Fisher’s exponent η η for a continuous spatial dimension d . Within this framework, we also provide empirical functions describing how the main critical exponents vary continuously as a function of the space dimension.
Abstract Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point. Focusing on two-dimensional percolation, we show that the size distribution of the largest cluster asymptotically approaches to a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and transitions within the critical finite-size scaling window. Numerical results indicate that, at consistently defined pseudocritical points, this distribution exhibits a universal form across various lattices and percolation models (bond or site), within error bars, yet differs from the distribution at the critical point. The critical polynomial, universally zero for two-dimensional percolation at the critical point, becomes nonzero at pseudocritical points. Nevertheless, numerical evidence suggests that the critical polynomial, along with other dimensionless quantities such as wrapping probabilities and Binder cumulants, assumes fixed values at the pseudocritical point that are independent of the percolation type (bond or site) but vary with lattice structures. These findings imply that while strict universality breaks down at the pseudocritical point, certain extreme-value statistics and dimensionless quantities exhibit quasi-universality, revealing a subtle connection between scaling behaviors at critical and pseudocritical points.
We investigate the geometry of a critical system undergoing a second-order thermal phase transition. Using a local description for the dynamics characterizing the system at the critical point T=T(c), we reveal the formation of clusters with fractal geometry, where the term cluster is used to describe regions with a nonvanishing value of the order parameter. We show that, treating the cluster as an open subsystem of the entire system, new instanton-like configurations dominate the statistical mechanics of the cluster. We study the dependence of the resulting fractal dimension on the embedding dimension and the scaling properties (isothermal critical exponent) of the system. Taking into account the finite-size effects, we are able to calculate the size of the critical cluster in terms of the total size of the system, the critical temperature, and the effective coupling of the long wavelength interaction at the critical point. We also show that the size of the cluster has to be identified with the correlation length at criticality. Finally, within the framework of the mean field approximation, we extend our local considerations to obtain a global description of the system.
We study the probability distribution function (PDF) of the order parameter of the three-dimensional O(N) model at criticality using the functional renormalization group. For this purpose, we generalize the method introduced by Balog et al. [Phys. Rev. Lett. 129, 210602 (2022)0031-900710.1103/PhysRevLett.129.210602] to the O(N) model. We study the large N limit, as well as the cases N=2 and N=3 at the level of the local potential approximation (LPA), and compare our results to Monte Carlo simulations. We compute the entire family of universal scaling functions, obtained in the limit where the system size L and the correlation length of the infinite system ξ_{∞} diverge, with the ratio ζ=L/ξ_{∞} constant. We also generalize our results to the approach of criticality from the low-temperature phase where another infinite family of universal PDF exists. We find that the LPA describes very well the functional form of the family of PDFs, once we correct for a global amplitude of the (logarithm of the) PDF and of ζ.
Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments rho(q) (r) of the local order parameter rho(r) scale with a set x(q) of nontrivial exponents x(q) not = qx(1). We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure w(r) normalized over space sum(r) w(r) = 1 occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments Y(q) = sum(r) w(q) (r), since they may scale with different generalized dimensions D(q) and D(q), and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents c = 1.5 (marginal disorder) and c = 1.75 (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value x(q-->infinity) = d and the minimal value alpha(min) = D(q-->infinity) = d - x(1) of the typical multifractal spectrum f(alpha).
Large-deviation tails of critical probability distributions provide a sensitive probe of universality beyond standard finite-size scaling. We study these tails for critical percolation and Fortuin--Kasteleyn Ising models on two-dimensional lattices, three-dimensional lattices, and complete graphs. We consider two rescaled order parameters: the magnetization-like variable xm=∣M∣/⟨∣M∣⟩, including a signed cluster-mass analogue for percolation, and the largest-cluster variable xC=C1/⟨C1⟩. For xm, we test the expected stretched-exponential large-deviation tail and show that the same form applies to the percolation analogue. For xC, guided by the exact complete-graph result and scaling arguments, we propose universal scaling forms for both tails of the cumulative distribution and test them by extensive Monte Carlo simulations. In the complete-graph FK-Ising model, the left tail is governed by rare configurations with percolation-like scaling rather than by the typical Ising scaling. Our results show that the tails of order-parameter distributions reveal universal features of critical fluctuations that are not captured by averaged observables alone.