Near critical scaling relations for planar Bernoulli percolation without differential inequalities

We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin-Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov but with the important difference that it describes the distribution of the volume of a cluster rather than of its radius. We apply this differential inequality to prove the following: The critical exponent inequalities $\gamma \leq \delta-1$ and $\Delta \leq \gamma +1$ hold for percolation and the random cluster model on any transitive graph. These inequalities are new even in the context of Bernoulli percolation on $\mathbb{Z}^d$, and are saturated in mean-field for Bernoulli percolation and for the random cluster model with $q \in [1,2)$. The volume of a cluster has an exponential tail in the entire subcritical phase of the random cluster model on any transitive graph. This proof also applies to infinite-range models, where the result is new even in the Euclidean setting.

We find the critical exponents for global in time solutions to differential inequalities with power nonlinearities, supplemented by an initial data condition. The operator for which the differential inequality is studied contains a Caputo or Riemann-Liouville time derivative of fractional order and a sum of homogeneous spatial partial differential operators. In the special case of a fractional diffusive equation, the obtained critical exponents are sharp. In particular, global existence of small data solutions to the fractional diffusive equation with Caputo and Riemann-Liouville time derivative of order in (0, 1) and in (1, 2), holds for supercritical powers. The existence result for the superdiffusive case (α ∈ (1, 2)), which interpolates a semilinear heat equation and a semilinear wave equation, was recently obtained in the general setting by the author and his collaborators. We use a simple representation of Mittag-Leffler functions to show that global existence of small data solutions for supercritical powers also holds for to the subdiffusive equation with Caputo and Riemann-Liouville time derivative (α ∈ (0, 1)).

Let us consider subcritical Bernoulli percolation on a connected, transitive, infinite and locally finite graph. In this paper, we propose a new (and short) proof of the exponential decay property for the volume of clusters. We do not rely on differential inequalities and rather use stochastic comparison techniques, which are inspired by several works including the paper An approximate zero-one law written by Russo in the early eighties.

We use the finiteness of the triangle diagram in order to establish that certain critical exponents take on their mean-field values. We again rely on the differential inequalities developed in chapter 3, and complement them with a differential inequality involving the triangle diagram. We then prove that, under the triangle condition, the critical exponents \(\delta \) and \(\beta \) take on their mean-field values \(\delta \) = 2 and \(\beta \) = 1.

On the critical behavior for inhomogeneous parabolic differential inequalities in the half-space

Let θ(J) be the order parameter of a (ferromagnetic) Potts or randomcluster process with bond-variables J = (J e : e ∈ K). We discuss differential inequalities of the form $$ \frac{{\partial \theta }}{{\partial {J_e}}} \leqslant \alpha (J)\quad for\,all\,e,f \in K $$ . Such inequalities may be established for all random-cluster processes that satisfy the FKG inequality, possibly in the presence of many-body interactions (subject to certain necessary and sufficient conditions on the sets of interactions). There are (at least) two principal consequences of this. First, for a process having ‘inverse-temperature’ β, the critical value s c = s c (J) is a strictly monotone function of J. Secondly, at any fixed point J lying on the critical surface of the process, the critical exponent of θ in the limit as J’ ↓ J is independent of the direction of approach of the limit. Such a conclusion should be valid for other critical exponents also; this amounts to a small amount of rigorous universality.

For a family of translation-invariant, ferromagnetic, one-component spin systems—which includes Ising and ϕ4 models—we prove that (i) the phase transition is sharp in the sense that at zero magnetic field the high- and low-temperature phases extend up to a common critical point, and (ii) the critical exponent β obeys the mean field bound β⩽1/2. The present derivation of these nonperturbative statements is not restricted to “regular” systems, and is based on a new differential inequality whose Ising model version isM⩽βhχ+M3+ βM2∂M/∂β. The significance of the inequality was recognized in a recent work on related problems for percolation models, while the inequality itself is related to previous results, by a number of authors, on ferromagnetic and percolation models.

We establish nonexistence results to systems of differential inequalities on the (2N + 1)‐Heisenberg group. The systems considered here are of the type (ESm). These nonexistence results hold for N less than critical exponents which depend on pi and γi, 1 ≤ i ≤ m. Our results improve the known estimates of the critical exponent.

The equality of two critical points - the percolation threshold pH and the point pτ where the cluster size distribution ceases to decay exponentially - is proven for all translation invariant independent percolation models on homogeneous d-dimensional lattices (d^ 1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter M(β, which for h = Q reduces to the percolation density P^ - at the bond density p = l—e~β in the single parameter case. These are: (1) M^hdM/dh + M2 + βMdM/dβ, and (2) dM/dβ^\J\MdM/dh. Inequality (1) is intriguing in that its derivation provides yet another hint of a φ3 structure in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents β and δ. One of these resembles an Ising model inequality of Frόhlich and Sokal and yields the mean field bound (5^2, and the other implies the result of Chayes and Chayes that β^ί. An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation /?((5 —1)^1 and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.

Consider critical Bernoulli percolation in the plane. We give a new proof of the sharp noise sensitivity theorem shown by Garban, Pete, and Schramm (Acta Math. 205 (2010), 19–104). Contrary to the previous approaches, we do not use any spectral tool. We rather study differential inequalities satisfied by a dynamical four‐arm event, in the spirit of Kesten's proof of scaling relations in Kesten (Comm. Math. Phys. 109 (1987), 109–156). We also obtain new results in dynamical percolation. In particular, we prove that the Hausdorff dimension of the set of times with both primal and dual percolation equals almost surely.

We state and prove the celebrated result that the critical value where the expected cluster size blows equals that where the probability of a vertex being in an infinite component agree. This uniqueness of the phase transition was independently proved by Menshikov and by Aizenman and Barsky. This theorem is the starting point of the investigation of high-dimensional percolation. We start with the recent and beautiful Duminil-Copin and Tassion proof, continuing with the original Aizenman and Barsky proof that relies on differential inequalities for the so-called percolation magnetization. The Aizenman-Barsky differential inequalities also play a pivotal role in the identification of mean-field critical exponents for percolation in high dimensions.

Known differential inequalities for certain ferromagnetic Potts models with pair interactions may be extended to Potts models with many-body interactions. As a major application of such differential inequalities, we obtain necessary and sufficient conditions on the set of interactions of such a Potts model in order that its critical point be astrictly monotonic function of the strengths of interactions. The method yields some ancillary information concerning the equality of certain critical exponents for Potts models; this amounts to a small amount of rigorous universality. These results are achieved in the context of a “Fortuin-Kasteleyn representation” of Potts models with many-body interactions. For such a Potts model, the corresponding random-cluster process is a (random) hypergraph.

Under a certain assumption (which is satisfied whenever there is a dense infinite cluster in the half-space), we prove a differential inequality for the infinite-cluster density, ${P}_{\ensuremath{\infty}}(p)$, in Bernoulli percolation. The principal implication of this result is that if ${P}_{\ensuremath{\infty}}(p)$ vanishes with critical exponent $\ensuremath{\beta}$, then $\ensuremath{\beta}$ obeys the mean-field bound $\ensuremath{\beta}<~1$. As a corollary, we also derive an inequality relating the backbone density, the truncated susceptibility, and the infinite-cluster density.

While undergoing gelation transition, a material passes through a distinctive state called the critical gel state. In the neighborhood of this critical gel state, how viscosity, equilibrium modulus, and relaxation times evolve are correlated by scaling relations, and their universality has been validated for materials undergoing the sol-gel transition. In this work, we extend this approach for the gel-sol transition of a thermoresponsive polymeric system of aqueous poly(vinyl alcohol) (PVOH) gel that passes through the critical state upon increasing temperature. We observe that, in the neighborhood of the critical gel state, the equilibrium modulus and viscosity demonstrate a power law dependence on the relative distance from the critical state in terms of normalized temperature. Furthermore, the relaxation times in the gel and the sol state shows symmetric power law divergence near the critical state. The corresponding critical power law exponents and the dynamic critical exponents computed at the critical gel-sol transition state validate the scaling and hyperscaling relations originally proposed for the critical sol-gel transition very well. Remarkably, the dependence of complex viscosity on frequency at different temperatures shows a comprehensive master curve irrespective of the temperature ramp rate independently in the gel and the sol state. This observation demonstrates how the shape of relaxation time spectrum is independent of both the temperature as well as the ramp rate. Since sol-gel and the gel-sol transitions are opposite to each other, the applicability of the scaling relations validated in this work suggests broader symmetry associated with how the structure evolves around the critical state irrespective of the direction.

GETELINA, J. C. A. On the critical behavior of the XX spin-1/2 chain under correlated quenched disorder. 2016. 82 p. Dissertation (Master in Science) Instituto de Fisica de Sao Carlos, Universidade de Sao Paulo, Sao Carlos, 2016. This work provides a full description of the critical behavior of the XX spin-1/2 chain under correlated quenched disorder. Previous investigations have shown that the introduction of correlation between couplings in the random XX model gives rise to a novel critical behavior, where the infinite-randomness critical point of the uncorrelated case is replaced by a family of finite-disorder critical points that depends on the disorder strength. Here it is shown that most of the critical exponents of the XX model with correlated randomness are equal to clean (without disorder) chain values and do not depend on disorder strength, except the critical dynamical exponent and the anomalous dimension. The former increases monotonically with disorder strength, whereas the results obtained for the latter are unreliable. Furthermore, the scaling relations between the critical exponents were also tested and it was found that those involving the system dimensionality, namely the hyperscaling and Fisher’s scaling relations, are not respected. Measurements of the Renyi entanglement entropy of the system at criticality have also been performed, and it is shown that the scaling behavior of the correlated-disorder case is similar to the theoretical prediction for the clean chain, displaying the same finite-size correction and a disorder-dependent effective central charge in the leading term of the scaling. Further corrections to the scaling of the entanglement entropy were also investigated, but the results are inconclusive. The model was studied via exact numerical diagonalization of the corresponding Hamiltonian.