scholarly journals DIRECTED PERCOLATION AND OTHER SYSTEMS WITH ABSORBING STATES: IMPACT OF BOUNDARIES

2001 ◽  
Vol 15 (12) ◽  
pp. 1761-1797 ◽  
Author(s):  
PER FRÖJDH ◽  
MARTIN HOWARD ◽  
KENT BÆKGAARD LAURITSEN
Keyword(s):  
Critical Behavior ◽  
Mean Field ◽  
Critical Dimension ◽  
Scaling Theory ◽  
Directed Percolation ◽  
Upper Critical Dimension ◽  
Duality Symmetry ◽  
Universality Classes ◽  
Absorbing States ◽  
The Impact

We review the critical behavior of nonequilibrium systems, such as directed percolation (DP) and branching-annihilating random walks (BARW), which possess phase transitions into absorbing states. After reviewing the bulk scaling behavior of these models, we devote the main part of this review to analyzing the impact of walls on their critical behavior. We discuss the possible boundary universality classes for the DP and BARW models, which can be described by a general scaling theory which allows for two independent surface exponents in addition to the bulk critical exponents. Above the upper critical dimension d c , we review the use of mean field theories, whereas in the regime d<d c , where fluctuations are important, we examine the application of field theoretic methods. Of particular interest is the situation in d=1, which has been extensively investigated using numerical simulations and series expansions. Although DP and BARW fit into the same scaling theory, they can still show very different surface behavior: for DP some exponents are degenerate, a property not shared with the BARW model. Moreover, a "hidden" duality symmetry of BARW in d=1 is broken by the boundary and this relates exponents and boundary conditions in an intricate way.

scholarly journals Power-law bounds for critical long-range percolation below the upper-critical dimension

2021 ◽  
Author(s):  
Tom Hutchcroft
Keyword(s):  
Long Range ◽  
Power Law ◽  
Upper Bound ◽  
Mean Field ◽  
Maximum Size ◽  
Finite Graph ◽  
Critical Dimension ◽  
Point Function ◽  
Upper Critical Dimension ◽  
Infinite Cluster

AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

ISING SPIN GLASS: RECENT PROGRESS IN THE FIELD THEORY APPROACH

1993 ◽  
Vol 07 (01n03) ◽  
pp. 986-992 ◽  
Author(s):  
C. DE DOMINICIS ◽  
I. KONDOR ◽  
T. TEMESVARI
Keyword(s):  
Field Theory ◽  
Mean Field Theory ◽  
Mean Field ◽  
Critical Dimension ◽  
Zero Field ◽  
Theory Approach ◽  
Replica Symmetry ◽  
Numerical Work ◽  
Upper Critical Dimension ◽  
Ising Spin

A loop expansion around Parisi's replica symmetry breaking mean field theory is constructed, in zero field. We obtain the equation of state (and associated Parisi's solution) below the upper critical dimension du=6, and, in particular, explicit corrections in εlnt and (εlnt)2 with t=(Tc−T)/Tc and ε=6−d. This allows us to verify that standard scaling is satisfied with β=1+ε/2+0(ε2). We have also investigated the transverse (replicon) correlation function, particularly at zero overlap. Near du, G00 (p) ~ t/P4 (for p~0) is the most obvious obstacle to a meaningful theory in d=3. If we make the twofold assumption that (i) scaling applies, and (ii) the replicon propagator is dominated by the spectrum of the (small) transverse masses, we obtain the softer behavior G00(p) ~ (1/p2–η). (t/p1/v)β and a prediction for a soft replicon spectrum in tγk2γ/β γ/β instead of tk2 at du. We have checked tγ to one loop and work is in progress to check k2γ/β to the same order. Taking the above divergence in p−(d+2−η)/2 as the leading divergence defines the lower critical dimension d ℓ by d ℓ=2−η (d ℓ). Known values of η (at d=6, from the ε expansion at Tc, or from numerical work at d=4, 3) are compatible with d ℓ ~ 2.5±.3.

scholarly journals NoBLE for Lattice Trees and Lattice Animals

2021 ◽  
Vol 185 (2) ◽  
Author(s):  
Robert Fitzner ◽  
Remco van der Hofstad
Keyword(s):  
Random Walk ◽  
Nearest Neighbor ◽  
Mean Field ◽  
Critical Dimension ◽  
Simple Random Walk ◽  
Computer Assisted ◽  
High Dimensions ◽  
Lace Expansion ◽  
Upper Critical Dimension ◽  
Lattice Trees

AbstractWe study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice $${\mathbb {Z}}^d$$ Z d in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension $$16$$ 16 and $$17$$ 17 , respectively. Such results have previously been obtained by Hara and Slade in sufficiently high dimensions. The dimension above which their results apply was not yet specified. We rely on the non-backtracking lace expansion (NoBLE) method that we have recently developed. The NoBLE makes use of an alternative lace expansion for LAs and LTs that perturbs around non-backtracking random walk rather than around simple random walk, leading to smaller corrections. The NoBLE method then provides a careful computational analysis that improves the dimension above which the result applies. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to $$d_c=8$$ d c = 8 for both models, as is known for sufficiently spread-out models by the results of Hara and Slade mentioned earlier. The main ingredients in this paper are (a) a derivation of a non-backtracking lace expansion for the LT and LA two-point functions; (b) bounds on the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) sharp numerical bounds on the coefficients. Our proof is complemented by a computer-assisted numerical analysis that verifies that the necessary bounds used in the NoBLE are satisfied.

SPATIAL MODULATION OF ORDER PARAMETERS AND CRITICAL DIMENSIONS

2008 ◽  
Vol 22 (07) ◽  
pp. 851-857 ◽  
Author(s):  
A. V. BABICH ◽  
S. V. BEREZOVSKY ◽  
V. F. KLEPIKOV
Keyword(s):  
Heat Capacity ◽  
Critical Behavior ◽  
Scale Invariance ◽  
Order Parameter ◽  
Space Dimension ◽  
Critical Dimension ◽  
Spatial Modulation ◽  
Upper Critical Dimension ◽  
Critical Dimensions ◽  
Lifshitz Point

Critical phenomena in systems which have a joint multicritical and Lifshitz-point-like behavior are investigated. An expression for the upper critical dimension of such systems is found. A condition of the variational scale invariance of the considered model is specified. If the space dimension coincides with the upper critical dimension, then the critical behavior of the fluctuation correction to the heat capacity is similar to that of the heat capacity without order parameter modulation.

scholarly journals Non-perturbative effects in spin glasses

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Michele Castellana ◽  
Giorgio Parisi
Keyword(s):  
Fixed Point ◽  
Ising Model ◽  
Spin Glasses ◽  
Numerical Study ◽  
Mean Field ◽  
Critical Dimension ◽  
Upper Critical Dimension ◽  
Ising Spin ◽  
Ferromagnetic Ising Model ◽  
The Mean

Abstract We present a numerical study of an Ising spin glass with hierarchical interactions—the hierarchical Edwards-Anderson model with an external magnetic field (HEA). We study the model with Monte Carlo (MC) simulations in the mean-field (MF) and non-mean-field (NMF) regions corresponding to d ≥ 4 and d < 4 for the d-dimensional ferromagnetic Ising model respectively. We compare the MC results with those of a renormalization-group (RG) study where the critical fixed point is treated as a perturbation of the MF one, along the same lines as in the "Equation missing"-expansion for the Ising model. The MC and the RG method agree in the MF region, predicting the existence of a transition and compatible values of the critical exponents. Conversely, the two approaches markedly disagree in the NMF case, where the MC data indicates a transition, while the RG analysis predicts that no perturbative critical fixed point exists. Also, the MC estimate of the critical exponent ν in the NMF region is about twice as large as its classical value, even if the analog of the system dimension is within only ~2% from its upper-critical-dimension value. Taken together, these results indicate that the transition in the NMF region is governed by strong non-perturbative effects.

LINE TENSION AT WETTING

1994 ◽  
Vol 08 (03) ◽  
pp. 309-345 ◽  
Author(s):  
J. O. INDEKEU
Keyword(s):  
Fundamental Problem ◽  
Mean Field Theory ◽  
Mean Field ◽  
Thermal Fluctuation ◽  
Line Tension ◽  
Critical Dimension ◽  
Intermolecular Forces ◽  
Wetting Transition ◽  
Upper Critical Dimension ◽  
Three Phase

A review is presented of recent theoretical advances on a fundamental problem in statistical mechanics that concerns the three-phase contact line ℒ and its tension τ near a wetting phase transition. In addition to answering the intriguing question whether or not ℒ and τ vanish at wetting, recent work has also revealed that τ displays universal singular behavior, reflecting critical phenomena associated with the wetting transition. Three factors are crucial for determining the fate of ℒ and τ at wetting: the order of the wetting transition, the range of the intermolecular forces, and the upper critical dimension du, above which mean-field theory holds and below which fluctuations dominate. For most systems studied experimentally, du < 3, so that the mean-field predictions should be correct in d = 3. In the thermal fluctuation regime, for d < du, hyperscaling predicts the value 2(d – 2)/(d – 1) for the critical exponent of τ(θ), in the limit that the contact angle θ approaches 0.

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