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3 The Near-Critical Scaling Limit
Using the full scaling limit, one can attempt to understand the geometry of the nearcritical scaling limit, where the percolation density tends to the critical one in an appropriate way as the lattice spacing δ tends to zero:
p = pc + λδα |
(1) |
where pc is the critical density, δ is the lattice spacing, λ (−∞, ∞), and α is set equal to 1/ν = 3/4 (where ν is the correlation length exponent) to get nontrivial λ-dependence in the limit δ → 0 (as prescribed by “scaling theory,” see also [1–3, 9]).
A heuristic analysis [16, 17] of the near-critical scaling limit leads to a oneparameter family of loop models (i.e., probability measures on random collections of loops), with the critical full scaling limit corresponding to a particular choice of the parameter (λ = 0). Except for the latter case, these measures are not scale invariant, but are mapped into one another by scale transformations.
The approach proposed in [16, 17] is based on a “Poissonian marking” of double points of the critical full scaling limit of [12, 13], i.e., points where two loops touch each other or a loop touches itself. These double points of the loop process in the plane are precisely the continuum limit of “macroscopically pivotal” lattice locations; each such site is microscopic, but such that a change in its state (e.g., black to white or closed to open) has a macroscopic effect on connectivity. For site percolation on the triangular lattice (or equivalently random black/white colorings of the hexagonal lattice – see Fig. 1), a macroscopically pivotal site is a hexagon at the center of four macroscopic arms with alternating colors – see Fig. 3.
Fig. 3 Schematic diagram of a macroscopically pivotal hexagon at the center of four macroscopic arms with alternating color. The full and dashed lines represent paths of white and black hexagons respectively
The Scaling Limit of (Near-)Critical 2D Percolation |
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The analysis presented in [16, 17] is based on the random marking of countably many double points, with each of these labelled by a number in (−∞, ∞) representing the value of λ at which that double point changes its state and hence correspondingly changes the macroscopic connectivity. This yields a realization on a single probability space of all the scaling limits as λ varies in (−∞, ∞). We point out that most double points are not marked since they do not change their state for a finite value of λ (in the limit δ → 0) – it is only the marked ones that change.
This approach can be used to define a renormalization group flow (under the action of dilations), and to describe the scaling limit of related models, such as invasion and dynamical percolation and the minimal spanning tree. In particular, this analysis helps explain why the scaling limit of the minimal spanning tree may be scale invariant but not conformally invariant, as first observed numerically by Wilson [37].
In [16, 17] some geometric properties of the near-critical scaling limit of twodimensional percolation are conjectured, including the fact that for any λ = 0, every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero (to be contrasted with the case of the critical full scaling limit, λ = 0, where there is no largest loop around any point).
The analysis done in [16, 17] is nonrigorous, and the purpose of the authors is not to prove theorems but rather to propose a conceptual framework rich enough to treat scaling limits of near-critical percolation and of related lattice objects like the minimal spanning tree.
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