Critical Two-Point Function for Long-Range O(n) Models Below the Upper Critical Dimension

Abstract

We consider the n-component |ϕ| 4 lattice spin model (n ≥ 1) and the weakly self-avoiding walk (n = 0) on Z d , in dimensions d = 1, 2, 3. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance r as r −(d+α) with α ∈ (0, 2). The upper critical dimension is d c = 2α. For ǫ > 0, and α = 1 2 (d+ǫ), the dimension d = d c −ǫ is below the upper critical dimension. For small ǫ, weak coupling, and all integers n ≥ 0, we prove that the two-point function at the critical point decays with distance as r −(d−α) . This "sticking" of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion. Introduction and main resultBroadly speaking, the mathematical understanding of critical phenomena for spin systems has progressed in dimension d = 2, where exact solutions and SLE are important tools; in dimensions d > 4, where infrared bounds and the lace expansion are useful; and in dimension d = 4, where renormalisation group (RG) methods have been applied. The physically most important case of d = 3 is more difficult, and mathematical methods are scarce.In the physics literature, the ǫ-expansion was introduced to study non-integer dimensions slightly below d = 4. An alternate approach is to consider long-range models, which change the upper critical dimension from d c = 4 to a lower value d c = 2α with α ∈ (0, 2). By choosing d = 1, 2, 3 and α = 1 2 (d + ǫ) with small ǫ, it is possible to study integer dimension d which is slightly below the upper critical dimension 2α = d + ǫ. In this paper, we consider n-component spins and the weakly self-avoiding walk in this long-range context, and prove that the critical twopoint function has mean-field decay r −(d−α) also below the upper critical dimension. Our method involves a RG analysis in the vicinity of a non-Gaussian fixed point.

Critical Exponents for Long-Range $${O(n)}$$ O ( n ) Models Below the Upper Critical Dimension

“…Interestingly, the critical exponent η was predicted to "stick" at the mean-field value η = 2 − α to all orders in ǫ [46,83] (we are interested here only in d = 1, 2, 3 and small ǫ and not in the range corresponding to crossover to short-range behaviour [19,23,63]). This has been proved very recently [71] for all n ≥ 0, using an extension of the methods we develop here. Earlier, a proof that η = 2 − α for small ǫ was announced by Mitter for a 1-component continuum model [76].…”

“…1. Very recently the |x| −(d−α) decay of the critical two-point function ϕ 0 · ϕ x νc has been proved for n ≥ 1, as well as for its n = 0 counterpart [71]. This required the introduction of observables into the renormalisation group analysis presented here.…”

Critical Two-Point Function for Long-Range Models with Power-Law Couplings: The Marginal Case for $${d\ge d_{\rm c}}$$

“…Suppose that the above physics prediction is true for the O(n) model as well, and that η > 0 for d < 4. Then, we can take a small ε > 0 to satisfy α = d+ε 2 ∈ ( d 2 , 2 − η) = ∅, hence d = 2α − ε < d c , and yet G pc (x) is proven to decay as |x| α−d [23]. This "sticking" at the mean-field behavior, even below the upper-critical dimension, has been proven by using a rigorous version of the ε-expansion.…”

“…If so, then sufficient conditions for the mean-field behavior, called the bubble condition for selfavoiding walk and the Ising model [1,24] and the triangle condition for percolation [6], hold for all dimensions above the model-dependent upper-critical dimension d c , which is 2m for short-range models, where m = 2 for self-avoiding walk and the Ising model and m = 3 for percolation. In recent years, long-range models defined by power-law couplings, D(x) ≈ |x| −d−α for some α > 0, have attracted more attention, due to unconventional critical behavior and crossover phenomena (e.g., [7,10,15,23]). Under some mild assumptions, we have shown [15, Proposition 2.1] that, for α = 2 and d > α ∧ 2, the random-walk Green function S 1 (x) is asymptotically γα vα |x| α∧2−d , where…”

The Tensor Track VI: Field Theory on Random Trees and SYK on Random Unicyclic Graphs

“…Regarding now inclusion of two-point subgraphs, there is evidence from the long history of the critical behaviour of long-range models 3 that since divergences and the required counterterms are analytic functions of the external momenta, the fractional propagator is not renormalized [19,20].…”