2018
DOI: 10.1103/physrevlett.121.185701
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Random-Length Random Walks and Finite-Size Scaling in High Dimensions

Abstract: We address a long-standing debate regarding the finite-size scaling of the Ising model in high dimensions, by introducing a random-length random walk model, which we then study rigorously. We prove that this model exhibits the same universal FSS behaviour previously conjectured for the self-avoiding walk and Ising model on finite boxes in high-dimensional lattices. Our results show that the mean walk length of the random walk model controls the scaling behaviour of the corresponding Green's function. We numeri… Show more

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Cited by 40 publications
(65 citation statements)
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“…The two-length scaling form ( 8 ) has been numerically confirmed for the 5D Ising model and self-avoiding random walk, with a geometric explanation based on the introduction of an unwrapped length on the torus [ 18 ]. It is also consistent with the rigorous calculations for the so-called random-length random-walk model [ 20 ]. It is noteworthy that the two-length scaling is able to explain both the FSS χ 0 ≡ χ ≍ L 5/2 for the susceptibility (the magnetic fluctuations at the zero Fourier mode) [ 14 ] and the FSS χ k ≍ L 2 for the magnetic fluctuations at nonzero modes [ 15 , 17 ].…”
Section: Introductionsupporting
confidence: 86%
“…The two-length scaling form ( 8 ) has been numerically confirmed for the 5D Ising model and self-avoiding random walk, with a geometric explanation based on the introduction of an unwrapped length on the torus [ 18 ]. It is also consistent with the rigorous calculations for the so-called random-length random-walk model [ 20 ]. It is noteworthy that the two-length scaling is able to explain both the FSS χ 0 ≡ χ ≍ L 5/2 for the susceptibility (the magnetic fluctuations at the zero Fourier mode) [ 14 ] and the FSS χ k ≍ L 2 for the magnetic fluctuations at nonzero modes [ 15 , 17 ].…”
Section: Introductionsupporting
confidence: 86%
“…In Cases (i) and (ii) we have κ n (y) → +∞ for all y ∈ R, and it then follows from (9), (22) and (28) that for all y ∈ R lim n→∞ P n,zn (L n > κ n (y)) =Φ(y) = P(X > y)…”
Section: Proof Of Theorem 12mentioning
confidence: 99%
“…For example, the RLRW picture predicts, and numerical evidence strongly suggests [17,22,23], that the Papathanakos scaling of the two-point function can be observed with free boundary conditions, at an appropriate pseudocritical point, thus clarifying a recent debate [24,14,25] on the matter. Moreover, by studying the complete-graph SAW on a general family of pseudocritical points, it was shown [18] that a continuous family of scalings for the mean walk length was possible, and numerical evidence [17,18] strongly suggested these results extend to tori with d ≥ 5. When inputted into the RLRW model, these asymptotic mean lengths then predict a continuous family of possible anomalous scalings of the two-point function on the torus, providing a broad generalisation of the Papathanakos conjecture.…”
Section: Introductionmentioning
confidence: 99%
“…For finite graphs of large girth, a dense phase was proven to exist in [12] (where, as mentioned above, the complete graph was also considered). Forthcoming work on the complete graph announced in [13] subsequently appeared in [3] (see Note Added at end of Section 1.4).…”
Section: Critical Behaviourmentioning
confidence: 99%