Random-length Random Walks and Finite-size scaling in high dimensions
Zongzheng Zhou, Jens Grimm, Sheng Fang, Youjin Deng, Timothy M. Garoni
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Random-length Random Walks and Finite-size Scaling in high dimensions
Zongzheng Zhou, Jens Grimm, ∗ Sheng Fang, Youjin Deng, † and Timothy M. Garoni ‡ ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS),School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China National Laboratory for Physical Sciences at Microscale,University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: November 2, 2018)We address a long-standing debate regarding the finite-size scaling of the Ising model in highdimensions, 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 theself-avoiding walk and Ising model on finite boxes in high-dimensional lattices. Our results show thatthe mean walk length of the random walk model controls the scaling behaviour of the correspondingGreen’s function. We numerically demonstrate the universality of our rigorous findings by extensiveMonte Carlo simulations of the Ising model and self-avoiding walk on five-dimensional hypercubiclattices with free and periodic boundaries.
Finite-size Scaling (FSS) [1, 2] is a fundamental the-ory which characterizes the asymptotic approach of finitesystems to the thermodynamic limit, close to a contin-uous phase transition. While critical systems above theupper critical dimension d c exhibit simple mean-field be-haviour in the thermodynamic limit [3], their FSS be-haviour above d c is surprisingly subtle and the subjectof long-standing debate; see e.g. [4–10]. In this work,we clarify a number of these subtleties by introducing asimple model, which can be studied rigorously.The n -vector model [11], which describes interactingspin systems on a lattice, plays a central role in vari-ous areas of physics such as statistical mechanics andcondensed matter physics. Prominent examples are theSelf-avoiding Walk (SAW) ( n →
0) in polymer physics,and the Ising ( n =
1) and XY ( n =
2) models of ferromag-netism. The latter can be related to the Bose-Hubbardmodel [12] which describes bosonic atoms in an opticallattice.On an infinite hypercubic lattice Z d , it is known rigor-ously [13, 14] that for sufficiently large dimension d , thetwo-point functions of the critical Ising and SAW modelsexhibit the same scaling behaviour as the Green’s func-tion of the Simple Random Walk (SRW). On finite lat-tices this connection breaks down because SRW is recur-rent, implying that its Green’s function does not exist.In this Letter, we argue that if one considers randomwalks with an appropriate random (finite) length N , thenthe Green’s function displays the same finite-size scalingas the two-point functions of the SAW and Ising models,defined on boxes in Z d of linear size L . For this Random-length Random Walk (RLRW) model, one can prove [15]that if d ≥ ⟨N ⟩ ≍ L µ with µ ≥
2, then the Green’sfunction scales as g ( x ) ≍ ⎧⎪⎪⎨⎪⎪⎩∥ x ∥ − d , ∥ x ∥ ≤ O ( L ( d − µ )/( d − ) ) L µ − d , ∥ x ∥ ≥ O ( L ( d − µ )/( d − ) ) . (1)In words, if µ > g ( x ) exhibits the standard infinite- lattice asymptotic decay ∥ x ∥ − d at moderate values of x , but then enters a plateau of order L µ − d which per-sists to the boundary. Since a typical RLRW will exploredistances of order √⟨N ⟩ from the origin, no plateau ex-ists for µ < g ( x ) decays significantlyfaster [15] than ∥ x ∥ − d for ∥ x ∥ ≫ √⟨N ⟩ .The above scaling behaviour of the Green’s functionholds on boxes with both free and periodic boundaries.As a consequence of this scaling [16], one can prove [15]that the corresponding susceptibility scales as χ ≍ L µ , for any µ >
0. (2)The mean walk length of SAW, restricted to a finitebox in Z d , depends strongly on the boundary conditionsimposed. For a given choice of SAW boundary condi-tions, one can consider a RLRW where ⟨N ⟩ is chosen toscale in the same way as it does for the SAW. Our numer-ical results below strongly suggest that the scaling of theGreen’s function of this RLRW model, given by Eq. (1),then correctly predicts the two-point function scaling ofthe corresponding SAW model. We therefore concludethat the SAW two-point function is only affected by ge-ometry via its effect on the mean walk length. Theseobservations are seen to hold not only at the thermody-namic critical point, but also at general pseudo-criticalpoints. We numerically demonstrate the universality ofthese predictions by showing that they also correctly de-scribe the FSS behaviour of the Ising two-point function.These observations shed light on a number of openquestions regarding the FSS behaviour of the Ising modelabove d c . For periodic boundary conditions (PBC) atcriticality, the scaling of the Ising two-point functionhas been actively debated in [6–8]. The known [17] be-haviour of the mean walk length of SAW on the completegraph [18],together with extensive Monte Carlo simula-tions in five dimensions, suggest that on high-dimensionaltori at criticality we have ⟨N ⟩ SAW ≍ L d / . We therefore -1 (a) g ( y L / ) L / y SAW, L =131SAW, L =191SAW, L =221Ising, L =31Ising, L =7110 -1 g ( y L ) L y y SAW, L =101SAW, L =121SAW, L =141Ising, L =31Ising, L =71 Figure 1: Appropriate scaled two-point functions of theIsing model and SAW on five-dimensional hypercubic lat-tices with periodic boundaries onto the scaling variable y =∥ x ∥/ L ( d − µ )/( d − ) . (a) Anomalous FSS scaling at z c onto theansatz in Eq. (1) with µ = d /
2. When ∥ x ∥ ≈ L /
2, the two-pointfunctions display the anomalous FSS behaviour g ( x ) ≍ L − d / ,in contrast to the standard mean field prediction g ( x ) ≍ L − d .(b) Standard mean-field scaling at the pseudo-critical point z L = z c − aL − onto the ansatz in Eq. (1) with µ =
2. In con-trast to the critical PBC case, the two-point functions displaythe standard mean-field scaling behaviour g ( x ) ≍ L − d when ∥ x ∥ ≈ L / predict that the critical SAW and Ising two-point func-tions should be given by Eq. (1) with µ = d /
2. This pre-diction is in agreement with the conjectured behaviourof the critical Ising two-point function given in [19], andis in excellent agreement with the numerical results pre-sented in [6].For free boundary conditions (FBC), the possible ex-istence of the FSS behaviour χ ≍ L d / at pseudo-critical points is the subject of ongoing debate [4, 7, 10]. Specif-ically, denoting by T L the temperature which maximizes χ ( T, L ) on a box of size L , it was observed numericallyin [10] that χ ( T L , L ) has the same L d / scaling observedat criticality for periodic systems. The results in [7] arein agreement with this observation, however, the more re-cent work [4] refuted this claim, and numerically observedonly the standard mean-field scaling L . From Eq. (2),we see that one can observe χ ≍ L d / in a RLRW model inwhich the mean walk length scales as L d / . Universalitythen suggests that this scaling should also be observablein SAW and Ising models, at appropriate pseudo-criticalpoints. Our numerical results below confirm this. Random-length Random Walk.—
Let ( S t ) t ∈ N be a sim-ple random walk on a box of side length L in Z d , centered at the origin. Let N be an N -valued random variable,independent of each choice of step in ( S t ) t ∈ N . We re-fer to ( S t ) N t = as the corresponding RLRW. We study itsGreen’s function g RLRW ( x ) ∶= E ( N ∑ n = P ( S n = x )) , which is the expected number of visits to x , and the cor-responding susceptibility χ RLRW ∶= ∑ x g RLRW ( x ) . Here, P ( S n = x ) denotes the probability that the RLRW is atsite x after n steps.Consider a RLRW with mean walk length N ∶= ⟨N ⟩ ≍ L µ on a d ≥ µ ≥
2, itcan then be proved [15] that the Green’s function ex-hibits the piecewise asymptotic behaviour in Eq. (1). Inparticular, the case µ > L µ − d for large distances, whilethis plateau is absent for 0 < µ <
2. The case µ = Numerical setup for n -vector models.— We study thetwo-point function g Ising ( x ) ∶= E ( s s x ) for the zero-fieldferromagnetic Ising model, defined by the Hamiltonian H = − ∑ ij s i s j . Here, s i = ± i of a hypercubic lattice of side length L , and the sum isover nearest neighbours. We simulate the Ising modelat fugacities z ∶= tanh ( β ) , where β is the inverse Isingtemperature, via the worm algorithm introduced in [20].We also investigate the SAW on a box with linear size L in the variable length ensemble. We study the two-point function g SAW ( x ) ∶= ∑ ω ∶ → x z ∣ ω ∣ , where the sum isover all SAWs starting at the origin 0 and ending at x .We simulated this ensemble using an irreversible versionof the Berretti-Sokal algorithm [21, 22]. For both mod-els we study the corresponding susceptibility, defined by χ Ising / SAW ∶= ∑ x g Ising / SAW ( x ) .We study our models on hypercubic lattices, in thecase of both free and periodic boundary conditions.The Ising model was simulated at the estimated lo-cation of the infinite-volume critical point z c,Ising,5d = .
113 915 0 ( ) [9] in five dimensions, and the simulationsfor the SAW were performed at the estimated infinite-volume critical point z c,SAW,5d = .
113 140 84 ( ) [22].We also simulated the FSS behaviour at pseudo-criticalpoints z L = z c − aL − λ for various a ∈ R and λ >
0. We sim-ulated linear system sizes up to L =
71 in the Ising modeland L =
201 for the SAW. To estimate the exponent valuefor a generic observable Y we performed least-squares fitsto the ansatz Y = a Y L b Y + c Y . A detailed analysis of au-tocorrelation times can be found in [23] for the wormalgorithm and in [22] for the irreversible Berretti-Sokalalgorithm. Universal scaling at criticality.—
We now argue thatEqs. (1) and (2) correctly predict the FSS behaviour of -10 -8 -6 -4 -2
1 2 4 8 16 32 64 128(a) g ( x ) || x || SAW, L =161SAW, L =201Ising, L =31Ising, L =71 x -3 -1 g ( y L / ) L / y y SAW, L =101SAW, L =141SAW, L =201Ising, L =31Ising, L =61 Figure 2: Two-point functions of the Ising model and SAW onfive-dimensional hypercubic lattices with free boundaries. (a)Standard mean-field scaling g ( x ) ≍ ∥ x ∥ − d at z c . (b) Anoma-lous FSS at the pseudo-critical point ˜ z L = z c + a L L − ontothe scaling variable y = ∥ x ∥/ L ( d − µ )/( d − ) with µ = d /
2. Thetwo-point functions collapse except at distances close to theboundary. This shows that g ( x ) displays the same FSS be-haviour as on periodic boundaries at criticality. the two-point functions and the susceptibility of the crit-ical SAW and Ising model, with either FBC or PBC.We first study the periodic case. It is expected thatmodels on high-dimensional tori should exhibit the samescaling as the corresponding model on the completegraph. It was proved in [17] that, on the complete graph, N SAW scales at criticality like the square root of the num-ber of vertices. On five-dimensional tori, our fits for N SAW at criticality lead to the exponent value 2 . ( ) , inexcellent agreement with the complete graph predictionof d /
2. Combining this scaling for N SAW with our resultsfor the RLRW, the two-point functions of the criticalIsing and SAW models on high-dimensional tori are thenpredicted to display the scaling in Eq. (1) with µ = d / y ∶= ∥ x ∥/ L ( d − µ )/( d − ) with µ = d /
2. Asa corollary of this two-point function scaling, we obtain χ ≍ L d / , in agreement with the numerical studies forthe Ising model in [24, 25], and with our direct exponentestimates for d = . ( ) for χ SAW , and 2 . ( ) for χ Ising .On free boundaries at criticality, our fits for N SAW leadto the exponent value 2 . ( ) , strongly suggesting that N SAW ≍ L . Combining this scaling for N SAW with ourresults for the RLRW, the two-point functions of the crit- ical Ising and SAW models on high-dimensional boxeswith free boundaries are then predicted to display thescaling in Eq. (1) with µ =
2. Figure 2(a) verifies thisprediction, showing an excellent data collapse for the two-point functions of the critical Ising and SAW models ontothe ansatz in Eq. (1) with µ =
2. Equation 2 then pre-dicts χ ≍ L , in agreement with the numerical study ofthe Ising model in [9], and with our direct exponent esti-mates for d = . ( ) for the Ising model and 1 . ( ) for the SAW. Universal scaling at pseudo-critical points.—
We nowturn to the actively debated question [4, 7, 10] of whetherone can observe the scaling behaviour χ ≍ L d / , corre-sponding to critical PBC behaviour, on free boundariesat pseudo-critical points. This also motivates the reversequestion, of whether it is possible to observe the stan-dard mean-field behaviour χ ≍ L , corresponding to crit-ical FBC behaviour, at pseudo-critical points on periodicboundaries. The above results for the RLRW suggestthat the FSS behaviour of the SAW two-point functionshould only depend on the boundary conditions throughtheir effect on N . We now numerically verify that this isindeed the case, and that analogous results also hold forthe Ising model.For periodic boundaries, we study FSS at pseudo-critical points z L ( λ ) = z c − aL − λ , with a chosen positiveso that the walk lengths are decreased compared withcriticality. On the complete graph, it can be shown [15]that at a pseudo-critical point z V ( ζ ) = z c − aV − ζ we have N SAW ≍ V / if ζ ≥ /
2, while N SAW ≍ V ζ if ζ ≤ / µ = ζd =∶ λ for any 0 < λ ≤ d /
2, and µ = d / λ ≥ d / z L ( λ ) on high-dimensional tori.Taking λ =
2, the above argument predicts that thepseudo-critical two-point functions display the mean-fieldbehaviour g ( x ) ≍ ∥ x ∥ − d . Fig. 1(b) shows an appropri-ately scaled version of the two-point functions of the Isingmodel and SAW onto the ansatz in Eq. (1) with µ = z L ( ) .We emphasize that, despite appearances, the two-pointfunctions in Fig. 1(a) and (b) do not display the sameFSS behaviour. In particular, it follows from the scalingansatz in Eq. (1) that if ∥ x ∥ ≈ L /
2, then the critical two-point functions scale as g ( x ) ≍ L − d / , while g ( x ) ≍ L − d at z L ( ) .Considering more general values of λ , Fig. 3(a) showsthe scaling of N SAW at z L ( λ ) on five-dimensional tori for λ = , . , , .
5. Our fits lead to the exponent values0 . ( ) for λ =
1, 1 . ( ) for λ = .
5, 2 . ( ) for λ = (a) λ = ∞ ( z c ) λ = 2.5 λ = 2 λ = 1.5 λ = 1 N
32 64 128(b) χ L Figure 3: FSS behaviour of the mean walk length N SAW (top) and the susceptibility χ Ising , SAW (bottom) with peri-odic boundary conditions in five dimensions. The diamonds(squares) display SAW (Ising) data. To emphasize universal-ity, the Ising and SAW data were translated onto the samecurve. In each figure, the line on the top corresponds tothe critical scaling behaviour
N, χ ≍ L d / . The remaininglines (top to bottom) correspond to the scaling behaviour N, χ ≍ L λ at pseudo-critical points z L ( λ ) = z c − aL λ ( a > λ = . , , . ,
16 32 64 128 N L SAW, λ =2SAW, z c
50 150 0.7 0.8 aL IsingSAW
Figure 4: FSS behaviour of the mean walk length N SAW with free boundary conditions in five dimensions. The inset showsthe convergence of a L in the SAW (circles) and Ising model(triangles). and 2 . ( ) for λ = .
5, in excellent agreement with thecorresponding results on the complete graph. Figure 3(b)then shows the scaling behaviour of the susceptibility for λ = , . , , .
5. Our fits for the SAW lead to theexponent values 1 . ( ) for λ =
1, 1 . ( ) for λ = . . ( ) for λ =
2, 2 . ( ) for λ = .
5. For the Ising model,our fits lead to 1 . ( ) for λ =
1, 1 . ( ) for λ = . . ( ) for λ =
2, and 2 . ( ) for λ = .
5. These estimatesare all in excellent agreement with above predictions.Finally, we consider pseudo-critical behaviour with freeboundary conditions. There has been considerable de- bate [4, 7, 10] concerning the existence of critical PBCFSS behaviour on lattices with FBC at a pseudo-criticalpoint which maximizes χ ( T, L ) on a box of linear size L . It has been numerically established that this pseudo-critical point has shift exponent λ = χ ≍ L d / at such a pseudo-critical point is to definea sequence a L such that χ FBC , ˜ z L ( L ) = χ PBC ,z c ( L ) with˜ z L ∶= z c + a L L − , and to then show that a L converges. Ifsuch a convergent sequence exists, this approach forces χ FBC ,z L to scale as L d / , where z L = z c + a ∞ L − . The in-set of Fig. 4 shows the sequence a L in the Ising and SAWmodels. For SAW, the series a L clearly appears to con-verge, and our fits predict a SAW , ∞ = . ( ) . The Isingdata are roughly consistent with the SAW data, albeitover a much smaller range of L values.Fitting the FBC data for N SAW at ˜ z L produces an ex-ponent estimate of 2 . ( ) , suggesting that N SAW ≍ L d / ,compared with N SAW ≍ L at z c ; see Fig. 4. Universalitythen suggests that the Ising and SAW two-point functionsshould follow Eq. (1) with µ = d /
2. Figure 2(b) showsthe appropriately re-scaled two-point functions. We ob-serve excellent data collapse, except at distances closeto the boundary. This strong boundary effect may ex-plain the apparent discrepancies [4, 7, 10] in determiningthe correct scaling behaviour for the pseudocritical Isingmodel with FBC. Regardless, we conclude from Fig. 2(b)that the anomalous FSS behaviour, observed on periodicboundaries at criticality, can be observed on free bound-aries, in agreement with [7, 10].
Discussion.—
In this Letter, we have introduced arandom-length random walk model to clarify a numberof open questions regarding the FSS behaviour of theIsing model above d c . For periodic boundaries, by com-bining the RLRW model with the scaling of the meanwalk length of SAW on the complete graph, we were ableto predict the asymptotic scaling of the Ising and SAWtwo-point functions on high-dimensional tori at a familyof pseudo-critical points z L ( λ ) = z c − aL − λ , and showedthat the scaling exponents vary continuously with λ when0 < λ ≤ d /
2. As special cases, at z c we recovered the be-haviour conjectured in [19], while at z L ( ) we showed theIsing two-point function displays standard mean-field be-haviour.On free boundaries, combining the RLRW model withthe numerical scaling of N SAW predicts that the criti-cal Ising two-point function displays standard mean-fielddecay. It follows that the susceptibility scales as L , inagreement with the numerical observation in [9]. Wealso studied the actively debated FSS behaviour at thepseudo-critical point z L = z c + aL − . We established thatthe Ising two-point function displays the same FSS be-haviour as on periodic boundaries at criticality, in agree-ment with the numerical observations in [7, 10].Recently, three-dimensional quantum spin models,which are related to the corresponding four-dimensionalclassical counterpart [26], have been the subject of inten-sive theoretical, experimental and numerical studies [27–29]. Our work has focused on the FSS behaviour of the n -vector model above d c =
4. Although at d c the situa-tion is likely complicated by logarithmic corrections, webelieve that our results for d > d c are a necessary firststep in understanding the correct scaling behaviour forapplications to three-dimensional quantum spin models.We would like to thank Andrea Collevecchio, Eren M.El¸ci and Kevin Leckey for fruitful discussions. This workwas supported under the Australian Research Coun-cil’s Discovery Projects funding scheme (Project NumberDP140100559). It was undertaken with the assistance ofresources from the National Computational Infrastruc-ture (NCI), which is supported by the Australian Gov-ernment. This research was supported in part by theMonash eResearch Centre and eSolutions-Research Sup-port Services through the use of the MonARCH HPCCluster. J. Grimm acknowledges Monash University andthe Australian Mathematical Society for their financialsupport. Y. Deng and S. Fang acknowledges the sup-port by the National Key R&D Program of China underGrant No. 2016YFA0301604 and by the National NaturalScience Foundation of China under Grant No. 11625522. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] M. E. Fisher,
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