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[6, internal are points three all when aus ra elo okhsbe oeoe the over done dimensions, been six has about of work expansions powers of in series finding deal lattice, in rational Bethe great years simple the A and or exactly graph values. known complete are the simi- exponents on where mean-field, that is becomes to dimension behavior lar the critical that upper meaning the 6, interest percolation, and of in 4D theoretically are in because, Tan, dimensions correlations [10], higher these paper These find- present study 5D. numerically the Jacobsen In and and 3D. Deng, 2D in for exponents the predictions confirm- ing 3D, exact and the 2D in ing correlations these studied cobsen remains. term logarithmic universal a and out ee h uhr oka orltosbtentwo between correlations at look authors the Here, nRf 8,Vser aosnadSlu introduced Saleur and Jacobsen Vasseur, [8], Ref. In narcn ae 9,Tn over egadJa- and Deng Couvreur, Tan, [9], paper recent a In L P ′ ( r in fXajnTn ojnDeng Youjin Tan, Xiaojun of tions ≫ in nfu-adfive-dimensional and four- in tions 1 R r , F ǫ L = 2 ( 6 = r , r N osuyteecreain,jiigtwo joining correlations, these study to eo)i hc h oe-a cancels power-law the which in below) ) p 3 stepoaiiyta he far- three that probability the is ) − P n )adanother and 3) and 2 = 7 ( ia Engineering, mical d r / n nigrslsi n di- 5 and 4 in results finding and , 2 1 3 r , − P 2 3 ( ) / A r P 4 1 π ( r , r 5 / 1 2 2 r , r , Γ(1 3 3 ) L Frontiers ) P / d L/ 3) ( for r L − 2 rmtefirst the from 2 r , 9 d / − d 3 2 1 ) r5. or 4 = ≈ × N 1 N L . ′ 29[6] 0299 . lattice neigh- neigh- 02201 (2) 2 separate systems together to form a layer in a bulk sys- where P0 is the probability that the four points all be- tem to study the correlations, and then returning to the long to different clusters, P1 is the probability that the un-joined systems to add additional layers. They use a points belong to three different clusters, with one clus- union-find algorithm [11] with a simple compression step ter linking one point in one set to a point in the other, and a flattening step to efficiently carry out this algo- and P=6 is the probability that the two points in a single rithm. One of the advantages of using this algorithm is set belong to different clusters. In 2D, we know that on that they could simulate a larger L than if the system a square lattice P=6 = 1/4 [13], but it is not known in were simply periodic of size Ld, because only one plane higher dimensions. Carrying out precise studies, the au- needs to be stored in the L′ direction. thors find δ =1.16(1) in 4D and 0.74(6) in 5D. These are The authors look at configurations where the N points new results, and together with their previous work in 2D in each of the two groups belong to different clusters– and 3D, paint a very complete picture on the universal already something of low probability—and then look at behavior of these quantities. various combinations of connectivites between the two sets of points along clusters joining them. Finally, by The exponent X2s is related to the correlation-length looking at certain symmetric PNs, asymmetric PNa, and exponent ν by X2s = d − yt = d − 1/ν, and their mea- mixed PNm combinations, they find quantities that also surements of this exponent yields substantially increased decay as a power-law in r, defined as the distance between precision of the value of this important exponent in 4D these two sets of point. This defines various exponents and 5D. They find yt =1.4610(12) or ν =0.6845(6) in 4D according to and yt = 1.737(2) or ν = 0.5757(6) in 5D, compared to other recent results ν =0.6852(28) by Koza and Pola[14] −2XNo PNo ∼ r (3) and 0.6845(23) by Zhang et al. [12], and consistent with older results 0.6782(50) of Adler et al. [15] and 0.689(10) where o = s, m, a and N = 2, 3. In their paper, they of Ballesteros et al. [16] in 4D, and ν = 0.5723(18) [14] present precise values of these exponents XNo. These are and 0.5737(33) [12] in recent works, and 0.571(3) in Adler new universal quantities that have never been measured et al. [15] from 1990 in 5D. These various results are gen- before. Because of the low probability of these events, ex- erally consistent with each other within error bars, de- tensive simulations were necessary to reveal the behavior. spite the variety of methods that were used, highlighting For example, for N = 2, they define the high level of understanding of percolation that has been developed over the last several decades. P2s = P (r1, r3|r2, r4)+ P (r1, r4|r2, r3) P2a = P (r1, r3|r2, r4) − P (r1, r4|r2, r3) (4) Besides Monte Carlo work, another classical approach has been through series expansions about 6D, and re- where the points r1, r2 are in one neighboring set, and cently Gracey [17] has extended those series to higher or- r3, r4 are in another neighboring set a distance L/2 away der and has made the following predictions: ν = 0.6920 from the first set. Here the vertical bar separates the in 4D and 0.5746 in 5D. These results are a bit lower points that are connected together. In [10], a graphical and outside the error bars of results found here, but not representation of these quantities is used. far off. Perhaps higher-order series and/or future Monte Likewise for N = 3, where there are 6 total points, Carlo work will show closer agreement. three function P3a, P3s, and P3m are defined. They can also extend this to N = 1, which is identical to Eq. (1) For future work, it might be nice to demonstrate that above with X1s = d − df . Here they quote values of X2s these results are indeed universal by looking at another from a recent paper of one of the authors, Ref. [12] system, such as site percolation on a hypercubic lattice or They also study the combination of correlations for site or bond percolation on a generalized FCC lattice, for N = 2 that is expected to show a logarithmic behavior example, although it is generally expected that univer- with a universal coefficient δ: sality should hold. Other combinations of correlations, P (r)+ P (r) − (P )2 such as the ratio of Eq. (2), might also be of interest in F (r)= 0 1 =6 ∼ δ ln r (5) higher dimensions. P2s(r)

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