March 28, 2017

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By Louis Comtet

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25, and is smaller still in more than six dimensions (Hara and Slade (unpublished)]. 23) ford~ 6. (ze- z)- 112, and hence g(z) ~ (zc- z)(d-4)/2. 5. 3, the bubble condition implies that x(z) ~ (ze- z)- 1 • Hence to prove the theorem it suffices to show that L 00 ·cN,,N2ZN,+N2 N,,N2=0 ~ (zc- z)-4. 5. 25) z,y,v w : 0 - tJ p:z-y In a nonzero contribution to this sum, let u be the first site along w where p intersect. Then the portion of w before u avoids p as well as the latter part of w, while the latter part of w avoids only the former part of w and may intersect p.

E->~ ...... (ze- z)-< 2 ->~)li. r = {2- 7J)v. 3 that = r, and we will shortly argue that j) = v, which then gives r = (2- q)v. 3). 1. rl is of the order of N". 7), gives c~l:rll/v G z. (0 , X ) N'l'-1-dv,.. ~ lxl('l'-dv)fv. 8) This implies -(d- 2 + 77) = ('Y- dv)Jv, which can be rewritten as 'Y = (2- 7J)V. 5), we now argue that ii = Vp = v. (zc- z)-(P+ 2-'7)ii. 10) and hence ii = Vp. 5) and the equality of and 'Y we conclude that ii = v. 24). 3) and require additional assumptions. Less numerical testing has been done of the hyperscaling relations than on the calculation of 'Y and v, but both the Monte Carlo and series extrapolation computations which have been done are consistent with them.

26), assuming that R(zc)- 1 < 1/4. 1. fN was first proven in Hammersley and Morton ( 1954); this paper essentially marks the beginning of rigorous results for the self-avoiding walk. /2 for the honeycomb lattice is due to Nienhuis (1982); see also Nienhuis (1984) and Nienhuis (1987). For high dimensions, it was shown in Kesten (1964) that Jl 2d -1- (2d)- 1 +O(d- 2 ), and this has recently been improved to Jl 2d- 1- (2d)- 1 - 3(2d)- 2 + 0( d- 3 ) using the lace expansion [Hara and Slade (unpublished)].

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