content_image/353257.jpg content_image/353256.jpg content_image/353258.jpg content_image/353255.jpg content_image/353254.jpg content_image/353253.jpg content_image/353252.jpg content_image/353251.jpg
for the m th moment of the probability distribution of k = 0 , . . . , n . Define also 

<img src='content_image/353257.jpg'>

so that the m th moment becomes 〈| x | $^{m}$〉 = ϱ$_{m}$/ϱ$_{0}$ . For m = 0 we have 

so that 

Next, for m = 1 

where − 1 < erf ( x ) < 1 is the error function. and thus 

In general we have for m ≥ 0 that 

where ϱ$_{m}$ is given by 

 $^{φ}$$^{−}$$^{(}$$^{m}$$^{+1)}$$^{/}$$^{4}$ φ − ( m +1) / 4 2 Γ( m +1 4 ) , a = 0 Here $_{1}$F$_{1}$ ( a, b, c ) and U( a, b, c ) denote the confluent hypergeometric functions of the first and second kind respectively. If we want to compute cumulant ratios we first need moment ratios which of course is easy now. For example, in the case of a = 0 we have 

<img src='content_image/353256.jpg'>

<img src='content_image/353258.jpg'>

<img src='content_image/353255.jpg'>

<img src='content_image/353254.jpg'>

<img src='content_image/353253.jpg'>

<img src='content_image/353252.jpg'>

<img src='content_image/353251.jpg'>

These moment ratios are the same as obtained in the 5-dimensional Ising model, see e.g. $^{11}$. 

