Moments of uniform random walks in the plane

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Numbers

$n$

$s$

$W_n(s)$

1.574597237551893657494692183076519690221666180758519170193693098301831180594454382131085313362241953

comment: Mean distance from the origin.

-1:

0.8964407887767628642327709000349704991387844034162414609834833987065596797806136031423376992276078122

comment: Mean reciprocal distance from the origin.

6.451679653132074204075792620988788055228991106790181333253751060035319162993506846802727096462854635

36.70518910262770539212833351083942280156566394634277292805674123053093515107189216917695516978563411

241.5438884325818909394572120653345661579642316234345510198866599376377900615201209124784525460868827

1.799092479842851033532602845846108910066282003291620456626641773598854266932120575241161930573474828

comment: Mean distance from the origin.

-1:

0.9027285783238348241903933987727629850304635036014195522559185553239931867554294224993878943710033573

comment: Mean reciprocal distance from the origin.

10.12068101030959857117828689311952623741652682870990046944160365931309708011713714090418407354722791

82.65146061218160253264341821152523717355689861471062139494090139831178199309483015283222906911803326

822.2733729627545473875906636736519876586871256155389079039288911570862095415026148804830247323282182

Definition

The raw radial moment $W_n(s)=\mathbb{E}[|\sum_{j=1}^n e^{i\theta_j}|^s]$ of an $n$-step planar random walk, where every step has length $1$ and the directions $\theta_j$ are independent and uniform on $[0,2\pi)$ [1].

Parameters

$n$

— number of steps ($n\geq 3$)

$s$

— radial moment order ($s\geq -1$)

Formulas

(1)

With $A=\dfrac{3\,2^{1/3}\Gamma(1/3)^6}{16\pi^4}$, where $\Gamma$ is Euler's gamma function, $W_3(-1)=A$ and $W_3(1)=A+\dfrac{27\,2^{2/3}\Gamma(2/3)^6}{4\pi^4}$ [1].

(2)

For $m_3=(2-\sqrt3)/4$ and $K_3=K(m_3)$, $W_3(-1)=2\sqrt3 K_3^2/\pi^2$ and $W_3(1)=2\sqrt3 K_3^2/\pi^2+\sqrt3/K_3^2$. The value $K_3$ is the elliptic integral at the third singular value [1].

(3)

For $J_p=\int_0^1 k^pK(1-k^2)^2\,dk$, $W_4(-1)=4J_0/\pi^3$ and $W_4(1)=16(J_0-3J_2)/\pi^3$ [1].

(4)

$(s+4)^2W_3(s+4)-2(5s^2+30s+46)W_3(s+2)+9(s+2)^2W_3(s)=0$ for integers $s\geq-1$ [2].

(5)

$(s+4)^3W_4(s+4)-4(s+3)(5s^2+30s+48)W_4(s+2)+64(s+2)^3W_4(s)=0$ for integers $s\geq-1$ [2].

(6)

$W_3(-1)=\tfrac23 W_{A_3}$, where $W_{A_3}$ is the face-centred cubic Watson integral [1].

Comments

(7)

The moments are taken about the origin, without centering the distance or rescaling by the number of steps. The order $s=1$ gives the mean distance and $s=-1$ gives the mean reciprocal distance. The defining integral is convergent for the stated integer domain $n\geq3$, $s\geq-1$ [1].

(8)

The complete elliptic integral is written in the parameter convention, $K(m)=\int_0^{\pi/2}(1-m\sin^2\theta)^{-1/2}\,d\theta$. The symbol $K^{\prime}(k)$ in [1] denotes the complementary-modulus integral, which is $K(1-k^2)$ in this table; it does not denote differentiation.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 50

def W(n, s):
    if n not in (3, 4) or s <= -2:
        raise ValueError("Require n=3 or 4 and s>-2")
    if n == 3:
        factor = mp.power(3, s+mp.mpf(3)/2) / (
            mp.pi * mp.power(2, s) * mp.gamma(mp.mpf(s)/2+1)**2)
    else:
        factor = mp.power(4, s+2) / (
            mp.pi**2 * mp.gamma(mp.mpf(s)/2+1)**2)
    return factor * mp.quad(
        lambda t: t**(s+1)*mp.besseli(0,t)*mp.besselk(0,t)**(n-1),
        [0, 1, 4, 16, 64, mp.inf])

print(W(3, 1))

References

[1]

J. M. Borwein, A. Straub and J. Wan, Three-step and four-step random walk integrals, Experimental Mathematics 22 (2013), 1–14. Equations (5), (7), (40) and (58) in the October 14, 2010 author preprint. (doi)

[2]

J. M. Borwein, D. Nuyens, A. Straub and J. Wan, Some arithmetic properties of short random walk integrals, The Ramanujan Journal 26 (2011), 109–132. Theorem 4 and Example 1. (doi)

[3]

J. M. Borwein, A. Straub, J. Wan and W. Zudilin, Densities of short uniform random walks, Canadian Journal of Mathematics 64 (2012), 961–990. Section 7, equations (75) and (76), Lemmas 1 and 3. (arXiv) (doi)

Links

[4]

Three-step and four-step random walk integrals

[5]

DLMF 19.12: convergent expansions for complete elliptic integrals

Similar tables

Mahler measures of short random walks — holds $W_n^{\prime}(0)$, the derivatives of the radial moment function

Elliptic integrals at singular values — contains the elliptic seed for the three-step odd moments

Watson integrals — contains the face-centred cubic lattice integral related to the reciprocal-distance moment

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the orders $s=-1,1,3,5,7$ for the three-step and four-step unit-length planar walks. These are the mean reciprocal distance, the mean distance and the next three positive odd radial moments studied through the short-walk recurrences [1] [2]; the cutoff $s=7$ is an editorial prefix of that discrete family. The mean distance is displayed first for each step count)

How they were obtained:

The generator evaluates the three-step gamma formulas and the four-step elliptic integrals in Arb ball arithmetic. It writes 100 significant decimal digits only after checking the enclosure widths. The complete computation is repeated at 150 and 200 working decimal digits, each with an additional 48-bit guard.

For the four-step seeds, the integral is split at $k=1/2$. The logarithmic expansion [5] is integrated term by term on $(0,1/2)$, with an explicit bound for the remainder of its square. Arb quadrature encloses the integral on $[1/2,1]$ with its analytic domain checked. The attached numerical-method.md gives the bounds; generate.py implements them. The proved moment recurrences propagate the seed enclosures.

Independent checks integrate the modified-Bessel representations [3] at 50 working digits for every stored pair $(n,s)$. The controls $W_n(0)=1$ and $W_n(2)=n$ are evaluated first. These quadratures provide agreement checks, while the enclosures come from the generator. The three-step elliptic seed identities, the face-centred cubic Watson identity, and all recurrence instances contained in the selected range are checked separately. The attached checks.py also checks the stored values.