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Generalization to arbitrary $n$: The desired distribution $P_n(r)$ of the distance $r$ from the origin after $n$ steps can be expressed as an integral over Bessel functions $J_0$, $$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$ $$F_n(r)\equiv\int_0^r P_n(r')\,dr'=\int_0^\infty [J_0(k/r)]^nJ_1(k )\,dk,\;\;0<r<n,$$ see theorem 2.1 in arXiv:1508.04729 for a derivation (and the generalization to $d$ dimensions). [A more tutorial exposition is given in A Short Walk can be Beautiful.]

For $n=2$ we recover $P_2(r)=(2/\pi)(4-r^2)^{-1/2}$. The cited paper contains closed form expressions for $n=3,4$, in terms of hypergeometric functions (eqs. 74, 79). Closed form expressions for the moments exist for any $n$.

There is also a remarkable exact result (first obtained by J.C. Kluyver in 1905) for the probability $p_n$ to return to the unit disc after $n$ steps, $$p_n=F_n(1)=\frac{1}{n+1},\;\;n>1.$$ A purely geometric derivation is given in arXiv:1508.04729.

For $n\gg 1$ we may approximate $J_0(k/r)^n\approx\exp\left(-\frac{nk^2}{4r^2}\right)$, see page 345 of Kluyvers paper, to obtain $$F_n(r)\approx\int_0^\infty \exp\left(-\frac{nk^2}{4r^2}\right)J_1(k)\,dk=1-e^{-r^2/n},$$ which gives the expected Maxwell distribution for $P_n(r)=dF_n(r)/dr$.

Generalization to arbitrary $n$: The desired distribution $P_n(r)$ of the distance $r$ from the origin after $n$ steps can be expressed as an integral over Bessel functions $J_0$, $$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$ $$F_n(r)\equiv\int_0^r P_n(r')\,dr'=\int_0^\infty [J_0(k/r)]^nJ_1(k )\,dk,\;\;0<r<n,$$ see theorem 2.1 in arXiv:1508.04729 for a derivation (and the generalization to $d$ dimensions). [A more tutorial exposition is given in A Short Walk can be Beautiful.]

For $n=2$ we recover $P_2(r)=(2/\pi)(4-r^2)^{-1/2}$. The cited paper contains closed form expressions for $n=3,4$, in terms of hypergeometric functions (eqs. 74, 79).

I note a remarkable exact result (first obtained by J.C. Kluyver in 1905) for the probability $p_n$ to return to the unit disc after $n$ steps, $$p_n=F_n(1)=\frac{1}{n+1},\;\;n>1.$$ A purely geometric derivation is given in arXiv:1508.04729.

For $n\gg 1$ we may approximate $J_0(k/r)^n\approx\exp\left(-\frac{nk^2}{4r^2}\right)$, see page 345 of Kluyvers paper, to obtain $$F_n(r)\approx\int_0^\infty \exp\left(-\frac{nk^2}{4r^2}\right)J_1(k)\,dk=1-e^{-r^2/n},$$ which gives the expected Maxwell distribution for $$P_n(r)=dF_n(r)/dr=(2r/n)e^{-r^2/n}.$$

Generalization to arbitrary $n$: The desired distribution $P_n(r)$ of the distance $r$ from the origin after $n$ steps can be expressed as an integral over Bessel functions $J_0$, $$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$ $$F_n(r)\equiv\int_0^r P_n(r')\,dr'=r\int_0^\infty [J_0(k)]^nJ_1(k r)\,dk,\;\;0<r<n,$$ see theorem 2.1 in arXiv:1508.04729 for a derivation (and the generalization to $d$ dimensions). [A more tutorial exposition is given in A Short Walk can be Beautiful.]

For $n=2$ we recover $P_2(r)=(2/\pi)(4-r^2)^{-1/2}$. The cited paper contains closed form expressions for $n=3,4$, in terms of hypergeometric functions (eqs. 74, 79). Closed form expressions for the moments exist for any $n$.

There is also a remarkable exact result (first obtained by J.C. Kluyver in 1905) for the probability $p_n$ to return to the unit disc after $n$ steps, $$p_n=F_n(1)=\frac{1}{n+1},\;\;n>1.$$ A purely geometric derivation is given in arXiv:1508.04729.

Generalization to arbitrary $n$: The desired distribution $P_n(r)$ of the distance $r$ from the origin after $n$ steps can be expressed as an integral over Bessel functions $J_0$, $$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$ $$F_n(r)\equiv\int_0^r P_n(r')\,dr'=\int_0^\infty [J_0(k/r)]^nJ_1(k )\,dk,\;\;0<r<n,$$ see theorem 2.1 in arXiv:1508.04729 for a derivation (and the generalization to $d$ dimensions). [A more tutorial exposition is given in A Short Walk can be Beautiful.]

For $n=2$ we recover $P_2(r)=(2/\pi)(4-r^2)^{-1/2}$. The cited paper contains closed form expressions for $n=3,4$, in terms of hypergeometric functions (eqs. 74, 79). Closed form expressions for the moments exist for any $n$.

There is also a remarkable exact result (first obtained by J.C. Kluyver in 1905) for the probability $p_n$ to return to the unit disc after $n$ steps, $$p_n=F_n(1)=\frac{1}{n+1},\;\;n>1.$$ A purely geometric derivation is given in arXiv:1508.04729.

For $n\gg 1$ we may approximate $J_0(k/r)^n\approx\exp\left(-\frac{nk^2}{4r^2}\right)$, see page 345 of Kluyvers paper, to obtain $$F_n(r)\approx\int_0^\infty \exp\left(-\frac{nk^2}{4r^2}\right)J_1(k)\,dk=1-e^{-r^2/n},$$ which gives the expected Maxwell distribution for $P_n(r)=dF_n(r)/dr$.

Generalization to arbitrary $n$: The desired distribution $P_n(r)$ of the distance $r$ from the origin after $n$ steps can be expressed as an integral over Bessel functions $J_0$, $$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$ see theorem 2.1 in arXiv:1508.04729 for a derivation (and the generalization to $d$ dimensions). [A more tutorial exposition is given in A Short Walk can be Beautiful.]

For $n=2$ we recover $P_2(r)=(2/\pi)(4-r^2)^{-1/2}$. The cited paper contains closed form expressions for $n=3,4$, in terms of hypergeometric functions (eqs. 74, 79). Closed form expressions for the moments exist for any $n$.

There is also a remarkable exact result (first obtained by J.C. Kluyver in 1905) for the probability $p_n$ to return to the unit disc after $n$ steps, $$p_n=\int_0^1 P_n(r)\,dr=\frac{1}{n+1}.$$ A purely geometric derivation is given in arXiv:1508.04729.

Generalization to arbitrary $n$: The desired distribution $P_n(r)$ of the distance $r$ from the origin after $n$ steps can be expressed as an integral over Bessel functions $J_0$, $$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$ $$F_n(r)\equiv\int_0^r P_n(r')\,dr'=r\int_0^\infty [J_0(k)]^nJ_1(k r)\,dk,\;\;0<r<n,$$ see theorem 2.1 in arXiv:1508.04729 for a derivation (and the generalization to $d$ dimensions). [A more tutorial exposition is given in A Short Walk can be Beautiful.]

For $n=2$ we recover $P_2(r)=(2/\pi)(4-r^2)^{-1/2}$. The cited paper contains closed form expressions for $n=3,4$, in terms of hypergeometric functions (eqs. 74, 79). Closed form expressions for the moments exist for any $n$.

There is also a remarkable exact result (first obtained by J.C. Kluyver in 1905) for the probability $p_n$ to return to the unit disc after $n$ steps, $$p_n=F_n(1)=\frac{1}{n+1},\;\;n>1.$$ A purely geometric derivation is given in arXiv:1508.04729.