Kleinert · 제15장 터널링

$$ P_{N}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)=\prod_{n=1}^{N}\left[\int d^{3} \Delta x_{n} \frac{1}{4 \pi a^{2}} \delta\left(\left|\Delta \mathbf{x}_{n}\right|-a\right)\right] \delta^{(3)}\left(\mathbf{x}_{b}-\mathbf{x}_{a}-\sum_{n=1}^{N} \Delta \mathbf{x}_{n}\right) . $$

$$ \begin{align*} \Delta(0,0) & =\Delta(L, L)=L / 3 \\ I_{1} & =\int_{0}^{L} d s \Delta(s, s)=L^{2} / 6 \\ I_{2} & =\int_{0}^{L} d s \cdot \Delta^{2}(s, s)=L / 12 \\ I_{3} & =\int_{0}^{L} d s \Delta^{2}(s, s)=L^{3} / 30 \\ I_{4} & =\int_{0}^{L} d s \Delta(s, s) \cdot \Delta^{2}(s, s)=7 L^{2} / 360 \\ I_{5} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta^{2}\left(s, s^{\prime}\right)=L^{4} / 90 \\ \Delta(0, L) & =\Delta(L, 0)=-L / 6 \\ I_{6} & =\int_{0}^{L} d s\left[\Delta^{2}(s, 0)+\Delta^{2}(s, L)\right]=2 L^{3} / 45 \\ I_{7} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta(s, s)^{\cdot} \Delta^{2}\left(s, s^{\prime}\right)=L^{3} / 45 \\ I_{8} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \cdot \Delta(s, s) \Delta\left(s, s^{\prime}\right) \cdot \Delta\left(s, s^{\prime}\right)=L^{3} / 180 \\ I_{9} & =\int_{0}^{L} d s \Delta(s, s)\left[\Delta^{2}(s, 0)+\Delta^{2}(s, L)\right]=11 L^{2} / 90 \\ I_{10} & =\int_{0}^{L} d s \cdot \Delta(s, s)[\Delta(s, 0) \cdot \Delta(s, 0)+\Delta(s, L) \cdot \Delta(s, L)]=17 L^{2} / 360 \end{align*} $$

$$ \begin{align*} H_{1}^{n(k)} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \rho_{n(k)}(s) \rho_{n(k)}\left(s^{\prime}\right) \Delta^{2}\left(s, s^{\prime}\right) \\ & =\delta_{n(k)}^{2} I_{5}+\delta_{n(k)} I_{6}+\frac{1}{2}\left[\Delta^{2}(0,0)+\Delta^{2}(0, L)\right] \\ H_{2}^{n(k)} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \rho_{n(k)}\left(s^{\prime}\right) \Delta(s, s)^{\cdot} \Delta^{2}\left(s, s^{\prime}\right)=\delta_{n(k)} I_{7}+\frac{I_{9}}{2} \\ H_{3}^{n(k)} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \rho_{n(k)}\left(s^{\prime}\right) \cdot \Delta d(s, s) \Delta^{2}\left(s, s^{\prime}\right)=\left[\delta_{n(k)} I_{5}+\frac{I_{6}}{2}\right] \delta(0), \\ H_{4}^{n(k)} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \rho_{n(k)}\left(s^{\prime}\right) \cdot \Delta(s, s) \cdot \Delta\left(s, s^{\prime}\right) \Delta\left(s, s^{\prime}\right)=\delta_{n(k)} I_{8}+\frac{I_{10}}{2} . \end{align*} $$

$$ \Delta\left(s, s^{\prime}\right)=\frac{L}{3} a-\frac{\left|s-s^{\prime}\right|}{2}-\frac{\left(s+s^{\prime}\right)}{2}+\frac{\left(s^{2}+s^{2}\right)}{2 L} $$

$$ P_{1}(\Delta \mathbf{x})=\frac{1}{4 \pi a^{2}} \delta(|\Delta \mathbf{x}|-a) $$

$$ \Delta_{\mathrm{F}}\left(s, s^{\prime}\right)=\Delta_{\mathrm{F}}\left(s-s^{\prime}\right)=\Delta\left(s, s^{\prime}\right)-\frac{1}{2} \Delta(s, s)-\frac{1}{2} \Delta\left(s^{\prime}, s^{\prime}\right)=-\frac{1}{2}\left(s-s^{\prime}\right) $$

$$ \int d^{3} x_{b} P_{1}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)=1 $$

$$ \begin{align*} & D_{1}\left(s, s^{\prime}\right)=\Delta^{2}\left(s, s^{\prime}\right)-\Delta(s, s) \Delta\left(s^{\prime}, s^{\prime}\right)=\left(s-s^{\prime}\right)\left[\frac{s(L-s)}{L}+\frac{\left(s-s^{\prime}\right)\left(s+s^{\prime}\right)^{2}}{4 L^{2}}-\frac{L a}{3}\right] \\ & D_{2}\left(s, s^{\prime}\right)=\Delta(s, s)-\Delta\left(s^{\prime}, s^{\prime}\right)=\frac{\left(s-s^{\prime}\right)\left(s+s^{\prime}-L\right)}{L} \end{align*} $$

$$ \int d^{3} x_{b} P_{N}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)=1 $$

$$ \delta^{(3)}\left(\mathbf{x}_{b}-\mathbf{x}_{a}-\sum_{n=1}^{N} \Delta \mathbf{x}_{n}\right)=\int \frac{d^{3} k}{(2 \pi)^{3}} e^{i \mathbf{k}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)-i \mathbf{k} \sum_{n=1}^{N} \Delta \mathbf{x}_{n}}, $$

$$ \begin{align*} H_{1}^{n} & =\frac{L^{4}}{90} \delta^{2}(0)+\frac{L^{3}}{45}(3-D-n) \delta(0)+\frac{L^{2}}{360}\left[\left(45-24 D+4 D^{2}\right)-4 n(6-2 D-n)\right] \\ H_{2}^{n} & =\frac{L^{3}}{45} \delta(0)+\frac{L^{2}}{180}(15-4 D-4 n) \\ H_{3}^{n} & =\frac{L^{4}}{90} \delta^{2}(0)+\frac{L^{3}}{90}(3-D-n) \delta(0) \\ H_{4}^{n} & =\frac{L^{3}}{180} \delta(0)+\frac{L^{2}}{720}(21-4 D-4 n) \end{align*} $$

$$ \begin{align*} J_{1}\left(s, s^{\prime}\right) & =\int_{0}^{L} d t \Delta(t, t) \cdot \Delta(t, s) \cdot \Delta\left(t, s^{\prime}\right)=\frac{\left(s^{4}+s^{4}\right)}{4 L^{2}}-\frac{\left(2 s^{3}+s^{3}\right)}{3 L} \\ & +\frac{\left((a+3) s^{2}+a s^{2}\right)}{6}-\frac{s a}{3} L+\frac{(20 a-9)}{180} L^{2} \end{align*} $$

$$ P_{N}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)=\int \frac{d^{3} k}{(2 \pi)^{3}} e^{i \mathbf{k}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)} \tilde{P}_{N}(\mathbf{k}) $$

$$ \begin{align*} J_{2}\left(s, s^{\prime}\right) & =\int_{0}^{L} d t \cdot \Delta(t, t) \Delta(t, s) \cdot \Delta\left(t, s^{\prime}\right)=\frac{\left(-2 s^{4}+6 s^{2} s^{\prime 2}+3 s^{\prime 4}\right)}{24 L^{2}} \\ & +\frac{\left(3 s^{3}-3 s^{2} s^{\prime}-6 s s^{\prime 2}-s^{\prime 3}\right)}{12 L}-\frac{\left(5 s^{2}-12 s s^{\prime}-4 a s^{\prime 2}\right)}{24}-\frac{s^{\prime} a}{6} L+\frac{(20 a-3)}{720} L^{2} \\ J_{3}\left(s, s^{\prime}\right) & =\int_{0}^{L} d t \Delta(t, s) \Delta\left(t, s^{\prime}\right)=-\frac{\left(s^{4}+6 s^{2} s^{\prime 2}+s^{\prime 4}\right)}{60 L}+\frac{\left(s^{2}+3 s^{\prime 2}\right) s}{6} \\ & -\frac{\left(s^{2}+s^{\prime 2}\right)}{6} L+\frac{\left(5 a^{2}-10 a+6\right)}{45} L^{3} \end{align*} $$

$$ \tilde{P}_{N}(\mathbf{k})=\prod_{n=1}^{N}\left[\int d^{3} \Delta x_{n} \frac{1}{4 \pi a^{2}} \delta\left(\left|\Delta \mathbf{x}_{n}\right|-a\right) e^{-i \mathbf{k} \Delta \mathbf{x}_{n}}\right] . $$

$$ \tilde{P}_{N}(\mathbf{k})=\left[\tilde{P}_{1}(\mathbf{k})\right]^{N} $$

$$ \begin{align*} & \left\langle f_{2}\left(q(s), q\left(s^{\prime}\right)\right)\right\rangle=\frac{(d-1)}{2}\left[D_{1}\left(s, s^{\prime}\right)-\frac{(d+1)}{4} D_{2}^{2}\left(s, s^{\prime}\right)\right]=-\frac{(d-1)\left(s-s^{\prime}\right)}{4} \\ & \quad \times\left[\frac{2 L a}{3}+\frac{(d-3) s-(d+1) s^{\prime}}{2}-\frac{(d-1) s^{2}-(d+1) s^{\prime 2}}{L}+\frac{d\left(s-s^{\prime}\right)\left(s+s^{\prime}\right)^{2}}{2 L^{2}}\right] \end{align*} $$

$$ \tilde{P}_{1}(\mathbf{k})=\int d^{3} \Delta \mathbf{x} \frac{1}{4 \pi a^{2}} \delta(|\Delta \mathbf{x}|-a) e^{-i \mathbf{k} \Delta \mathbf{x}}=\frac{\sin k a}{k a} $$

$$ \begin{align*} H_{1}^{k} & =\frac{L^{4}}{90} \delta^{2}(0)+\frac{L^{3}}{45}[2+(-i k)] \delta(0)+\frac{L^{2}}{90}(-i k)[4+(-i k)]+\frac{5 L^{2}}{72} \\ H_{2}^{k} & =\frac{L^{3}}{45} \delta(0)+\frac{L^{2}}{180}[11+4(-i k)] \\ H_{3}^{k} & =\frac{L^{4}}{90} \delta^{2}(0)+\frac{L^{3}}{90}[2+(-i k)] \delta(0) \\ H_{4}^{k} & =\frac{L^{3}}{180} \delta(0)+\frac{L^{2}}{720}[17+4(-i k)] \end{align*} $$

$$ \begin{align*} K_{1}\left(s, s^{\prime}\right) & =J_{1}\left(s, s^{\prime}\right)-\frac{1}{2} J_{1}(s, s)-\frac{1}{2} J_{1}\left(s^{\prime}, s^{\prime}\right) \\ & =-\left(s-s^{\prime}\right)\left[\frac{L a}{6}-\frac{\left(s+s^{\prime}\right)}{4}+\frac{\left(s^{2}+s s^{\prime}+s^{2}\right)}{6 L}\right] \\ K_{2}\left(s, s^{\prime}\right) & =J_{2}\left(s, s^{\prime}\right)+J_{2}\left(s^{\prime}, s\right)-J_{2}(s, s)-J_{2}\left(s^{\prime}, s^{\prime}\right) \\ & =-\left(s-s^{\prime}\right)^{2}\left[\frac{1}{4}-\frac{\left(5 s+7 s^{\prime}\right)}{12 L}+\frac{\left(s+s^{\prime}\right)^{2}}{4 L^{2}}\right] \\ K_{3}\left(s, s^{\prime}\right) & =J_{3}\left(s, s^{\prime}\right)-\frac{1}{2} J_{3}(s, s)-\frac{1}{2} J_{3}\left(s^{\prime}, s^{\prime}\right)=-\left(s-s^{\prime}\right)^{2}\left[\frac{\left(s+2 s^{\prime}\right)}{6}-\frac{\left(s+s^{\prime}\right)^{2}}{8 L}\right] \\ K_{4}\left(s, s^{\prime}\right) & =\Delta(0, s) \Delta\left(0, s^{\prime}\right)-\frac{1}{2} \Delta^{2}(0, s)-\frac{1}{2} \Delta^{2}\left(0, s^{\prime}\right)=-\frac{\left(s-s^{\prime}\right)^{2}\left(s+s^{\prime}-2 L\right)^{2}}{8 L^{2}} \\ K_{5}\left(s, s^{\prime}\right) & =\Delta(L, s) \Delta\left(L, s^{\prime}\right)-\frac{1}{2} \Delta^{2}(L, s)-\frac{1}{2} \Delta^{2}\left(L, s^{\prime}\right)=-\frac{\left(s^{2}-s^{\prime 2}\right)^{2}}{8 L^{2}} \end{align*} $$

$$ \begin{align*} P_{N}(\mathbf{R}) & =\int \frac{d^{3} k}{(2 \pi)^{3}}\left[\tilde{P}_{1}(\mathbf{k})\right]^{N} e^{i \mathbf{k} \mathbf{R}} \\ & =\frac{1}{2 \pi^{2} R} \int_{0}^{\infty} d k k \sin k R\left[\frac{\sin k a}{k a}\right]^{N} \end{align*} $$

$$ \mathbf{R} \equiv \mathbf{x}_{b}-\mathbf{x}_{a} $$

$$ P_{1}(\Delta \mathbf{x})=\frac{1}{S_{D} a^{D-1}} \delta(|\Delta \mathbf{x}|-a) $$

$$ \tilde{P}_{1}(\mathbf{k})=\int d^{D} \Delta \mathbf{x} \frac{1}{S_{D} a^{D-1}} \delta(|\Delta \mathbf{x}|-a) e^{-i \mathbf{k} \Delta \mathbf{x}} $$

$$ \begin{align*} \Delta(0,0) & =-\Delta(L, L)=-1 / 2 \\ I_{11} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta(s, s) \Delta^{\cdot}\left(s, s^{\prime}\right) \Delta\left(s^{\prime}, s^{\prime}\right)=L^{3} / 360 \\ I_{12} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta\left(s, s^{\prime}\right) \Delta^{2}\left(s, s^{\prime}\right)=L^{3} / 90 \end{align*} $$

$$ H_{5}=\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta(s, s)^{\cdot} \Delta^{2}\left(s, s^{\prime}\right)^{\cdot} \Delta d\left(s^{\prime}, s^{\prime}\right)=\delta(0) I_{7}=\frac{L^{3}}{45} \delta(0) $$

$$ I_{\nu}\left(e^{-i \pi / 2} z\right)=e^{-i \pi / 2} J_{\nu}(z) $$

$$ \begin{align*} H_{6} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta d(s, s) \Delta^{2}\left(s, s^{\prime}\right) \Delta d\left(s^{\prime}, s^{\prime}\right)=\delta^{2}(0) I_{5}=\frac{L^{4}}{90} \delta^{2}(0) \\ H_{7} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta(s, s) \cdot \Delta\left(s, s^{\prime}\right) \Delta\left(s, s^{\prime}\right) \cdot \Delta d\left(s^{\prime}, s^{\prime}\right)=\delta(0) I_{8}=\frac{L^{3}}{180} \delta(0) \\ H_{8} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta(s, s) \cdot \Delta\left(s, s^{\prime}\right) \Delta^{\cdot}\left(s, s^{\prime}\right) \cdot \Delta\left(s^{\prime}, s^{\prime}\right)=-\frac{L^{2}}{720} \\ H_{9} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \cdot \Delta(s, s) \Delta\left(s, s^{\prime}\right) \cdot \Delta d\left(s, s^{\prime}\right)^{\cdot} \Delta\left(s^{\prime}, s^{\prime}\right)=\frac{I_{10}}{2}-\frac{I_{8}}{L}-H_{8}=\frac{7 L^{2}}{360} \\ H_{10} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta(s, s) \cdot \Delta\left(s, s^{\prime}\right) \cdot \Delta d\left(s, s^{\prime}\right) \cdot \Delta\left(s^{\prime}, s^{\prime}\right)=\frac{I_{9}}{4}-\frac{I_{7}}{2 L}=\frac{7 L^{2}}{360} \\ H_{11} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta(s, s) \cdot \Delta d^{2}\left(s, s^{\prime}\right) \Delta\left(s^{\prime}, s^{\prime}\right)=\delta(0) I_{3}+\left(\frac{I_{1}}{L}\right)^{2}-\frac{2\left(I_{3}-I_{11}\right)}{L}-2 I_{4} \\ & +2\left[\Delta^{2}(L, L) \Delta^{\cdot}(L, L)-\Delta^{2}(0,0) \Delta^{\cdot}(0,0)\right]-2 H_{10}=\frac{L^{3}}{30} \delta(0)+\frac{L^{2}}{9} \\ H_{12} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta^{2}\left(s, s^{\prime}\right) \Delta^{\cdot 2}\left(s, s^{\prime}\right)=\frac{L^{2}}{90} \\ H_{13} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta\left(s, s^{\prime}\right) \cdot \Delta\left(s, s^{\prime}\right) \Delta^{\cdot}\left(s, s^{\prime}\right) \cdot \Delta d\left(s^{\prime}, s\right)=\frac{I_{4}}{2}-\frac{I_{12}}{2 L}-\frac{H_{12}}{2}=-\frac{L^{2}}{720} \\ H_{14} & =\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \Delta^{2}\left(s, s^{\prime}\right)^{\cdot} \Delta d^{2}\left(s, s^{\prime}\right)=\delta(0) I_{3}-\frac{2\left(I_{3}-I_{12}\right)}{L}-2 I_{4}+\frac{I_{5}}{L^{2}} \\ & +2\left[\Delta^{2}(L, L) \Delta^{\cdot}(L, L)-\Delta^{2}(0,0) \Delta^{\cdot}(0,0)\right]-2 H_{13}=\frac{L^{3}}{30} \delta(0)+\frac{11 L^{2}}{72} \end{align*} $$

$$ \tilde{P}_{1}(\mathbf{k})=\frac{\Gamma(D / 2)}{(k a / 2)^{D / 2-1}} J_{D / 2-1}(k a) $$

$$ \tilde{P}_{N}(\mathbf{k})=\left[\tilde{P}_{1}(\mathbf{k})\right]^{N}=\sum_{l=0}^{\infty} P_{N, 2 l} \frac{(k a)^{2 l}}{(2 l)!} $$

$$ \left\langle R^{2 l}\right\rangle \equiv \int d^{D} R R^{2 l} P_{N}(\mathbf{R}) $$

$$ \tilde{P}_{N}(\mathbf{k})=\int d^{D} R e^{-i \mathbf{k R}} P_{N}(\mathbf{R})=\sum_{n=0}^{\infty} \int d^{D} R \frac{(-i \mathbf{k} \mathbf{R})^{n}}{n!} P_{N}(\mathbf{R}) $$

$$ \left\langle(\mathbf{k R})^{n}\right\rangle=k^{n}\left\langle R^{n}\right\rangle\left\{\begin{array}{cc} 0 & n=\text { odd } \\ \frac{(n-1)!!(D-2)!!}{(D+n-2)!!}, & n=\text { even } \end{array}\right. $$

$$ \left\langle R_{i} R_{j}\right\rangle^{(0)}=\frac{1}{D} \delta_{i j} a^{2} N $$

$$ \left\langle R_{i_{1}} R_{i_{2}} \cdots R_{i_{n}}\right\rangle^{(0)}=\frac{1}{D^{n / 2}} \delta_{i_{1} i_{2} i_{3} \ldots i_{n}} a^{n} N^{n / 2} $$

$$ \delta_{i_{1} i_{2} i_{3} \ldots i_{n}}=\delta_{i_{1} i_{2}} \delta_{i_{3} i_{4} \ldots i_{n}}+\delta_{i_{1} i_{3}} \delta_{i_{2} i_{4} \ldots i_{n}}+\ldots \delta_{i_{1} i_{n}} \delta_{i_{2} i_{3} \ldots i_{n-1}} $$

$$ \left\langle R^{n}\right\rangle^{(0)}=\frac{(D+n-2)!!}{(D-2)!!D^{n / 2}} a^{n} N^{n / 2}=\frac{\Gamma(D / 2+n / 2)}{\Gamma(D / 2)} \frac{2^{n / 2}}{D^{n / 2}} a^{n} N^{n / 2} $$

$$ \left\langle R^{4}\right\rangle^{(0)}=\frac{(D+2)}{D} a^{4} N^{2}, \quad\left\langle R^{6}\right\rangle^{(0)}=\frac{(D+2)(D+4)}{D^{2}} a^{6} N^{3} . $$

$$ \left\langle(\mathbf{k R})^{n}\right\rangle^{(0)}=(n-1)!!\frac{1}{D^{n / 2}}(k a)^{n} N^{n / 2}=k^{n}\left\langle(R)^{n}\right\rangle^{(0)} d_{n} $$

$$ d_{n}=\frac{(n-1)!!(D-2)!!}{(D+n-2)!!} $$

$$ P_{N, 2 l}=(-1)^{l} d_{2 l}\left\langle R^{2 l}\right\rangle $$

$$ \tilde{P}_{N}(\mathbf{k})=\sum_{l=0}^{\infty} \frac{(-1)^{l}(k)^{2 l}}{(2 l)!} d_{2 l}\left\langle R^{2 l}\right\rangle $$

$$ \log \tilde{P}_{N}(\mathbf{k})=N \log \tilde{P}_{1}(\mathbf{k})=N \log \left(\frac{\sin k a}{k a}\right)=N \sum_{l=1}^{\infty} \frac{2^{2 l}(-1)^{l} B_{2 l}}{(2 l)!2 l}(k a)^{2 l} $$

$$ y(x)=\sum_{n=1}^{\infty} \frac{a_{n}}{n!} x^{n} $$

$$ e^{y(x)}=\sum_{n=1}^{\infty} \frac{b_{n}}{n!} x^{n} $$

$$ \frac{b_{n}}{n!}=\sum_{\left\{m_{i}\right\}} \prod_{i=0}^{n} \frac{1}{m_{i}!}\left(\frac{a_{i}}{i!}\right)^{m_{i}} $$

$$ n=\sum_{i=1}^{n} i \cdot m_{i} $$

$$ a_{n}=\left\{\begin{array}{cl} -N 2^{2 l}(-1)^{l} B_{2 l} / 2 l & \text { for } n=2 l \\ 0 & \text { for } n=2 l+1 \end{array}\right. $$

$$ \left\langle R^{2 l}\right\rangle=a^{2 l}(-1)^{l}(2 l+1)!\sum_{\left\{m_{i}\right\}} \prod_{i=1}^{l} \frac{1}{m_{i}!}\left[\frac{N 2^{2 i}(-1)^{i} B_{2 i}}{(2 i)!2 i}\right]^{m_{i}} $$

$$ l=\sum_{i=1}^{l} i \cdot m_{i} . $$

$$ \left\langle R^{2}\right\rangle=a^{2} N, \quad\left\langle R^{4}\right\rangle=\frac{5}{3} a^{4} N^{2}\left(1-\frac{2}{5 N}\right) . $$

$$ \left\langle R^{2}\right\rangle \propto a^{2} N^{2 \nu} $$

$$ P_{L}(\mathbf{R})=\frac{1}{S_{D} R^{D-1}} \sum_{n=0}^{\infty}\left\langle R^{n}\right\rangle \frac{(-1)^{n}}{n!} \partial_{R}^{n} \delta(R) $$

$$ \int d z z^{n} \partial_{z}^{n} \delta(z)=(-1)^{n} n! $$

$$ P_{N}(\mathbf{R})=\frac{i}{4 \pi^{2} a^{2} R} \int_{-\infty}^{\infty} d \eta \eta e^{-i \eta R / a}\left(\frac{\sin \eta}{\eta}\right)^{N} $$

$$ \sin ^{N} \eta=\frac{1}{(2 i)^{N}} \sum_{n=0}^{N}(-1)^{n}\binom{N}{n} \exp [i(N-2 n) \eta], $$

$$ P_{N}(\mathbf{R})=\frac{1}{2^{N+2} i^{N-1} \pi^{2} a^{2} R} \sum_{n=0}^{N}(-1)^{n}\binom{N}{n} I_{N}(N-2 n-R / a), $$

$$ I_{N}(x) \equiv \int_{-\infty}^{\infty} d \eta \frac{e^{i \eta x}}{\eta^{N-1}} $$

$$ I_{N}(x)=\int_{-\infty}^{\infty} d \eta \frac{e^{i x(\eta-i \epsilon)}}{(\eta-i \epsilon)^{N-1}} $$

$$ I_{N}(x)=0, \quad x<0 $$

$$ I_{N}(x)=\frac{2 \pi i^{N-1}}{(N-2)!} x^{N-2}, \quad x>0 $$

$$ P_{N}(\mathbf{R})=\frac{1}{2^{N+1}(N-2)!\pi a^{2} R} \sum_{0 \leq n \leq(N-R / a) / 2}(-1)^{n}\binom{N}{n}(N-2 n-R / a)^{N-2} . $$

$$ P_{N}^{(0)}(\mathbf{R})={\sqrt{\frac{3}{2 \pi N a^{2}}}}^{3} \exp \left\{-\frac{3 R^{2}}{2 N a^{2}}\right\} \rightarrow P_{L}^{(0)}(\mathbf{R})=\sqrt{\frac{D}{2 \pi a L}}^{D} e^{-D R^{2} / 2 a L} $$

$$ \left[\tilde{P}_{1}(\mathbf{k})\right]^{N} \sim e^{-N k^{2} a^{2} / 2 D} $$

$$ P_{N}(\mathbf{R})=P_{N}^{(0)}(\mathbf{k})\left[1+\sum_{n=1}^{\infty} \frac{1}{N^{n}} C_{n}\left(R^{2} / N a^{2}\right)\right], $$

$$ \tilde{C}(\mathbf{k}) \equiv \exp \left[N \sum_{l=2}^{\infty} \frac{2^{2 l}(-1)^{l} B_{2 l}}{(2 l)!2 l}\left(k^{2} a^{2}\right)^{l}\right] $$

$$ \tilde{P}_{N}(\mathbf{k})=e^{-N a^{2} k^{2} / 6} \tilde{C}(\mathbf{k}) $$

$$ \tilde{C}(\mathbf{k})=1+\sum_{\substack{n=1,2, \ldots \\ l=2 n, 2 n+1, \ldots}} \tilde{C}_{n, l} N^{n}\left(a^{2} k^{2}\right)^{l} $$

$$ \begin{array}{ll} \tilde{C}_{1,2}=-\frac{1}{180}, & \tilde{C}_{1,3}=-\frac{1}{2835}, \\ \tilde{C}_{1,4}=-\frac{1}{37800}, & \tilde{C}_{2,4}=\frac{1}{64800}, \quad \ldots . \end{array} $$

$$ \tilde{P}_{N}(\mathbf{k})=e^{-N a^{2} k^{2} / 2 D} \tilde{C}(\mathbf{k}) $$

$$ \begin{align*} \tilde{C}_{1,2} & =-1 / 4 D^{2}(D+2) \\ \tilde{C}_{1,3} & =-1 / 3 D^{3}(D+2)(D+4) \\ \tilde{C}_{1,4} & =-(5 D+12) / 8 D^{4}(D+2)^{2}(D+4)(D+6) \\ \tilde{C}_{2,4} & =1 / 32 D^{4}(D+2)^{2} \end{align*} $$

$$ P_{N}^{(0)}(\mathbf{R})=\sqrt{\frac{D}{2 \pi N a^{2}}} e^{-D R^{2} / 2 N a^{2}} $$

$$ P_{N}^{(0)}(\mathbf{R})=\sqrt{\frac{D}{2 \pi N a^{2}}} e^{-D N \rho^{2} / 2} $$

$$ P_{N}(\mathbf{R})=\int \frac{d^{3} k}{(2 \pi)^{3}} e^{i \mathbf{k R}} P_{N}(\mathbf{k})=\int \frac{d^{3} k}{(2 \pi)^{3}} e^{i \mathbf{k R}} e^{-N k^{2} a^{2} / 2 D} \bar{C}\left(k^{2} a^{2}\right) $$

$$ P_{N}(\mathbf{R})=\bar{C}\left(-2 D \partial_{N}\right) \int \frac{d^{3} k}{(2 \pi)^{3}} e^{i \mathbf{k R}} e^{-N k^{2} a^{2} / 2 D}=\bar{C}\left(-2 D \partial_{N}\right) P_{N}^{(0)}(\mathbf{R}) $$

$$ P_{N}(\mathbf{R})=\left(\frac{D}{2 \pi N a^{2}}\right)^{D / 2} e^{-D N \rho^{2} / 2} C(R) $$

$$ C(R)=1+\sum_{\substack{n=1 \\ l=0, \ldots, 2 n}}^{\infty} C_{n, l} N^{-n}\left(N \rho^{2}\right)^{l} $$

$$ C_{1, l}=\left(-\frac{3}{4}, \frac{3}{2},-\frac{9}{20}\right), \quad C_{2, l}=\left(\frac{29}{160},-\frac{69}{40}, \frac{981}{400},-\frac{1341}{1400}, \frac{81}{800}\right) . $$

$$ \begin{align*} C_{1, l}= & \left(-\frac{D}{4}, \frac{D}{2},-\frac{D^{2}}{4(D+2)}\right) \\ C_{2, l}= & \left(\frac{\left(3 D^{2}-2 D+8\right) D}{96(D+2)},-\frac{\left(D^{2}+2 D+8\right) D}{8(D+2)}, \frac{\left(3 D^{2}+14 D+40\right) D^{2}}{16(D+2)^{2}},\right. \\ & \left.-\frac{\left(3 D^{2}+22 D+56\right) D^{3}}{24(D+2)^{2}(D+4)}, \frac{D^{4}}{32(D+2)^{2}}\right) . \end{align*} $$

$$ \int_{-\infty}^{\infty} d \eta \eta e^{-N f(\eta)} $$

$$ f(\eta)=i \frac{R}{N a} \eta-\log \left(\frac{\sin \eta}{\eta}\right) $$

$$ \operatorname{coth}(i \bar{\eta})-\frac{1}{i \bar{\eta}}=\frac{R}{N a} $$

$$ L(x) \equiv \operatorname{coth} x-\frac{1}{x} $$

$$ L(\bar{x})=\frac{R}{N a} $$

$$ f^{\prime \prime}(\bar{\eta})=L^{\prime}(\bar{x})=-\frac{1}{\sinh ^{2} \bar{x}}+\frac{1}{\bar{x}^{2}}>0 . $$

$$ \begin{align*} P_{N}(\mathbf{R}) & \approx-\frac{1}{4 i \pi^{2} a^{2} R} e^{-N f(\bar{\eta})} \int_{-\infty}^{\infty} d \eta(-i \bar{x}+\eta) \exp \left\{-\frac{N}{2} f^{\prime \prime}(\bar{\eta}) \eta^{2}\right\} \\ & =\frac{\bar{\eta}}{4 \pi^{2} a^{2} R} \sqrt{\frac{2 \pi}{N f^{\prime \prime}(\bar{x})}} e^{-N f(\bar{\eta})} \end{align*} $$

$$ P_{N}(\mathbf{R}) \approx \frac{1}{\left(2 \pi N a^{2}\right)^{3 / 2}} \frac{L^{\mathrm{i}}(\rho)^{2}}{\rho\left\{1-\left[L^{\mathrm{i}}(\rho) / \sinh L^{\mathrm{i}}(\rho)\right]^{2}\right\}^{1 / 2}}\left\{\frac{\sinh L^{\mathrm{i}}(\rho)}{L^{\mathrm{i}}(\rho) \exp \left[\rho L^{\mathrm{i}}(\rho)\right]}\right\}^{N} . $$

$$ \bar{x}=L^{i}(\rho) $$

$$ P_{N}(\mathbf{R})=\mathcal{N}\left(\frac{3}{2 \pi N a^{2}}\right)^{3 / 2} \exp \left(-\frac{3 R^{2}}{2 N a^{2}}\right)\left(1+\frac{3 R^{2}}{2 N^{2} a^{2}}-\frac{9 R^{4}}{20 N^{3} a^{4}}+\ldots\right) $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\frac{1}{{\sqrt{2 \pi\left(\tau_{b}-\tau_{a}\right) / M}}^{D}} \exp \left[-\frac{M}{2} \frac{\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)^{2}}{\tau_{b}-\tau_{a}}\right] . $$

$$ \mathbf{x}_{b}-\mathbf{x}_{a} \equiv \mathbf{R} $$

$$ \begin{align*} \tau_{b}-\tau_{a} & \rightarrow N a \\ M & \rightarrow D / a \end{align*} $$

$$ P_{L}(\mathbf{R})=\int \mathcal{D}^{D} x \exp \left\{-\frac{D}{2 a} \int_{0}^{L} d s\left[\mathbf{x}^{\prime}(s)\right]^{2}\right\}=\sqrt{\frac{D}{2 \pi a}} e^{-D R^{2} / 2 L a} $$

$$ \tilde{P}_{L}(\mathbf{q})=\int d^{D} R e^{-i \mathbf{q} \cdot \mathbf{R}} P_{L}(\mathbf{R}) $$

$$ \tilde{P}_{L}(\mathbf{q})=e^{-L a q^{2} / 2 D} $$

$$ \tilde{P}_{L}(\mathbf{q})=\sum_{l=0}^{\infty}(-1)^{l} \frac{q^{2 l}}{l!}\left(\frac{L a}{2 D}\right)^{l} $$

$$ \tilde{P}_{L}(\mathbf{q})=\sum_{l=0}^{\infty}(-1)^{l} \frac{q^{2 l}}{l!} \frac{\Gamma(D / 2)}{2^{2 l} \Gamma(D / 2+l)}\left\langle R^{2 l}\right\rangle $$

$$ S(\mathbf{q})=\frac{1}{L^{2}} \int_{0}^{L} d s \int_{0}^{L} d s^{\prime}\left\langle e^{i \mathbf{q} \cdot\left[\mathbf{x}(s)-\mathbf{x}\left(s^{\prime}\right)\right]}\right\rangle $$

$$ \begin{align*} \left\langle e^{i \mathbf{q} \cdot\left[\mathbf{x}(s)-\mathbf{x}\left(s^{\prime}\right)\right]}\right\rangle & =\int d^{D} x(L) \int d^{D}\left(x\left(s^{\prime}\right)-x(s)\right) \int d^{D} x(0) \\ & \times P_{L-s^{\prime}}\left(\mathbf{x}(L)-\mathbf{x}\left(s^{\prime}\right)\right) e^{-i \mathbf{q} \cdot \mathbf{x}\left(s^{\prime}\right)} P_{s^{\prime}-s}\left(\mathbf{x}\left(s^{\prime}\right)-\mathbf{x}(s)\right) e^{i \mathbf{q} \cdot \mathbf{x}(s)} P_{s-0}(\mathbf{x}(s)-\mathbf{x}(0)) \end{align*} $$

$$ \left\langle e^{i \mathbf{q} \cdot\left[\mathbf{x}(s)-\mathbf{x}\left(s^{\prime}\right)\right]}\right\rangle=\int d^{D} R e^{-i \mathbf{q} \cdot \mathbf{R}} P_{s^{\prime}-s}(\mathbf{R}) $$

$$ S(\mathbf{q})=\frac{2}{L^{2}} \int_{0}^{L} d L^{\prime}\left(L-L^{\prime}\right) \int d^{D} R e^{i \mathbf{q} \cdot \mathbf{R}\left(L^{\prime}\right)} P_{L^{\prime}}(\mathbf{R}) $$

$$ S(\mathbf{q})=\frac{2}{L^{2}} \int_{0}^{L} d L^{\prime}\left(L-L^{\prime}\right) \tilde{P}_{L^{\prime}}(\mathbf{q}) $$

$$ S^{\mathrm{Gauss}}(\mathbf{q})=\frac{2}{x^{2}}\left(x-1+e^{-x}\right), \quad x \equiv \frac{q^{2} a L}{2 D} $$

$$ S(\mathbf{q})=\sum_{l=0}^{\infty}(-1)^{l} q^{2 l} \frac{\Gamma(D / 2)}{2^{2 l} l!\Gamma(l+D / 2)} \frac{2}{L^{2}} \int_{0}^{L} d L^{\prime}\left(L-L^{\prime}\right)\left\langle R^{2 l}\right\rangle $$

$$ \mathcal{A}^{N}=a \sum_{n=1}^{N} \frac{M}{2} \frac{\left(\Delta \mathbf{x}_{n}\right)^{2}}{a^{2}} $$

$$ \left\langle\left(\Delta \mathbf{x}_{n}\right)^{2}\right\rangle_{0}=\frac{a}{M}=\frac{a^{2}}{D} $$

$$ \left\langle R^{2}\right\rangle=a L, \quad\left\langle R^{2 l}\right\rangle=\frac{(D+2 l-2)!!}{(D-2)!!D^{l}}(a L)^{l} $$

$$ \left\langle R^{2}\right\rangle \equiv L^{2}, \quad\left\langle R^{2 l}\right\rangle \equiv L^{2 l} $$

$$ P_{L}^{\mathrm{rod}}(\mathbf{R})=\frac{1}{S_{D} R^{D-1}} \delta(R-L) $$

$$ \left\langle R^{n}\right\rangle=\int d^{D} R R^{n} P_{L}^{\mathrm{rod}}(\mathbf{R})=\int_{0}^{\infty} d R R^{n} \delta(R-L)=L^{n} $$

$$ P_{L}^{\mathrm{rod}}(\mathbf{R})=\frac{1}{S_{D} R^{D-1}} \sum_{n=0}^{\infty} L^{n}\left\langle R^{n}\right\rangle \frac{(-1)^{n}}{n!} \partial_{R}^{n} \delta(R) $$

$$ \tilde{P}_{L}^{\mathrm{rod}}(\mathbf{q})=\tilde{P}^{\mathrm{rod}}(q L) \equiv \frac{\Gamma(D / 2)}{(q L / 2)^{D / 2-1}} J_{D / 2-1}(q L) $$

$$ \tilde{P}_{L}(\mathbf{q})=S_{D} \int_{0}^{\infty} d R R^{D-1} \tilde{P}^{\mathrm{rod}}(q L) P_{L}(R) $$

$$ \tilde{P}_{L}(\mathbf{q})=4 \pi \int_{0}^{\infty} d R R^{2} \frac{\sin q R}{q R} P_{L}(R) . $$

$$ J_{\nu}(z)=\left(\frac{z}{2}\right)^{\nu} \sum_{l=0}^{\infty} \frac{(-1)^{k}(z / 2)^{2 l}}{l!\Gamma(\nu+l+1)} $$

$$ \tilde{P}_{L}(\mathbf{q})=\sum_{l=0}^{\infty}(-1)^{l}\left(\frac{q}{2}\right)^{2 l} \frac{\Gamma(D / 2)}{l!\Gamma(D / 2+l)} S_{D} \int_{0}^{\infty} d R R^{D-1} R^{2 l} P_{L}(R) $$

$$ S^{\mathrm{rod}}(q L)=\frac{4-2 D}{q^{2} L^{2}}+\left(\frac{2}{q L}\right)^{D / 2} \Gamma(D / 2) J_{D / 2-2}(q L)+2 F\left(1 / 2 ; 3 / 2, D / 2 ;-q^{2} L^{2} / 4\right) $$

$$ S^{\mathrm{rod}}(z)=\frac{2}{z^{2}}[\cos z-1+z \operatorname{Si}(z)], \quad \operatorname{Si}(z) \equiv \int_{0}^{z} \frac{d t}{t} \sin t $$

$$ S_{L}(\mathbf{q})=S_{D} \int_{0}^{\infty} d R R^{D-1} S^{\mathrm{rod}}(\mathbf{q} R) P_{L}(R) $$

$$ E_{\mathrm{bend}}^{N}=\frac{\kappa}{2 a} \sum_{n=1}^{N}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)^{2} $$

$$ \left(\mathbf{u}_{b} L \mid \mathbf{u}_{a} 0\right)=\frac{1}{A} \prod_{n=1}^{N-1}\left[\int \frac{d \mathbf{u}_{n}}{A}\right] \exp \left[-\frac{\kappa}{2 a k_{B} T} \sum_{n=1}^{N}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)^{2}\right], $$

$$ \frac{M r^{2}}{\hbar \epsilon}=\frac{\kappa}{a k_{B} T} $$

$$ A={\sqrt{2 \pi a k_{B} T / \kappa}}^{D-1} $$

$$ \left(\mathbf{u}_{b} L \mid \mathbf{u}_{a} 0\right)=\sum_{l=0}^{\infty}\left[\tilde{I}_{l+D / 2-1}(h)\right]^{N} \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\mathbf{u}_{b}\right) Y_{l \mathbf{m}}^{*}\left(\mathbf{u}_{a}\right), \quad h \equiv \frac{\kappa}{a k_{B} T}, $$

$$ \begin{align*} Z_{N} & =\int \frac{d \mathbf{u}_{a}}{S_{D}} \prod_{n=1}^{N}\left[\int \frac{d \mathbf{u}_{n}}{A}\right] \exp \left[-\frac{\kappa}{2 a k_{B} T} \sum_{n=1}^{N}\left(\mathbf{u}_{n}-\mathbf{u}_{n-1}\right)^{2}\right] \\ & =\int d \mathbf{u}_{b} \int \frac{d \mathbf{u}_{a}}{S_{D}}\left(\mathbf{u}_{b} L \mid \mathbf{u}_{a} 0\right) \end{align*} $$

$$ Z_{N}=\left[\tilde{I}_{D / 2-1}\left(\frac{\kappa}{a k_{B} T}\right)\right]^{N}=\left[\sqrt{\frac{2 \pi \kappa}{a k_{B} T}} e^{-\kappa / a k_{B} T} I_{D / 2-1}\left(\frac{\kappa}{a k_{B} T}\right)\right]^{N} . $$

$$ P_{N}\left(\mathbf{u}_{b}, \mathbf{u}_{a}\right)=\frac{1}{Z_{N}}\left(\mathbf{u}_{b} L \mid \mathbf{u}_{a} 0\right) $$

$$ \int d \mathbf{u}_{b} P_{N}\left(\mathbf{u}_{b}, \mathbf{u}_{a}\right)=\int d \mathbf{u}_{a} P_{N}\left(\mathbf{u}_{b}, \mathbf{u}_{a}\right)=1 $$

$$ Z_{N}^{\mathrm{Heis}} \equiv \int \frac{d \mathbf{u}_{a}}{S_{D}} \prod_{n=1}^{N}\left[\int d \mathbf{u}_{n}\right] \exp \left[\frac{J}{k_{B} T} \sum_{n=1}^{N} \mathbf{u}_{n} \cdot \mathbf{u}_{n-1}\right] $$

$$ Z_{N}^{\text {Heis }}=\left[{\sqrt{{\frac{2 \pi J}{k_{B} T}}^{2-D}}}^{2} I_{D / 2-1}\left(\frac{J}{k_{B} T}\right)\right]^{N} . $$

$$ Z_{N}^{\mathrm{Heis}}\left(J_{1}, \ldots, J_{N}\right)=\prod_{n=1}^{N}\left[{\sqrt{{\frac{2 \pi J_{n}}{k_{B} T}}^{2-D}}}^{2} I_{D / 2-1}\left(\frac{J_{n}}{k_{B} T}\right)\right] $$

$$ \left\langle\mathbf{u}_{n} \cdot \mathbf{u}_{n-1}\right\rangle=\left.\left(k_{B} T\right) \frac{d Z_{N}^{\mathrm{Heis}}\left(J_{1}, \ldots, J_{N}\right)}{d J_{n}}\right|_{J_{n} \equiv J}=\frac{I_{D / 2}\left(J / k_{B} T\right)}{I_{D / 2-1}\left(J / k_{B} T\right)} . $$

$$ E_{N}=N\left\langle\mathbf{u}_{n} \cdot \mathbf{u}_{n-1}\right\rangle=N \frac{I_{D / 2}\left(J / k_{B} T\right)}{I_{D / 2-1}\left(J / k_{B} T\right)} $$

$$ \left\langle\left(\mathbf{u}_{n+1} \cdot \mathbf{u}_{n}\right)\left(\mathbf{u}_{n} \cdot \mathbf{u}_{n-1}\right)\right\rangle=\left.\left(k_{B} T\right)^{2} \frac{d^{2} Z_{N}^{\mathrm{Heis}}\left(J_{1}, \ldots, J_{N}\right)}{d J_{n+1} d J_{n}}\right|_{J_{n} \equiv J}=\left[\frac{I_{D / 2}\left(J / k_{B} T\right)}{I_{D / 2-1}\left(J / k_{B} T\right)}\right]^{2} . $$

$$ \left\langle\mathbf{u}_{n+1} \cdot \mathbf{u}_{n-1}\right\rangle=\left\langle\mathbf{u}_{n+1} \cdot \mathbf{u}_{n}\right\rangle\left\langle\mathbf{u}_{n} \cdot \mathbf{u}_{n-1}\right\rangle=\left[\frac{I_{D / 2}\left(J / k_{B} T\right)}{I_{D / 2-1}\left(J / k_{B} T\right)}\right]^{2}, $$

$$ \left\langle\mathbf{u}_{l} \cdot \mathbf{u}_{k}\right\rangle=\left[\frac{I_{D / 2}\left(J / k_{B} T\right)}{I_{D / 2-1}\left(J / k_{B} T\right)}\right]^{|l-k|} $$

$$ \left\langle\mathbf{u}_{l} \cdot \mathbf{u}_{k}\right\rangle=e^{-|l-k| a / \xi} $$

$$ \xi=-a / \log \left[\frac{I_{D / 2}\left(\kappa / a k_{B} T\right)}{I_{D / 2-1}\left(\kappa / a k_{B} T\right)}\right] $$

$$ \xi=-\frac{a}{\log \left[\operatorname{coth}\left(\kappa / a k_{B} T\right)-a k_{B} T / \kappa\right]} $$

$$ \mathbf{M}=a \sum_{n=0}^{N} \mathbf{u}_{n}, $$

$$ \left\langle\mathbf{M}^{2}\right\rangle=a^{2}(N+1) \frac{1+e^{-a / \xi}}{1-e^{-a / \xi}}-2 a^{2} e^{-a / \xi} \frac{1-e^{-(N+1) a / \xi}}{\left(1-e^{-a / \xi}\right)^{2}} $$

$$ \mathbf{R}=\mathbf{x}_{b}-\mathbf{x}_{a}=a \sum_{n=1}^{N} \mathbf{u}_{n} $$

$$ \begin{align*} P_{N}\left(\mathbf{u}_{b}, \mathbf{u}_{a} ; \mathbf{R}\right) & =\frac{1}{Z_{N}} \frac{1}{A} \prod_{n=2}^{N-1}\left[\int \frac{d \mathbf{u}_{n}}{A}\right] \delta^{(D)}\left(\mathbf{R}-a \sum_{n=1}^{N} \mathbf{u}_{n}\right) \\ & \times \exp \left[-\frac{\kappa}{2 a k_{B} T} \sum_{n=1}^{N-1}\left(\mathbf{u}_{n+1}-\mathbf{u}_{n}\right)^{2}\right] \end{align*} $$

$$ \int d^{D} R P_{N}\left(\mathbf{u}_{b}, \mathbf{u}_{a} ; \mathbf{R}\right)=P_{N}\left(\mathbf{u}_{b}, \mathbf{u}_{a}\right) $$

$$ P_{N}(\mathbf{R})=\int d \mathbf{u}_{b} \int \frac{d \mathbf{u}_{a}}{S_{D}} P_{N}\left(\mathbf{u}_{b}, \mathbf{u}_{a} ; \mathbf{R}\right) $$

$$ \left\langle R^{2 l}\right\rangle=\int d^{D} R R^{2 l} P_{N}(\mathbf{R}) $$

$$ \begin{align*} \left\langle R^{2 l}\right\rangle & =\frac{1}{Z_{N}} \int d^{D} R \frac{1}{A} \int d \mathbf{u}_{b} \prod_{n=2}^{N-1}\left[\int \frac{d \mathbf{u}_{n}}{A}\right]\left[\int \frac{d \mathbf{u}_{a}}{S_{D}}\right] \delta^{(D)}\left(\mathbf{R}-\sum_{n=1}^{N} a \mathbf{u}_{n}\right) \\ & \times R^{2 l} \exp \left[-\frac{\kappa}{2 a k_{B} T} \sum_{n=1}^{N-1}\left(\mathbf{u}_{n+1}-\mathbf{u}_{n}\right)^{2}\right] \end{align*} $$

$$ \begin{align*} &\left\langle R^{2 l}\right\rangle=\frac{1}{Z_{N}} \frac{1}{A} \int d \mathbf{u}_{b} \prod_{n=2}^{N-1}\left[\int \frac{d \mathbf{u}_{n}}{A}\right]\left[\int \frac{d \mathbf{u}_{a}}{S_{D}}\right]\left(a \sum_{n=1}^{N} \mathbf{u}_{n}\right)^{2 l} \\ & \times \exp \left[-\frac{\kappa}{2 a k_{B} T} \sum_{n=1}^{N-1}\left(\mathbf{u}_{n+1}-\mathbf{u}_{n}\right)^{2}\right] \end{align*} $$

$$ \langle 1\rangle=\int d^{D} R P_{N}(\mathbf{R})=\int d \mathbf{u}_{b} \int \frac{d \mathbf{u}_{a}}{S_{D}} P_{N}\left(\mathbf{u}_{b}, \mathbf{u}_{a} \mid L\right)=1 $$

$$ E_{\mathrm{bend}}=\frac{\kappa}{2} \int_{0}^{L} d s\left(\partial_{s} \mathbf{u}\right)^{2} $$

$$ \mathbf{u}(s)=\frac{d}{d s} \mathbf{x}(s) $$

$$ \left(\mathbf{u}_{b} L \mid \mathbf{u}_{a} 0\right)=\int \mathcal{D} \mathbf{u} e^{-\left(\kappa / 2 k_{B} T\right) \int_{0}^{L} d s\left[\mathbf{u}^{\prime}(s)\right]^{2}} $$

$$ P\left(\mathbf{u}_{b}, \mathbf{u}_{a} \mid L\right)=\sum_{l=0}^{\infty} \exp \left(-L \frac{k_{B} T}{2 \kappa} L_{2}\right) \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\mathbf{u}_{b}\right) Y_{l \mathbf{m}}^{*}\left(\mathbf{u}_{a}\right), $$

$$ L_{2}=(D / 2-1+l)^{2}-1 / 4 . $$

$$ L_{2} \rightarrow \hat{L}^{2}=l(l+D-2) $$

$$ \int d \mathbf{u}_{b} P\left(\mathbf{u}_{b}, \mathbf{u}_{a} \mid L\right)=1 $$

$$ \int d \mathbf{u}_{b} \sum_{\mathbf{m}} Y_{l \mathbf{m}}^{*}\left(\mathbf{u}_{b}\right) Y_{l \mathbf{m}}\left(\mathbf{u}_{a}\right)=\delta_{l 0} $$

$$ P\left(\mathbf{u}_{b}, \mathbf{u}_{a} \mid L\right)=\sum_{l=0}^{\infty} \exp \left(-L \frac{k_{B} T}{2 \kappa} \hat{L}^{2}\right) \frac{1}{S_{D}} \frac{2 l+D-2}{D-2} C_{l}^{(D / 2-1)}\left(\mathbf{u}_{2} \mathbf{u}_{1}\right) . $$

$$ R^{2 l}=\left[\int_{0}^{L} d s \mathbf{u}(s)\right]^{2 l} $$

$$ \left\langle R^{2}\right\rangle=\int_{0}^{L} d s_{2} \int_{0}^{L} d s_{1}\left\langle\mathbf{u}\left(s_{2}\right) \mathbf{u}\left(s_{1}\right)\right\rangle=2 \int_{0}^{L} d s_{2} \int_{0}^{s_{2}} d s_{1}\left\langle\mathbf{u}\left(s_{2}\right) \mathbf{u}\left(s_{1}\right)\right\rangle $$

$$ \begin{align*} \left\langle\mathbf{u}\left(s_{2}\right) \mathbf{u}\left(s_{1}\right)\right\rangle & =\int d \mathbf{u}_{b} \int \frac{d \mathbf{u}_{a}}{S_{D}} \int d \mathbf{u}_{2} \int d \mathbf{u}_{1} \\ & \times P\left(\mathbf{u}_{b}, \mathbf{u}_{2} \mid L-s_{2}\right) \mathbf{u}_{2} P\left(\mathbf{u}_{2}, \mathbf{u}_{1} \mid s_{2}-s_{1}\right) \mathbf{u}_{1} P\left(\mathbf{u}_{1}, \mathbf{u}_{a} \mid s_{1}\right) \end{align*} $$

$$ \left\langle\mathbf{u}\left(s_{2}\right) \mathbf{u}\left(s_{1}\right)\right\rangle=\int d \mathbf{u}_{2} \int \frac{d \mathbf{u}_{1}}{S_{D}} \mathbf{u}_{2} \mathbf{u}_{1} P\left(\mathbf{u}_{2}, \mathbf{u}_{1} \mid s_{2}-s_{1}\right) $$

$$ \begin{align*} & \left\langle\mathbf{u}\left(s_{2}\right) \mathbf{u}\left(s_{1}\right)\right\rangle=\int d \mathbf{u}_{2} \mathbf{u}_{2} \mathbf{u}_{1} P\left(\mathbf{u}_{2}, \mathbf{u}_{1} \mid s_{2}-s_{1}\right) \\ & \quad=\sum_{l} e^{-\left(s_{2}-s_{1}\right) k_{B} T \hat{L}^{2} / 2 \kappa}\left[\int d \mathbf{u}_{2} \mathbf{u}_{2} \mathbf{u}_{1} \frac{1}{S_{D}} \frac{2 l+D-2}{D-2} C_{l}^{(D / 2-1)}\left(\mathbf{u}_{2} \mathbf{u}_{1}\right)\right] \end{align*} $$

$$ z C_{l}^{(\nu)}(z)=\frac{1}{2(\nu+l)}\left[(2 \nu+l-1) C_{l-1}^{(\nu)}(z)+(l+1) C_{l+1}^{\nu}(z)\right] $$

$$ \int d \mathbf{u}_{2} C_{0}^{(D / 2-1)}(\cos \theta)=S_{D} $$

$$ \left\langle\mathbf{u}\left(s_{2}\right) \mathbf{u}\left(s_{1}\right)\right\rangle=\exp \left[-\left(s_{2}-s_{1}\right) \frac{k_{B} T}{2 \kappa}(D-1)\right], $$

$$ \xi \equiv 2 \kappa / k_{B} T(D-1) $$

$$ \left\langle R^{2}\right\rangle=2\left\{\xi L-\xi^{2}\left[1-e^{-L / \xi}\right]\right\} . $$

$$ \left\langle R^{2}\right\rangle=L^{2}\left[1-\frac{1}{3} \frac{L}{\xi}+\frac{1}{12}\left(\frac{L}{\xi}\right)^{2}-\frac{1}{60}\left(\frac{L}{\xi}\right)^{3}+\ldots\right] $$

$$ \left\langle R^{2}\right\rangle \approx 2 \xi L\left(1-\frac{\xi}{L}\right)+\ldots $$

$$ a_{\mathrm{eff}}=2 \xi=\frac{4}{D-1} \frac{\kappa}{k_{B} T} $$

$$ \left\langle R^{4}\right\rangle=8 \int_{0}^{L} d s_{4} \int_{0}^{s_{4}} d s_{3} \int_{0}^{s_{3}} d s_{2} \int_{0}^{s_{2}} d s_{1} \delta_{i_{4} i_{3} i_{2} i_{1}}\left\langle u_{i_{4}}\left(s_{4}\right) u_{i_{3}}\left(s_{3}\right) u_{i_{2}}\left(s_{2}\right) u_{i_{1}}\left(s_{1}\right)\right\rangle $$

$$ \delta_{i_{4} i_{3} i_{2} i_{1}} \equiv\left(\delta_{i_{4} i_{3}} \delta_{i_{2} i_{1}}+\delta_{i_{4} i_{2}} \delta_{i_{3} i_{1}}+\delta_{i_{4} i_{1}} \delta_{i_{3} i_{2}}\right) $$

$$ R^{4}=\int_{0}^{L} d s_{4} \int_{0}^{L} d s_{3} \int_{0}^{L} d s_{2} \int_{0}^{L} d s_{1}\left(\mathbf{u}\left(s_{4}\right) \mathbf{u}\left(s_{3}\right)\right)\left(\mathbf{u}\left(s_{2}\right) \mathbf{u}\left(s_{1}\right)\right) $$

$$ \begin{align*} & \left\langle\mathbf{u}_{i_{4}}\left(s_{4}\right) \mathbf{u}_{i_{3}}\left(s_{3}\right) \mathbf{u}_{i_{2}}\left(s_{2}\right) \mathbf{u}_{i_{1}}\left(s_{1}\right)\right\rangle=\int d \mathbf{u}_{4} \int d \mathbf{u}_{3} \int d \mathbf{u}_{2} \int \frac{d \mathbf{u}_{1}}{S_{D}} \\ & \quad \times u_{i_{4}} u_{i_{3}} u_{i_{2}} u_{i_{1}} P\left(\mathbf{u}_{4}, \mathbf{u}_{3} \mid s_{4}-s_{3}\right) P\left(\mathbf{u}_{3}, \mathbf{u}_{2} \mid s_{3}-s_{2}\right) P\left(\mathbf{u}_{2}, \mathbf{u}_{1} \mid s_{2}-s_{1}\right) \end{align*} $$

$$ \begin{align*} \left\langle R^{4}\right\rangle= & \frac{4(D+2)}{D} L^{2} \xi^{2}-8 L \xi^{3}\left(\frac{D^{2}+6 D-1}{D^{2}}-\frac{D-7}{D+1} e^{-L / \xi}\right) \\ & +4 \xi^{4}\left[\frac{D^{3}+23 D^{2}-7 D+1}{D^{3}}-2 \frac{(D+5)^{2}}{(D+1)^{2}} e^{-L / \xi}+\frac{(D-1)^{5}}{D^{3}(D+1)^{2}} e^{-2 D L /(D-1) \xi}\right] \end{align*} $$

$$ \left\langle R^{4}\right\rangle=L^{4}\left[1-\frac{2}{3} \frac{L}{\xi}+\frac{25 D-17}{90(D-1)}\left(\frac{L}{\xi}\right)^{2}-4 \frac{7 D^{2}-8 D+3}{315(D-1)^{2}}\left(\frac{L}{\xi}\right)^{3}+\ldots\right] $$

$$ \left\langle R^{4}\right\rangle=4 \frac{D+2}{D} L^{2} \xi^{2}\left[1-2 \frac{D^{2}+6 D-1}{D(D+2)} \frac{\xi}{L}+\frac{D^{3}+23 D^{2}-7 D+1}{D^{2}(D+2)}\left(\frac{\xi}{L}\right)^{2}\right]+\ldots, $$

$$ P_{L}(\mathbf{R})=\sqrt{\frac{D}{4 \pi L \xi}}^{D} e^{-D R^{2} / 4 L \xi}\left\{1-\frac{2 D-1}{4} \frac{\xi}{L}+\frac{3 D-1}{4} \frac{R^{2}}{L^{2}}-\frac{D(4 D-1)}{16(D+2)} \frac{R^{4}}{\xi L^{3}}\right\} . $$

$$ \frac{1-7 D+23 D^{2}+D^{3}}{D+1}\left[\frac{D+2}{8 D} \frac{\xi^{2}}{L^{2}}\left(1+\frac{R^{2}}{\xi L}\right)+\frac{1}{32} \frac{R^{4}}{L^{4}}\right] . $$

$$ \left\langle R^{n}\right\rangle=\frac{2^{n} \Gamma(D / 2+n / 2)}{D^{n / 2} \Gamma(D / 2)} L^{n} \xi^{n}\left[1+A_{1} \frac{\xi}{L}+A_{2}\left(\frac{\xi}{L}\right)^{2}+\ldots\right] $$

$$ A_{1}=n \frac{n-2-2 d^{2}-4 d(n-1)}{4 d(2+d)}, \quad A_{2}=n(n-2) \frac{1-7 d+23 d^{2}+d^{3}}{8 d^{2}(1+d)} $$

$$ P_{L}(\mathbf{R}) \propto \int d \mathbf{u}_{b} \int d \mathbf{u}_{a} \int \mathcal{D}^{D-1} \mathbf{u} \delta^{(D)}\left(\mathbf{R}-\int_{0}^{L} d s \mathbf{u}(s)\right) e^{-(\bar{\kappa} / 2) \int_{0}^{L} d s\left[\mathbf{u}^{\prime}(s)\right]^{2}} $$

$$ \bar{\kappa}=\frac{\kappa}{k_{B} T}=(D-1) \frac{\xi}{2} $$

$$ P_{L}(\mathbf{R}) \propto \int_{-i \infty}^{i \infty} \frac{d^{D} \lambda}{2 \pi i} e^{\bar{\kappa} \lambda \cdot \mathbf{R} / 2} \int d \mathbf{u}_{b} \int d \mathbf{u}_{a}\left(\mathbf{u}_{b} L \mid \mathbf{u}_{a} 0\right)^{\boldsymbol{\lambda}} $$

$$ \left(\mathbf{u}_{b} L \mid \mathbf{u}_{a} 0\right)^{\boldsymbol{\lambda}} \equiv \int_{\mathbf{u}(0)=\mathbf{u}_{a}}^{\mathbf{u}(L)=\mathbf{u}_{b}} \mathcal{D}^{D-1} \mathbf{u} e^{-(\bar{\kappa} / 2) \int_{0}^{L} d s\left\{\left[\mathbf{u}^{\prime}(s)\right]^{2}+\boldsymbol{\lambda} \cdot \mathbf{u}(s)\right\}} $$

$$ \left(-\frac{1}{2} \Delta_{\mathbf{u}}+\frac{1}{2} \boldsymbol{\lambda} \cdot \mathbf{u}+\frac{d}{d \tau}\right)\left(\mathbf{u} \tau \mid \mathbf{u}_{a} 0\right)^{\boldsymbol{\lambda}}=0 $$

$$ \psi(z, \tau ; \lambda) \equiv \int d \mathbf{u}_{a}\left(\mathbf{u} \tau \mid \mathbf{u}_{a} 0\right)^{\boldsymbol{\lambda}} $$

$$ \hat{H} \psi(z, \tau ; \lambda)=-\frac{d}{d \tau} \psi(z, \tau ; \lambda) $$

$$ \begin{align*} \hat{H} & \equiv \hat{H}_{0}+\lambda \hat{H}_{I}=-\frac{1}{2} \Delta+\lambda z \\ & =-\frac{1}{2}\left[\left(1-z^{2}\right) \frac{d^{2}}{d z^{2}}-(D-1) z \frac{d}{d z}\right]+\frac{1}{2} \lambda z \end{align*} $$

$$ f(L ; \lambda) \equiv \int_{-1}^{1} d z \psi(z, L ; \lambda) $$

$$ f(L ; \lambda)=\sum_{l=0}^{\infty} \frac{\int_{-1}^{1} d z \varphi^{(l) \dagger}(z) \exp \left(-E^{(l)} L\right) \int_{-1}^{1} d z_{a} \varphi^{(l)}\left(z_{a}\right)}{\int_{-1}^{1} d z \varphi^{(l) \dagger}(z) \varphi^{(l)}(z)} $$

$$ E^{(l)}=\sum_{j=0}^{\infty} \epsilon_{j}^{(l)} \lambda^{j}, \quad\left|\varphi^{(l)}\right\rangle=\sum_{l^{\prime}, i=0}^{\infty} \gamma_{l^{\prime}, i}^{(l)} \lambda^{i} \alpha_{l^{\prime}}\left|l^{\prime}\right\rangle $$

$$ \gamma_{l, i}^{(l)}=\delta_{i, 0} \quad \gamma_{k, 0}^{(l)}=\delta_{l, k} $$

$$ \gamma_{k, i}^{(l)} \epsilon_{0}^{(k)}+\sum_{j=0}^{\infty} \frac{\alpha_{j}}{\alpha_{k}} V_{k, j} \gamma_{j, i-1}^{(l)}=\sum_{j=0}^{i} \epsilon_{j}^{(l)} \gamma_{k, i-j}^{(l)} $$

$$ \epsilon_{i}^{(l)}=\sum_{n= \pm 1} \gamma_{l+n, i-1}^{(l)} W_{n}^{(l)} $$

$$ \gamma_{k, i}^{(l)}=\frac{\sum_{j=1}^{i-1} \epsilon_{j}^{(l)} \gamma_{k, i-j}^{(l)}-\sum_{n= \pm 1} \gamma_{k+n, i-1}^{(l)} W_{n}^{(l)}}{\epsilon_{0}^{(k)}-\epsilon_{0}^{(l)}} $$

$$ W_{n}^{(l)} \equiv \frac{\alpha_{l+n}}{\alpha_{l}}\langle l| z|l+n\rangle=0, \text { for } n \neq \pm 1 $$

$$ \langle l| z|l+n\rangle=\frac{\{l|z| l+n\}}{\sqrt{\{l \mid l\}\{l+n \mid l+n\}}}, $$

$$ \{k|F(z)| l\} \equiv \int_{-1}^{1} C_{k}^{D / 2-1}(z) F(z) C_{l}^{D / 2-1}(z)\left(1-z^{2}\right)^{(D-3) / 2} d z $$

$$ \{l \mid l\}=\frac{2^{4-D} \Gamma(l+D-2) \pi}{l!(2 l+D-2) \Gamma(D / 2-1)^{2}} $$

$$ (l+1) \mid l+1\}=(2 l+D-2) z \mid l\}-(l+D-3) \mid l-1\} $$

$$ \begin{align*} \{l+1|z| l\} & =\frac{l+1}{2 l+D-2}\{l+1 \mid l+1\} \\ \{l-1|z| l\} & =\frac{l+D-3}{2 l+D-2}\{l-1 \mid l-1\} \end{align*} $$

$$ \langle l| z|l-1\rangle=\sqrt{\frac{l(l+D-3)}{(2 l+D-2)(2 l+D-4)}}, $$

$$ W_{1}^{(l)}=\frac{\alpha_{l+1}}{\alpha_{l}}\langle l| z|l+1\rangle=1 $$

$$ \frac{\alpha_{l}}{\alpha_{l+1}}=\langle l| z|l+1\rangle=\sqrt{\frac{(l+1)(l+D-2)}{(2 l+D)(2 l+D-2)}} $$

$$ \alpha_{l}=\left[\prod_{j=1}^{l} \frac{(2 l+D-2)(2 l+D-4)}{l(l+D-3)}\right]^{1 / 2} $$

$$ W_{-1}^{(l)}=\frac{l(l+D-3)}{(2 l+D-2)(2 l+D-4)} $$

$$ \varphi_{\mathrm{symm}}^{(l)}(z)=\left\langle z \mid \varphi_{\mathrm{symm}}^{(l)}\right\rangle=\sum_{i=0}^{\infty} \gamma_{0, i}^{(l)} \lambda^{i}\langle z \mid 0\rangle $$

$$ \frac{\left\langle R^{n}\right\rangle}{L^{n}}=1-\frac{n}{6} \frac{L}{\xi}+\frac{n(-13-n+5 D(1+n))}{360(D-1)} \frac{L^{2}}{\xi^{2}}-a_{3} \frac{L^{3}}{\xi^{3}}+a_{4} \frac{L^{4}}{\xi^{4}}+\ldots $$

$$ \begin{align*} & a_{3}=n \frac{444-63 n+15 n^{2}+7 D^{2}\left(4+15 n+5 n^{2}\right)+2 D\left(-124-141 n+7 n^{2}\right)}{45360(D-1)^{2}} \\ & a_{4}=\frac{n}{5443200(d-1)^{3}}\left(D_{0}+D_{1} d+D_{2} d^{2}+D_{3} d^{3}\right) \end{align*} $$

$$ \begin{array}{ll} D_{0}=3\left(-5610+2921 n-822 n^{2}+67 n^{3}\right), D_{1}=8490+12103 n-3426 n^{2}+461 n^{3} \\ D_{2}=45\left(-2-187 n-46 n^{2}+7 n^{3}\right), \end{array} \quad D_{4}=35\left(-6+31 n+30 n^{2}+5 n^{3}\right) . $$

$$ P_{L}(\mathbf{R}) \propto r^{k}\left(1-r^{\beta}\right)^{m} $$

$$ \left\langle r^{2 l}\right\rangle=\frac{\Gamma\left(\frac{3+k+2 l}{\beta}\right) \Gamma\left(\frac{3+k}{\beta}+m+1\right)}{\Gamma\left(\frac{3+k}{\beta}\right) \Gamma\left(\frac{3+k+2 l}{\beta}+m+1\right)} . $$

$$ \mathbf{u}(s)=\mathbf{u}_{0}+\boldsymbol{\eta}(s)=\mathbf{u}_{0}+\sum_{n=1}^{\infty} \mathbf{u}_{n} \cos \nu_{n} s, \quad \nu_{n}=n \pi / L $$

$$ \sigma \equiv \sqrt{1-q^{2}} \approx 1-q^{2} / 2-\left(q^{2}\right)^{2} / 8+\ldots $$

$$ \begin{align*} \mathcal{A} & =\mathcal{A}^{(0)}+\mathcal{A}^{\mathrm{int}}=\frac{\bar{\kappa}}{2} \int_{0}^{L} d s\left[\mathbf{u}^{\prime}(s)\right]^{2}+\frac{1}{2} \delta(0) \log \left(1-q^{2}\right) \\ & \approx \frac{\bar{\kappa}}{2} \int_{0}^{L} d s\left[q^{\prime}(s)\right]^{2}-\frac{1}{2} \delta(0) \int_{0}^{L} d s q^{2} \end{align*} $$

$$ \begin{align*} \delta^{(D)}\left(\mathbf{R}-\int_{0}^{L} d s \mathbf{u}(s)\right) & =\delta\left(R-L+\int_{0}^{L} d s\left\{\frac{1}{2} q^{2}(s)+\frac{1}{8}\left[q^{2}(s)\right]^{2}+\ldots\right\}\right) \\ & \times \delta^{(D-1)}\left(\int_{0}^{L} d s q(s)\right) \end{align*} $$

$$ \bar{q}=L^{-1} \int_{0}^{L} d s q^{\mu}(s)=0, \quad \mu=1, \ldots, d-1 $$

$$ \mathcal{A}_{\mathrm{e}}^{\mathrm{FP}}=\frac{D-1}{2 L} \int_{0}^{L} d s q^{2} $$

$$ P_{L}(\mathbf{R}) \propto \int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q \delta\left(R-L+\int_{0}^{L} d s \frac{1}{2} q^{2}(s)\right) e^{-(\bar{\kappa} / 2) \int_{0}^{L} d s\left[q^{\prime}(s)\right]^{2}} $$

$$ P_{L}(\mathbf{R}) \propto \bar{\kappa} \int_{-i \infty}^{i \infty} \frac{d \omega^{2}}{2 \pi i} e^{\bar{\kappa} \omega^{2}(L-R)} \int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q \exp \left[-\frac{\bar{\kappa}}{2} \int_{0}^{L} d s\left(q^{\prime 2}+\omega^{2} q^{2}\right)\right] $$

$$ \int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q \exp \left[-\frac{\bar{\kappa}}{2} \int_{0}^{L} d s\left(q^{\prime 2}+\omega^{2} q^{2}\right)\right] \propto\left(\frac{\omega L}{\sinh \omega L}\right)^{(D-1) / 2} $$

$$ P_{L}(\mathbf{R}) \propto \bar{\kappa} \int_{-i \infty}^{i \infty} \frac{d \omega^{2}}{2 \pi i} e^{\bar{\kappa} \omega^{2}(L-R)}\left(\frac{\omega L}{\sinh \omega L}\right)^{(D-1) / 2} $$

$$ P_{L}(\mathbf{R}) \propto \bar{\kappa} \int_{-i \infty}^{i \infty} \frac{d \omega^{2}}{2 \pi i} e^{\bar{\kappa} \omega^{2}(L-R)} \prod_{n=1}^{\infty}\left(1+\frac{\omega^{2}}{\nu_{n}^{2}}\right)^{-1} $$

$$ \sum_{k=1}^{\infty} \nu_{k}^{2} \prod_{n(\neq k)=1}^{\infty}\left(1-\frac{k^{2}}{n^{2}}\right)^{-1} $$

$$ \begin{align*} \left\{\prod_{n=1}^{\infty}\left[1-\frac{(k+\epsilon)^{2}}{n^{2}}\right]^{-1}\right\}\left[1-\frac{(k+\epsilon)^{2}}{k^{2}}\right] & \rightarrow \frac{(k+\epsilon) \pi}{\sin (k+\epsilon) \pi} \frac{-2 \epsilon}{k} \rightarrow \frac{-2 \epsilon \pi}{\sin (k+\epsilon) \pi} \\ & \rightarrow-\frac{2 \epsilon \pi}{\cos k \pi \sin \epsilon \pi} \rightarrow 2(-1)^{k+1} \end{align*} $$

$$ P_{L}(\mathbf{R}) \propto \sum_{k=1}^{\infty} 2(-1)^{k+1} \bar{\kappa} \nu_{k}^{2} e^{-\bar{\kappa} \nu_{k}^{2}(L-R)} $$

$$ P_{L}(\mathbf{R})=\mathcal{N} L(\mathbf{R})=\mathcal{N} \sum_{k=1}^{\infty}(-1)^{k+1} k^{2} \pi^{2} e^{-k^{2} \pi^{2}(1-r) / l} $$

$$ \int d^{3} R P_{L}(\mathbf{R})=4 \pi L^{3} \int_{0}^{\infty} d r r^{2} P_{L}(\mathbf{R})=1 $$

$$ \left(\frac{\omega L}{\sinh \omega L}\right)^{(D-1) / 2}=(\omega L)^{(D-1) / 2} \sum_{k=0}^{\infty}(-1)^{k}\binom{-(D-1) / 2}{k} e^{-(2 k+(D-1) / 2) \omega L}, $$

$$ P_{L}(\mathbf{R}) \propto \sum_{k=0}^{\infty}(-1)^{k}\binom{-(D-1) / 2}{k} I_{k}(R / L) $$

$$ I_{k}(r) \equiv \int_{-i \infty}^{i \infty} \frac{d \bar{\omega}}{2 \pi i} \bar{\omega}^{(D+1) / 2} e^{-[2 k+(D-1) / 2] \bar{\omega}+(D-1) \bar{\omega}^{2}(1-r) / 2 l}, $$

$$ \int_{-i \infty}^{i \infty} \frac{d x}{2 \pi i} x^{\nu} e^{\beta x^{2} / 2-q x}=\frac{1}{\sqrt{2 \pi} \beta^{(\nu+1) / 2}} e^{-q^{2} / 4 \beta} D_{\nu}(q / \sqrt{\beta}), $$

$$ D_{n}(z)=\frac{1}{\sqrt{2}^{n}} e^{-z^{2} / 4} H_{n}(z / \sqrt{2}) . $$

$$ I_{k}(r)=\frac{1}{\sqrt{2 \pi}}\left[\frac{l}{(D-1)(1-r)}\right]^{(D+3) / 4} e^{-\frac{[2 k+(D-1) / 2]^{2}}{4(D-1)(1-r) / l}} D_{(D+1) / 2}\left(\frac{2 k+(D-1) / 2}{\sqrt{(D-1)(1-r) / l}}\right) $$

$$ I_{k}(r)=\frac{1}{2 \sqrt{2 \pi}} \frac{1}{\sqrt{2(1-r) / l}^{3}} e^{-\frac{(2 k+1)^{2}}{4(1-r) / l}} H_{2}\left(\frac{2 k+1}{2 \sqrt{(1-r) / l}}\right) $$

$$ f\left(\bar{\omega}^{2}\right) \equiv \sqrt{\frac{\bar{\omega}}{\sinh \bar{\omega}}}^{D-1} $$

$$ f\left(\bar{\omega}^{2}\right)=1-\frac{D-1}{2^{2} \cdot 3} \bar{\omega}^{2}+\frac{(D-1)(5 D-1)}{2^{5} \cdot 3^{2} \cdot 5} \bar{\omega}^{4}-\frac{(D-1)\left(15+14 D+35 D^{2}\right)}{2^{7} \cdot 3^{4} \cdot 5 \cdot 7} \bar{\omega}^{6}+\ldots $$

$$ \bar{\omega}^{2} \rightarrow \hat{\bar{\omega}}^{2} \equiv-\frac{L}{\bar{\kappa}} \frac{d}{d r}=-\frac{2 l}{D-1} \frac{d}{d r} $$

$$ P_{L}(\mathbf{R}) \propto\left[1+\frac{l}{6} \frac{d}{d r}+\frac{(-1+5 D) l^{2}}{360(D-1)} \frac{d^{2}}{d r^{2}}+\frac{\left(15+14 D+35 D^{2}\right) l^{3}}{45360(D-1)^{2}} \frac{d^{3}}{d r^{3}}+\ldots\right] \delta(r-1) $$

$$ \left\langle R^{m}\right\rangle=\int d^{D} R R^{m} P_{L}(\mathbf{R}) \propto \int_{0}^{\infty} d r r^{D-1} r^{m} P_{L}(\mathbf{R}) $$

$$ \left\langle(r-1)^{n}\right\rangle_{1} \propto \int_{0}^{\infty} d r(r-1)^{n} P_{L}(\mathbf{R}) $$

$$ \begin{align*} & \left\langle R^{2}\right\rangle=L^{2}\left[1-\frac{1}{3} l+\frac{13 D-9}{180(D-1)} l^{2}-\frac{8}{945} \quad l^{3}+\ldots\right] \\ & \left\langle R^{4}\right\rangle=L^{4}\left[1-\frac{2}{3} l+\frac{23 D-11}{90(D-1)} l^{2}-\frac{123 D^{2}-98 D+39}{1890(D-1)^{2}} l^{3}+\ldots\right] \end{align*} $$

$$ \begin{align*} & \left\langle R^{2}\right\rangle=L^{2}\left[1-\frac{1}{3} l+\frac{1}{12} l^{2}-\frac{8}{945} l^{3}+\ldots\right] \\ & \left\langle R^{4}\right\rangle=L^{4}\left[1-\frac{2}{3} l+\frac{29}{90} l^{2}-\frac{71}{630} l^{3}+\ldots\right] \end{align*} $$

$$ P(r ; L)=S_{D}^{-1} \int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q(s) \delta\left(r-L^{-1} \int_{0}^{L} d s \sqrt{1-q^{2}(s)}\right) e^{-\mathcal{A}^{\mathrm{tot}}[q]-\mathcal{A}^{\mathrm{cor}}\left[q_{b}, q_{a}\right]} $$

$$ \mathcal{A}_{\mathrm{tot}}[q]=\frac{1}{2 \varepsilon} \int_{0}^{L} d s\left[g_{\mu \nu}(q) \dot{q}^{\mu}(s) \dot{q}^{\nu}(s)-\varepsilon \delta(s, s) \log g(q(s))\right]+\mathcal{A}^{\mathrm{FP}}-\varepsilon L \frac{R}{8} $$

$$ Z=S_{D} \int_{0}^{\infty} d r r^{D-1} P(r ; L)=1 $$

$$ \mathcal{A}^{\mathrm{FP}}[q]=-(D-1) \log \left(L^{-1} \int_{0}^{L} d s \sqrt{1-q^{2}(s)}\right) $$

$$ \mathcal{A}^{\operatorname{cor}}\left[q_{b}, q_{a}\right]=-\log J\left[q_{b}, q_{a}\right]=-\left[q^{2}(0)+q^{2}(L)\right] / 4 $$

$$ Z=\int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q(s) \exp \left\{-\mathcal{A}_{\mathrm{tot}}[q]-\mathcal{A}^{\mathrm{cor}}\left[q_{b}, q_{a}\right]-\mathcal{A}^{\mathrm{FP}}[q]\right\} $$

$$ \mathbf{R}^{2}=\int_{0}^{L} d s \int_{0}^{L} d s^{\prime} \mathbf{u}(s) \cdot \mathbf{u}\left(s^{\prime}\right)=\left(\int_{0}^{L} d s \sqrt{1-q^{2}(s)}\right)^{2}=R^{2} $$

$$ \left\langle\left(\mathbf{R}^{2}\right)^{n}\right\rangle=\left\langle\left[\int_{0}^{L} \int_{0}^{L} d s d s^{\prime} \mathbf{u}(s) \cdot \mathbf{u}\left(s^{\prime}\right)\right]^{n}\right\rangle $$

$$ \left\langle R^{n}\right\rangle=\int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q(s) \exp \left\{-\mathcal{A}_{\mathrm{tot}}[q]-\mathcal{A}_{\mathrm{cor}}\left[q_{b}, q_{a}\right]-\mathcal{A}_{n}^{\mathrm{FP}}[q]\right\} $$

$$ \mathcal{A}_{n}^{\mathrm{FP}}[q]=-(n+D-1) \log \left(L^{-1} \int_{0}^{L} d s \sqrt{1-q^{2}(s)}\right) $$

$$ \Delta_{\mathrm{N}}^{\prime}\left(s, s^{\prime}\right)=\frac{L}{3}-\frac{\left|s-s^{\prime}\right|}{2}-\frac{\left(s+s^{\prime}\right)}{2}+\frac{\left(s^{2}+s^{\prime 2}\right)}{2 L} . $$

$$ \int_{0}^{L} d s \Delta_{\mathrm{N}}^{\prime}\left(s, s^{\prime}\right)=0 $$

$$ \left\langle R^{n}\right\rangle=\int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q(s) \exp \left\{-\mathcal{A}_{\mathrm{tot}, n}[q ; \varepsilon] \cdot\right\} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{tot}, n}[q ; \varepsilon] & =\int_{0}^{L} d s\left[\frac{1}{2}\left(\dot{q}^{2}+\varepsilon \frac{(q \dot{q})^{2}}{1-\varepsilon q^{2}}\right)+\frac{1}{2} \delta(0) \log \left(1-\varepsilon q^{2}\right)\right] \\ & -\sigma_{n} \log \left[\frac{1}{L} \int_{0}^{L} d s \sqrt{1-\varepsilon q^{2}}\right]-\frac{\varepsilon}{4}\left[q^{2}(0)+q^{2}(L)\right]-\varepsilon L \frac{R}{8} \end{align*} $$

$$ \sigma_{n} \equiv n+(d-1) $$

$$ \mathcal{A}_{\mathrm{tot}, n}[q ; \varepsilon]=\mathcal{A}^{(0)}[q]+\mathcal{A}_{n}^{\mathrm{int}}[q ; \varepsilon] $$

$$ \mathcal{A}^{(0)}[q]=\frac{1}{2} \int_{0}^{L} d s \dot{q}^{2}(s) $$

$$ \mathcal{A}_{n}^{\mathrm{int}}[q ; \varepsilon]=\varepsilon \mathcal{A}_{n}^{\mathrm{int} 1}[q]+\varepsilon^{2} \mathcal{A}_{n}^{\mathrm{int} 2}[q]+\ldots . $$

$$ Z^{(0)} \equiv \int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q(s) e^{-\mathcal{A}^{(0)}[q]}=\int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q(s) e^{-(1 / 2) \int_{0}^{L} d s \dot{q}^{2}(s)}=1 $$

$$ \mathcal{A}_{n}^{\mathrm{int} 1}[q]=\frac{1}{2} \int_{0}^{L} d s\left\{[q(s) \dot{q}(s)]^{2}-\rho_{n}(s) q^{2}(s)\right\}-L \frac{R}{8} $$

$$ \rho_{n}(s) \equiv \delta_{n}+[\delta(s)+\delta(s-L)] / 2, \quad \delta_{n} \equiv \delta(0)-\sigma_{n} / L . $$

$$ \begin{align*} \mathcal{A}_{n}^{\mathrm{int} 2}[q] & =\frac{1}{2} \int_{0}^{L} d s\left\{[q(s) \dot{q}(s)]^{2}-\frac{1}{2}\left[\delta(0)-\frac{\sigma_{n}}{2 L}\right] q^{2}(s)\right\} q^{2}(s) \\ & +\frac{\sigma_{n}}{8 L^{2}} \int_{0}^{L} d s \int_{0}^{L} d s^{\prime} q^{2}(s) q^{2}\left(s^{\prime}\right) \end{align*} $$

$$ \langle F[q]\rangle_{0} \equiv \int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q(s) F[q] e^{-(1 / 2) \int_{0}^{L} d s \dot{q}^{2}(s)} $$

$$ \begin{align*} \left\langle R^{n}\right\rangle / L^{n} & =1-\left\langle\mathcal{A}_{n}^{\mathrm{int}}[q ; \varepsilon]\right\rangle_{0}+\frac{1}{2}\left\langle\mathcal{A}_{n}^{\mathrm{int}}[q ; \varepsilon]^{2}\right\rangle_{0}-\ldots \\ & =1-\varepsilon\left\langle\mathcal{A}_{n}^{\mathrm{int} 1}[q]\right\rangle_{0}+\varepsilon^{2}\left(-\left\langle\mathcal{A}_{n}^{\mathrm{int} 2}[q]\right\rangle_{0}+\frac{1}{2}\left\langle\mathcal{A}_{n}^{\mathrm{int} 1}[q]^{2}\right\rangle_{0}\right)-\ldots \end{align*} $$

$$ \left\langle q^{\mu}(s) q^{\nu}\left(s^{\prime}\right)\right\rangle_{0}=\delta^{\mu \nu} \Delta\left(s, s^{\prime}\right) $$

$$ \begin{align*} \left\langle\mathcal{A}_{n}^{\mathrm{int} 1}[q]\right\rangle_{0} & =\frac{(D-1)}{2}\left[\frac{\sigma_{n}}{L} I_{1}+D I_{2}-\frac{1}{2} \Delta(0,0)-\frac{1}{2} \Delta(L, L)\right]-L \frac{R}{8}=L \frac{(D-1) n}{12} \\ \left\langle\mathcal{A}_{n}^{\mathrm{int} 2}[q]\right\rangle_{0} & =\frac{\left(D^{2}-1\right)}{4}\left[\left(\delta(0)+\frac{\sigma_{n}}{2 L}\right) I_{3}+2(D+2) I_{4}\right]+\frac{(D-1) \sigma_{n}}{8 L^{2}}\left[(D-1) I_{1}^{2}+2 I_{5}\right] \\ & =L^{3} \frac{\left(D^{2}-1\right)}{120} \delta(0)+L^{2} \frac{(D-1)}{1440}\left[\left(25 D^{2}+36 D+23\right)+n(11 D+5)\right] \\ \frac{1}{2}\left\langle\mathcal{A}_{n}^{\mathrm{int} 1}[q]^{2}\right\rangle_{0} & =\frac{L^{2}}{2}\left\{\frac{(D-1) L}{12}\left[\left(\delta(0)+\frac{D}{2 L}\right)-\left(\delta_{n}+\frac{2}{L}\right)\right]-\frac{R}{8}\right\}^{2} \\ & +\frac{(D-1)}{4}\left[H_{1}^{n}-2\left(H_{2}^{n}+H_{3}^{n}-H_{5}\right)+H_{6}-4 D\left(H_{4}^{n}-H_{7}-H_{10}\right)\right. \\ & \left.+H_{11}+2 D^{2}\left(H_{8}+H_{9}\right)\right]+\frac{(D-1)}{4}\left[D H_{12}+2(D+2) H_{13}+D H_{14}\right] \\ & =\frac{L^{2}}{2}\left[\frac{(D-1) n}{12}\right]^{2}+L^{3} \frac{(D-1)}{120} \delta(0) \\ & +L^{2} \frac{(D-1)}{1440}\left[\left(25 D^{2}-22 D+25\right)+4(n+4 D-2)\right] \\ & +L^{3} \frac{(D-1) D}{120} \delta(0)+L^{2} \frac{(D-1)}{720}(29 D-1) \end{align*} $$

$$ \begin{align*} \left\langle R^{n}\right\rangle / L^{n} & =1-\varepsilon L \frac{(D-1) n}{12}+\varepsilon^{2} L^{2}\left[\frac{(D-1)^{2} n^{2}}{288}+\frac{(D-1)(4 n+5 D-13) n}{1440}\right]-\mathcal{O}\left(\varepsilon^{3}\right) \\ & =1-\frac{n}{6} l+\left[\frac{n^{2}}{72}+\frac{(4 n+5 D-13) n}{360(D-1)}\right] l^{2}-\mathcal{O}\left(l^{3}\right) \end{align*} $$

$$ G\left(s, s^{\prime}\right)=e^{-\left|s-s^{\prime}\right| / \xi}=e^{-\left|s-s^{\prime}\right| l / L} $$

$$ G\left(s, s^{\prime}\right)=\left\langle\mathbf{u}(s) \cdot \mathbf{u}\left(s^{\prime}\right)\right\rangle=\int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q(s) f\left(s, s^{\prime}\right) \exp \left\{-\mathcal{A}_{\mathrm{tot}}^{(0)}[q ; \varepsilon]\right\} $$

$$ f\left(q(s), q\left(s^{\prime}\right)\right) \equiv \mathbf{u}(s) \cdot \mathbf{u}\left(s^{\prime}\right)=\sqrt{1-q^{2}(s)} \sqrt{1-q^{2}\left(s^{\prime}\right)}+q(s) q\left(s^{\prime}\right) $$

$$ f\left(q(s), q\left(s^{\prime}\right)\right)=1+\varepsilon f_{1}\left(q(s), q\left(s^{\prime}\right)\right)+\varepsilon^{2} f_{2}\left(q(s), q\left(s^{\prime}\right)\right)+\ldots $$

$$ \begin{align*} f_{1}\left(q(s), q\left(s^{\prime}\right)\right) & =q(s) q\left(s^{\prime}\right)-\frac{1}{2} q^{2}(s)-\frac{1}{2} q^{2}\left(s^{\prime}\right) \\ f_{2}\left(q(s), q\left(s^{\prime}\right)\right) & =\frac{1}{4} q^{2}(s) q^{2}\left(s^{\prime}\right)-\frac{1}{8}\left[q^{2}(s)\right]^{2}-\frac{1}{8}\left[q^{2}\left(s^{\prime}\right)\right]^{2} \end{align*} $$

$$ f\left(q(s), q\left(s^{\prime}\right)\right) \equiv e^{-\mathcal{A}^{f}[q ; \varepsilon]} $$

$$ \begin{align*} \mathcal{A}^{f}[q ; \varepsilon] & =-\log f\left(q(s), q\left(s^{\prime}\right)\right) \\ & =-\varepsilon f_{1}\left(q(s), q\left(s^{\prime}\right)\right)+\varepsilon^{2}\left[-f_{2}\left(q(s), q\left(s^{\prime}\right)\right)+\frac{1}{2} f_{1}^{2}\left(s, s^{\prime}\right)\right]-\ldots \end{align*} $$

$$ G\left(s, s^{\prime}\right)=1-\left\langle\left(\mathcal{A}_{0}^{\mathrm{int}}[q ; \varepsilon]+\mathcal{A}^{f}[q ; \varepsilon]\right)\right\rangle_{0}+\frac{1}{2}\left\langle\left(\mathcal{A}_{0}^{\mathrm{int}}[q ; \varepsilon]+\mathcal{A}^{f}[q ; \varepsilon]\right)^{2}\right\rangle_{0}-\ldots $$

$$ G\left(s, s^{\prime}\right)=1+\varepsilon\left\langle f_{1}\left(q(s), q\left(s^{\prime}\right)\right)\right\rangle_{0}+\varepsilon^{2}\left[\left\langle f_{2}\left(q(s), q\left(s^{\prime}\right)\right)\right\rangle_{0}-\left\langle f_{1}\left(q(s), q\left(s^{\prime}\right)\right) \mathcal{A}_{0}^{\mathrm{int} 1}[q]\right\rangle_{0}\right]+\ldots, $$

$$ \begin{align*} & \left\langle f_{1}\left(q(s), q\left(s^{\prime}\right)\right)\right\rangle_{0}=-\frac{(D-1)}{2}\left|s-s^{\prime}\right| \\ & \left\langle f_{2}\left(q(s), q\left(s^{\prime}\right)\right)\right\rangle_{0}-\left\langle f_{1}\left(q(s), q\left(s^{\prime}\right)\right) \mathcal{A}_{0}^{\mathrm{int} 1}[q]\right\rangle_{0} \\ & =(D-1)\left[\frac{1}{2} D_{1}-\frac{1}{8}(D+1) D_{2}^{2}-K_{1}-D K_{2}-\frac{(D-1)}{L} K_{3}+\frac{1}{2} K_{4}+\frac{1}{2} K_{5}\right] \\ & \quad=\frac{1}{8}(D-1)^{2}\left(s-s^{\prime}\right)^{2} \end{align*} $$

$$ G\left(s, s^{\prime}\right)=1-\varepsilon \frac{D-1}{2}\left|s-s^{\prime}\right|+\varepsilon^{2} \frac{(D-1)^{2}}{8}\left(s-s^{\prime}\right)^{2}+\ldots=1-\frac{\left|s-s^{\prime}\right|}{\xi}+\frac{\left(s-s^{\prime}\right)^{2}}{2 \xi^{2}}-\ldots $$

$$ P(k ; L)=\int d r e^{i k(r-1)} P(r ; L) $$

$$ P(k ; L)=\int_{\mathrm{NBC}} \mathcal{D}^{\prime D-1} q(s) \exp \left\{-\mathcal{A}_{k}^{\mathrm{tot}}[q ; \varepsilon]\right\}, $$

$$ \begin{align*} \mathcal{A}_{k}^{\mathrm{tot}}[q ; \varepsilon] & =\int_{0}^{L} d s\left\{\frac{1}{2}\left[\dot{q}^{2}+\varepsilon \frac{(q \dot{q})^{2}}{1-\varepsilon q^{2}}\right]+\frac{1}{2} \delta(0) \log \left(1-\varepsilon q^{2}\right)-\frac{i k}{L}\left(\sqrt{1-\varepsilon q^{2}}-1\right)\right\} \\ & -\frac{1}{4} \varepsilon\left[q^{2}(0)+q^{2}(L)\right]-\varepsilon L \frac{R}{8} \equiv \mathcal{A}^{0}[q]+\mathcal{A}_{k}^{\mathrm{int}}[q ; \varepsilon] \end{align*} $$

$$ \rho_{k}(s)=\delta_{k}+[\delta(s)+\delta(s-L)] / 2, \quad \delta_{k}=\delta(0)-i k / L $$

$$ \mathcal{A}_{k}^{\mathrm{int} 1}[q]=\int_{0}^{L} d s \frac{1}{2}\left\{[q(s) \dot{q}(s)]^{2}-\rho_{k}(s) q^{2}(s)\right\}-L \frac{R}{8} $$

$$ \mathcal{A}_{k}^{\mathrm{int} 2}[q]=\int_{0}^{L} d s \frac{1}{2}\left\{[q(s) \dot{q}(s)]^{2}-\frac{1}{2}\left(\delta(0)-\frac{i k}{2 L}\right) q^{2}(s)\right\} q^{2}(s) $$

$$ P(k ; L)=1-\varepsilon\left\langle\mathcal{A}_{k}^{\mathrm{int} 1}[q]\right\rangle_{0}+\varepsilon^{2}\left(-\left\langle\mathcal{A}_{k}^{\mathrm{int} 2}[q]\right\rangle_{0}+\frac{1}{2}\left\langle\mathcal{A}_{k}^{\mathrm{int} 1}[q]^{2}\right\rangle_{0}\right)-\ldots $$

$$ \begin{align*} \left\langle\mathcal{A}_{, k}^{\mathrm{int} 1}[q]\right\rangle_{0} & =\frac{(D-1)}{2}\left[\frac{i k}{L} I_{1}+D I_{2}-\frac{1}{2} \Delta(0,0)-\frac{1}{2} \Delta(L, L)\right]-L \frac{R}{8} \\ & =-L \frac{(D-1)[(D-1)-i k]}{12} \\ \left\langle\mathcal{A}_{, k}^{\mathrm{int} 2}[q]\right\rangle_{0} & =\frac{\left(D^{2}-1\right)}{4}\left[\left(\delta(0)+\frac{i k}{2 L}\right) I_{3}+2(D+2) I_{4}\right] \\ & =L^{3} \frac{\left(D^{2}-1\right)}{120} \delta(0)+L^{2} \frac{\left(D^{2}-1\right)[7(D+2)+3 i k]}{720} \\ \frac{1}{2}\left\langle\mathcal{A}_{, k}^{\mathrm{int} 1}[q]^{2}\right\rangle_{0} & =\frac{L^{2}}{2}\left\{\frac{(D-1) L}{12}\left[\left(\delta(0)+\frac{D}{2 L}\right)-\left(\delta_{k}+\frac{2}{L}\right)\right]-\frac{R}{8}\right\}^{2} \\ & +\frac{(D-1)}{4}\left[H_{1}^{k}-2\left(H_{2}^{k}+H_{3}^{k}-H_{5}\right)+H_{6}-4 D\left(H_{4}^{k}-H_{7}-H_{10}\right)\right. \\ & \left.+H_{11}+2 D^{2}\left(H_{8}+H_{9}\right)\right]+\frac{(D-1)}{4}\left[D H_{12}+2(D+2) H_{13}+D H_{14}\right] \\ & =L^{2} \frac{(D-1)^{2}[(D-1)-i k]^{2}}{2 \cdot 12^{2}} \\ & +L^{3} \frac{(D-1)}{120} \delta(0)+L^{2} \frac{(D-1)}{1440}\left[\left(13 D^{2}-6 D+21\right)+4 i k(2 D+i k)\right] \\ & +L^{3} \frac{(D-1) D}{120} \delta(0)+L^{2} \frac{(D-1)}{720}(29 D-1) \end{align*} $$

$$ \begin{align*} P(k ; L) & =1+\varepsilon L \frac{(D-1)}{12}[(D-1)-i k]+\varepsilon^{2} L^{2} \frac{(D-1)}{1440} \\ & \times\left[(i k)^{2}(5 D-1)-2 i k\left(5 D^{2}-11 D+8\right)+(D-1)\left(5 D^{2}-11 D+14\right)\right]+\mathcal{O}\left(\varepsilon^{3}\right) \end{align*} $$

$$ P(k ; L)=P_{1 \text { loop }}(k ; L)\left\{1+\frac{(D-1)}{6} l+\left[\frac{(D-3)}{180(D-1)} i k+\frac{\left(5 D^{2}-11 D+14\right)}{360}\right] l^{2}+\mathcal{O}\left(l^{3}\right)\right\}, $$

$$ P_{1 \text { loop }}(k ; L)=1-\varepsilon L \frac{(D-1)}{2^{2} \cdot 3}(i k)+\varepsilon^{2} L^{2} \frac{(D-1)(5 D-1)}{2^{5} \cdot 3^{2} \cdot 5}(i k)^{2}-\ldots $$

$$ \begin{align*} P(r ; l) & =\delta(r-1)+\frac{l}{6}\left[\delta^{\prime}(r-1)+(d-1) \delta(r-1)\right]+\frac{l^{2}}{360(d-1)}\left[(5 d-1) \delta^{\prime \prime}(r-1)\right. \\ & \left.+2\left(5 d^{2}-11 d+8\right) \delta^{\prime}(r-1)+(d-1)\left(5 d^{2}-11 d+14\right) \delta(r-1)\right]+\mathcal{O}\left(l^{3}\right) \end{align*} $$

$$ \left\langle R^{n}\right\rangle=L^{n} \int d r r^{n+(D-1)} P(r ; l) $$

$$ P(r ; l) \propto \int_{-\infty}^{\infty} \frac{d \hat{\omega}^{2}}{2 \pi} e^{-i \hat{\omega}^{2}(r-1)(D-1) / 2 l}\left(\frac{\bar{\omega}}{\sinh \bar{\omega}}\right)^{(D-1) / 2} e^{-V\left(l, \hat{\omega}^{2}\right)} $$

$$ V\left(l, \hat{\omega}^{2}\right) \equiv V_{0}(l)+\bar{V}\left(l, \hat{\omega}^{2}\right)=V_{0}(l)+V_{1}(l) \frac{\hat{\omega}^{2}}{l}+V_{2}(l) \frac{\hat{\omega}^{4}}{l^{2}}+V_{3}(l) \frac{\hat{\omega}^{6}}{l^{3}}+\ldots . $$

$$ \begin{align*} V_{0}(l) & =-\frac{d-1}{6} l+\frac{d-9}{360} l^{2}+\frac{(d-1)\left(32-13 d+5 d^{2}\right)}{6480} l^{3} \\ & -\frac{34-272 d+259 d^{2}-110 d^{3}+25 d^{4}}{259200} l^{4}+\ldots \end{align*} $$

$$ \begin{align*} & V_{1}(l)=-\frac{d-3}{360} l^{2}+\frac{(-5+9 d)}{7560(-1+d)} l^{3}+\frac{\left(-455+431 d+91 d^{2}+5 d^{3}\right)}{907200(d-1)^{2}} l^{4}+\ldots \\ & V_{2}(l)=-\frac{(5-3 d) l^{3}}{7560}-\frac{\left(-31+42 d+25 d^{2}\right) l^{4}}{907200(d-1)}+\ldots \\ & V_{3}(l)=-\frac{(d-1) l^{4}}{18900}+\ldots \end{align*} $$

$$ \left\langle\hat{\omega}^{2} / l\right\rangle=a_{k}^{2} \equiv \frac{2 k+(D-1) / 2}{(D-1)(1-r)}, \quad\left\langle\hat{\omega}^{2} / l\right\rangle^{2}=3 a_{k}^{4}, \quad\left\langle\hat{\omega}^{2} / l\right\rangle^{3}=15 a_{k}^{6} $$

$$ f_{k}=V_{1}(l) a_{k}^{2}+\left[3 V_{2}(l)-V_{1}^{2}(l)\right] a_{k}^{4}+\left[15 V_{3}(l)-12 V_{1}(l) V_{2}(l)+\frac{4}{3} V_{1}^{3}(l)\right] a_{k}^{6}+\ldots, $$

$$ \begin{align*} 3 V_{2}(l)-V_{1}^{2}(l) & =\frac{3 D-5}{2520} l^{3}+\frac{156-231 D-26 D^{2}-7 D^{3}}{907200(D-1)} l^{4}+\ldots \\ 15 V_{3}(l)-12 V_{1}(l) V_{2}(l)+\frac{4}{3} V_{1}^{3}(l) & =-\frac{D-1}{1260} l^{4}+\ldots \end{align*} $$

$$ \left\langle R^{2}\right\rangle \propto L^{2 \nu} $$

$$ P_{N}(\mathbf{R})=\frac{1}{\sqrt{2 \pi a / M}^{D}} \prod_{n=1}^{N-1}\left[\int \frac{d^{D} x_{n}}{\sqrt{2 \pi a / M}^{D}}\right] \exp \left(-\mathcal{A}^{N} / \hbar\right) $$

$$ \mathcal{A}^{N}=a \sum_{n=1}^{N} \frac{M}{2} \frac{\left(\Delta \mathbf{x}_{n}\right)^{2}}{a^{2}} $$

$$ P_{L}(\mathbf{R})=\int \mathcal{D}^{D} x e^{-\mathcal{A}^{L}[\mathbf{x}]} $$

$$ \mathcal{A}_{\mathrm{int}}=\frac{1}{2} \int_{0}^{L} d \tau \int_{0}^{L} d \tau^{\prime} V\left(\mathbf{x}(\tau), \mathbf{x}\left(\tau^{\prime}\right)\right) $$

$$ \mathcal{A}_{\mathrm{int}}^{\varphi}=\int_{0}^{L} d \tau \varphi(\mathbf{x}(\tau))-\frac{1}{2} \int d^{D} x d^{D} x^{\prime} \varphi(\mathbf{x}) V^{-1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \varphi\left(\mathbf{x}^{\prime}\right) $$

$$ \int d^{D} x^{\prime} V^{-1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) V\left(\mathbf{x}^{\prime}, \mathbf{x}^{\prime \prime}\right)=\delta^{(D)}\left(\mathbf{x}-\mathbf{x}^{\prime \prime}\right) $$

$$ \mathcal{A}_{\mathrm{int}}^{\varphi}=\int d^{D} x \rho(\mathbf{x}) \varphi(\mathbf{x})-\frac{1}{2} \int d^{D} x d^{D} x^{\prime} \varphi(\mathbf{x}) V^{-1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \varphi\left(\mathbf{x}^{\prime}\right) $$

$$ \rho(\mathbf{x}) \equiv \int_{0}^{L} d \tau \delta^{(D)}(\mathbf{x}-\mathbf{x}(\tau)) $$

$$ \mathcal{A}_{\mathrm{int}}^{\varphi}=-\frac{1}{2} \int d^{D} x d^{D} x^{\prime}\left[\varphi^{\prime}(\mathbf{x}) V^{-1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \varphi^{\prime}\left(\mathbf{x}^{\prime}\right)-\rho(\mathbf{x}) V\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \rho\left(\mathbf{x}^{\prime}\right)\right] $$

$$ \varphi^{\prime}(\mathbf{x}) \equiv \varphi(\mathbf{x})-\int d^{D} x^{\prime} V\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \rho\left(\mathbf{x}^{\prime}\right) $$

$$ \int \mathcal{D} \varphi(\mathbf{x}) e^{-\mathcal{A}_{\mathrm{int}}^{\varphi}} $$

$$ P_{L}(\mathbf{R}) \propto \int \mathcal{D}^{D} x(\tau) \int \mathcal{D} \varphi(\mathbf{x}) e^{-\mathcal{A}} $$

$$ \mathcal{A}=\mathcal{A}^{L}[\mathbf{x}, \dot{\mathbf{x}}, \varphi]+\mathcal{A}[\varphi] $$

$$ \begin{align*} \mathcal{A}^{L}[\mathbf{x}, \varphi] & \equiv \int_{0}^{L} d \tau\left[\frac{M}{2} \dot{\mathbf{x}}^{2}+\varphi(\mathbf{x}(\tau))\right] \\ \mathcal{A}[\varphi] & \equiv-\frac{1}{2} \int d^{D} x d^{D} x^{\prime} \varphi(\mathbf{x}) V^{-1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \varphi\left(\mathbf{x}^{\prime}\right) \end{align*} $$

$$ P_{L}(\mathbf{R}) \propto \int \mathcal{D} \varphi(\mathbf{x}) e^{-\mathcal{A}[\varphi]} P_{L}^{\varphi}(\mathbf{R}, \mathbf{0}) $$

$$ P_{L}^{\varphi}(\mathbf{R}, \mathbf{0})=\int \mathcal{D}^{D} x(\tau) e^{-\mathcal{A}^{L}[\mathbf{x}, \varphi]} $$

$$ \left[\frac{\partial}{\partial L}-\frac{1}{2 M} \partial_{\mathbf{R}}^{2}+\varphi(\mathbf{R})\right] P_{L}^{\varphi}(\mathbf{R}, \mathbf{0})=\delta^{(D)}(\mathbf{R}-\mathbf{0}) \delta(L) $$

$$ \hat{H}^{\varphi}=-\frac{1}{2 M} \partial_{\mathbf{R}}^{2}+\varphi(\mathbf{R}) $$

$$ P_{L}^{\varphi}(\mathbf{R}, \mathbf{0})=\int d E e^{-E L} \psi_{E}^{\varphi}(\mathbf{R}) \psi_{E}^{\varphi}(\mathbf{0}), \quad L>0 $$

$$ V\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=v a^{D} \delta^{(D)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ V^{-1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=v^{-1} a^{-D} \delta^{(D)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \mathcal{A}[\varphi]=-\frac{v^{-1} a^{-D}}{2} \int d^{D} x \varphi^{2}(\mathbf{x}) $$

$$ v^{-1} a^{-D} \varphi(\mathbf{x})=\frac{\delta}{\delta \varphi(\mathbf{x})} \log P_{L}^{\varphi}(\mathbf{R}, \mathbf{0}) $$

$$ v^{-1} a^{-D}\langle\varphi(\mathbf{x})\rangle=\langle\rho(\mathbf{x})\rangle \equiv\left\langle\int_{0}^{L} d \tau \delta^{(D)}(\mathbf{x}-\mathbf{x}(\tau))\right\rangle_{\mathbf{x}} $$

$$ \frac{\delta}{\delta \varphi(\mathbf{x})} P_{L}^{\varphi}(\mathbf{R})=\int \mathcal{D} \varphi \frac{\delta}{\delta \varphi(\mathbf{x})} \int \mathcal{D}^{D} x e^{-\mathcal{A}^{L}[\mathbf{x}, \varphi]-\mathcal{A}[\varphi]}=0 $$

$$ \left\langle\int_{0}^{L} d \tau \delta^{(D)}(\mathbf{x}-\mathbf{x}(\tau))\right\rangle_{\mathbf{x}}=\int_{0}^{L} d L^{\prime} P_{L^{\prime}}^{\varphi}(\mathbf{x}) P_{L-L^{\prime}}^{\varphi}(\mathbf{R}-\mathbf{x}) $$

$$ v^{-1} a^{-D}\langle\varphi(\mathbf{x})\rangle_{\mathbf{x}}=\int_{0}^{L} d L^{\prime} P_{L^{\prime}}^{\varphi}(\mathbf{x}) P_{L-L^{\prime}}^{\varphi}(\mathbf{R}-\mathbf{x}) $$

$$ P_{L}(\mathbf{R}) \sim P_{L}^{\varphi}(\mathbf{R}, \mathbf{0})=\int \mathcal{D}^{D} x \exp \left\{-\int_{0}^{L} d \tau\left[\frac{M}{2} \dot{\mathbf{x}}^{2}+\varphi(r(\tau))\right]\right\} $$

$$ \frac{M}{2} \dot{\mathbf{x}}^{2}-\varphi(r)=E=\mathrm{const} . $$

$$ d \tau=\frac{d r}{\sqrt{2[E+\varphi(r)] / M}} $$

$$ L=\int_{0}^{R} \frac{d r}{\sqrt{2[E+\varphi(r)] / M}} $$

$$ E=E_{L}[\varphi] $$

$$ \begin{align*} \mathcal{A}_{\mathrm{cl}}[\mathbf{x}, \varphi] & =\int_{0}^{L} d \tau\left[\frac{M}{2} \dot{\mathbf{x}}^{2}+\varphi(r(\tau))\right] \\ & =-\int_{0}^{L} d \tau\left[\frac{M}{2} \dot{\mathbf{x}}^{2}-\varphi(r(\tau))\right]+\int_{0}^{L} d \tau M \dot{\mathbf{x}}^{2} \\ & =-E L+\int_{0}^{R} d r \sqrt{2 M[E+\varphi(r)]} \end{align*} $$

$$ \frac{\partial}{\partial E} \mathcal{A}_{\mathrm{cl}}[\mathbf{x}, \dot{\mathbf{x}}, \varphi]=0 $$

$$ \mathcal{A}_{\mathrm{cl}}=-E L+\int_{0}^{r} d r^{\prime} \sqrt{2 M\left[E+\varphi\left(r^{\prime}\right)\right]}-\frac{1}{2} v^{-1} a^{-D} \int d^{D} x \varphi^{2}(r) $$

$$ \varphi\left(r^{\prime}\right)=\left\{\begin{array}{l} 0 \\ M v a^{D} S_{D}^{-1} r^{\prime 1-D} / \sqrt{2 M\left[E+\varphi\left(r^{\prime}\right)\right]} \quad \text { for } \quad r^{\prime}>r \\ r^{\prime}