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Saddle point method and end-to-end distribution of polymer models

Introduction Saddle point methods are widely used in estimation of integrals with form $$ I = \int \exp\left(-f(x)\right)\mathrm{d}x $$ where function $f$ can be approximated by first 2 terms of its Taylor series around some $x_0$, i.e. $$ f(x)\approx f(x_0) + f^\prime (x_0)(x-x_0) + \frac{1}{2}f^{\prime\prime}(x_0)(x-x_0)^2 $$ The integral is thus approximated by its saddle point, where $f^\prime (x_0)=0$ and $f^{\prime\prime}(x_0)>0$: $$ \begin{align} I&\approx \int \exp\left(-f(x_0) - \frac{1}{2}f^{\prime\prime}(x_0)(x-x_0)^2\right) \mathrm{d}x\\ &=\exp(-f(x_0))\sqrt{\frac{2\pi}{f^{\prime\prime}(x_0)}} \end{align}$$ Examples Stirling's formula: With the knowledge of $\Gamma$ function we know that $$N!=\int_0^\infty \exp(-x)x^N\mathrm{d}x$$ let $f(x):=x-N\ln(x)$, with large $N$, the negative part is negligible, solving $f^\prime (x) = 0$, we have: $$N!\approx\exp(-N+N\ln(N))\sqrt{2\pi{}N}=\sqrt{2\pi{}N}\left(\frac{N}{e}\right)^N$$ Partition function: