这是一份leeds利兹大学MATH314301作业代写的成功案例
Let $y \in G$, and write $y$ in the form (1). We can write
$$
\varphi(x)-\varphi(x y)=\left[\varphi(x 8)-\varphi\left(x s_{1}\right)\right]+\ldots+\left[\varphi\left(x s_{1} \ldots s_{\ell-1}-\varphi(x y)\right] .\right.
$$
It follows, for example, by the Canchy-Schwarz inequality that
$$
(\varphi(x)-\varphi(x y))^{2} \leq \ell^{*} \sum^{\ell}\left(\varphi\left(x s_{1} \ldots s_{i-1}\right)-\varphi\left(x s_{1} \ldots s_{i}\right)\right)^{2}
$$
where $\ell^{*}$ is the number of nonzero terms in the sum, and is bounded above by $d=\operatorname{diam}(\bar{C})$, since $\gamma$ is geodesic. Summing this inequality over $x \in G$ we get
$$
\sum_{x \in G}(\varphi(x)-\varphi(x y))^{2} \leq d \sum_{t \in C, s \in S} N_{\gamma}(s, \bar{C})(\varphi(z)-\varphi(z s))^{2}
$$
Since this holds for all $y \in G$, we may average the left hand side with respect to $y$ with weights $\approx(y)$ to get
$$
\sum_{x, y \in C}(\varphi(x)-\varphi(x y))^{2} \tilde{\pi}(y) \leq d \sum_{z \in G, s \in S} N_{\gamma}(s, \bar{C})(\varphi(z)-\varphi(z s))^{2}
$$
MATH314301 COURSE NOTES ：
Last time, we proved
Theorem 1 Let $C$ be a subset of the group $G, S=S^{-1}$ a symmetric generating set, $\pi=U(S)$ the uniform distribution on $S$, and $p_{\pi}$ the “one-step evolution” of the random walk (i.e. $p_{\pi} \varphi=U(S) * \varphi$ ). Then for any probability distribution $\varphi$,
$$
\left|p_{n} \varphi\right|^{2} \leq\left(1-\frac{|G \backslash C|}{2 \cdot A \cdot|G|}\right) \cdot|\varphi|^{2}
$$
where $A=d \cdot|S| \cdot \max {s \in S} \max {g \in C} \mu_{s}(g), d=\operatorname{diam}(\bar{C}, G)$.
We will use this theorem to bound the escape time of a random walk $X_{\mathrm{t}}$ generated by $S$. For a subset $C$ of $G$, set
$$
\varphi_{t}(g)=\operatorname{Pr}\left[X_{t}=g \text { and } X_{i} \in C \forall i=1 \ldots t\right]
$$
Obviously, supp $\varphi_{t} \subset C,\left|\varphi_{0}\right| \leq 1$ ( 1 if $C$ contains $1_{G}, 0$ otherwise) and