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Suppose I have a connected graph $G = (V,E)$, and a subset of vertices $U$, such that $W$ is the subgraph of $G$ induced by the vertices $V setminus U$, and that $W$ is also connected.

Now, let $X_n$ be a random walk on $G$ and let $P$ be its transition probability matrix. Now, let $Q$ be the submatrix of $P$ that corresponds to the vertices $W$. So, $Q$ is a sub-stochastic matrix (i.e. at least $1$ row sums to less than $1$)

I have shown that for any $k,j in W$, $(K^n)_{kj} = P_k({X_n = j} cap {X_i in W, forall i < n})$, where $P_k$ refers to the probability with respect to the random walk started at $k$.

I want to show that, for all $k, j in W$, $$ sum_{n=0}^infty (K^n)_{kj} < infty,$$ and also that $$ sum_{n=0}^infty x^n(K^n)_{kj}$$ has a radius of convergence strictly greater than $1$.

I can show both using a lot of linear algebra, most of which comes down to showing that $K$ has spectral radius less than $1$.

I was wondering if there is any probability arguments I can use to show this instead, as the solutions I have now seem extremely non-illustrative. Moreover, we haven't learned Perron-Frobenius and other similar results, which to me indicates that there must be a slicker, probabilistic solution to this problem.