On the spectral radius of the $(L,\kappa )$-lazy Markov chain

We consider an $(L,\kappa )$-lazy operation on an irreducible Markov transition probability $P$ with state space $S$ where $L\subset S$ and $\kappa \in [0,1)$. For each $x\in L$ and $y\in S$, this $(L,\kappa )$-operation replaces $P(x,y)$, the transition probability from $x$ to $y$, by $\kappa 1_{\{x=y\}}+(1-\kappa )P(x,y)$. We are interested in how $L$ and $\kappa$ influence the spectral radius ${\rho }_{\kappa }^L$ of this new transition probability. We first show that ${\rho }_{\kappa }^L$ is non-decreasing and continuous in $\kappa$. We then show that: (1) If $L$ is nonempty and finite, then $P$ being rho-transient is equivalent to that the growth of ${\displaystyle ({\rho }_{\kappa }^L{)}_{\kappa \in [0,1)}}$ exhibits a phase transition: There exists a critical value ${\kappa }_c(L)\in (0,1)$ such that $\kappa \mapsto {\rho }_{\kappa }^L$ is a constant on $[0,{\kappa }_c(L)]$ and increases strictly on $[{\kappa }_c(L),1)$; (2) For every $\kappa \in (0,1)$, if $S\setminus L$ is nonempty and finite, then ${\rho }_{\kappa }^L={\rho }_{\kappa }^S$ if and only if $P$ is not strictly rho-recurrent.