We consider an $(L,κ)$-lazy operation on an irreducible Markov transition probability $P$ with state space $S$ where $L \subset S$ and $κ\in[0,1)$. For each $x \in L$ and $y\in S$, this $(L,κ)$-operation replaces $P(x,y)$, the transition probability from $x$ to $y$, by $κ1_{\{x=y\}} + (1-κ)P(x,y)$. We are interested in how $L$ and $κ$ influence the spectral radius $ρ^L_κ$ of this new transition probability. We first show that $ρ^L_κ$ is non-decreasing and continuous in $κ$. We then show that: (1) If $L$ is nonempty and finite, then $P$ being rho-transient is equivalent to that the growth of $\displaystyle (ρ^L_κ)_{κ\in[0,1)}$ exhibits a phase transition: There exists a critical value $κ_c(L) \in (0,1)$ such that $κ\mapsto ρ^L_κ$ is a constant on $[0,κ_c(L)]$ and increases strictly on $[κ_c(L),1)$; (2) For every $κ\in(0,1)$, if $S\setminus L$ is nonempty and finite, then $ρ^L_κ=ρ^S_κ$ if and only if $P$ is not strictly rho-recurrent.