The authors consider a reversible Markov chain on a state space \(\Omega\) and estimate its special gap. Let \(P(x,dy)\) be the transition probability kernel of a Markov chain that is reversible with respect to a probability density. The authors describe the “pieces” of the chain \(P\). Let \(A_1,\dots, A_m\) be subsets of \(\Omega\) such that \(\bigcup A_i=\Omega\). For each \(i= 1,\dots, m\), the authors define a new Markov chain on \(A_i\). The transition kernel \(P_{[A_i]}\) of the new chain is \[ P_{[A_i]}(x,B)= P(x,B)+ 1_{[x\in B]} P(x, A^c_i),\qquad x\in A_i,\;B\subset A_i.\tag{1} \] Now they define \[ Z:= \sum^m_{i=1} \pi[A_i]\tag{2} \] and the “maximum overlap” \(\Theta\) of the covering \([A_1,\dots, A_m]\) by \[ \Theta:= \max_{x\in\Omega}\|i: x\in A_i\|.\tag{3} \] Then \[ 1\leq Z\leq\Theta.\tag{4} \] Also they introduce a crude model of the movement of the original chain among the “pieces”. They consider a state space \(a_1,\dots, a_m\) of \(m\) points representing our \(m\) pieces and define the transition probabilities for a discrete Markov chain on this finite state space: \[ P_H(a_i, a_j)= {\pi[A_i, A_j]\over \Theta\pi[A_i]}\quad\text{and}\quad P_H(a_i, a_i)= 1-\sum_{j\neq i} P_H(a_i, a_j).\tag{5} \] To describe the rate of convergence to equilibrium, the authors use the spectral gap. For example the first theorem presented is
Theorem 1.1 (State decomposition theorem). With the equations (1)–(5) it results \[ \text{Gap}(P)\geq {1\over\Theta^2} \text{Gap}(P_H(\min\text{ Gap } P_{[A_i]})),\qquad i= 1,\dots, m. \] Section 2 treats simulated tempering. Section 3 is devoted to the Metropolis algorithm and umbrella sampling. Section 4 is entitled “State decomposition”. Section 5 treats density decomposition. The last part of the note contains an appendix, in which the authors consider the Caraciolo-Pelissetto-Sokal result and the Madras-Piccioni result.