1 Finite index sets
Throughout this chapter, \(I\) is a finite set, which is the set \([N]\) of the indices of the particles in the later chapters. The results of the paper which hold given the particle positions are stated here for every probability vector, so that they hold for all particle positions.
For weights \(p\) on a finite set \(I\), a function \(\varphi \colon I \to \mathbb {R}\), and a matrix \(K\) indexed by \(I\), \(p(\varphi ) \mathrel {\vcenter {:}}=\sum _{i \in I} p_i \varphi (i)\) and \(\langle \varphi , K\varphi \rangle _p \mathrel {\vcenter {:}}=\sum _{i, j \in I} p_i K(i, j) \varphi (i) \varphi (j)\).
For all \(p\), \(\varphi \), and \(\psi \), \(p(\varphi + \psi ) = p(\varphi ) + p(\psi )\).
Distribute the sum.
A probability vector on \(I\) has nonnegative entries which sum to one. A Markov kernel on \(I\) is a matrix with nonnegative entries whose rows sum to one, that is, an element of Matrix.rowStochastic. A matrix \(K\) is reversible with respect to \(p\) if \(p_i K(i, j) = p_j K(j, i)\) for all \(i, j \in I\). The Gibbs kernel with respect to \(p\) is defined by \(K(i, \cdot ) = p\) for every \(i \in I\).
Let \(p\) be a probability vector on \(I\) and \(i \in I\). Then \(p_j \leq 1\) for all \(j \in I\), \(\sum _{j \neq i} p_j = 1 - p_i\), and \(p_j = 0\) for all \(j \neq i\) if \(p_i = 1\). A nonnegative vector which is not zero has a positive sum, and, divided by this sum, is a probability vector.
The entries are nonnegative and sum to one.
Let \(p\) be a probability vector and \(K\) a Markov kernel on \(I\), with \(q_j \mathrel {\vcenter {:}}=\sum _i p_i K(i, j)\) for \(j \in I\). Then, for every \(\varphi \colon I \to \mathbb {R}\),
The coefficients \(p_i \{ K(i, j) - q_j\} \) sum to zero over \(j\) for every \(i\) and over \(i\) for every \(j\). The identity follows by expanding the square, as in the proof of Lemma 2 in Section S1.
In the setting of Lemma 5, if \(K(i, j) \geq q_j\) for all \(i \neq j\), then \(\langle \varphi , K\varphi \rangle _p \leq p(\varphi ) q(\varphi )\) for every \(\varphi \colon I \to \mathbb {R}\).
The terms with \(i = j\) of the sum of Lemma 5 vanish, whereas the others are nonnegative.
Of two Markov kernels on \(I\) which are reversible with respect to \(p \geq 0\), the one with the larger off-diagonal entries has the smaller quadratic form \(\langle \varphi , K\varphi \rangle _p\), for every \(\varphi \colon I \to \mathbb {R}\).
For a Markov kernel \(K\) reversible with respect to \(p\), \(\langle \varphi , K\varphi \rangle _p = \sum _i p_i \varphi (i)^2 - \frac{1}{2} \sum _{i \neq j} p_i K(i, j) \{ \varphi (i) - \varphi (j)\} ^2\).
The Gibbs kernel \(K(i, \cdot ) = p\) and the identity kernel are reversible with respect to \(p\). The Gibbs kernel is a Markov kernel when \(p\) is a probability vector. The entries of a Markov kernel are at most one. For the Gibbs kernel \(K(i, \cdot ) = q\), \(\langle \varphi , K\varphi \rangle _p = p(\varphi ) q(\varphi )\). For the identity kernel, \(\langle \varphi , \varphi \rangle _p = p(\varphi ^2)\). For every matrix \(K\) indexed by \(I\) and every \(i \in I\), \(\langle \mathbf{1}_{\{ i\} }, K\mathbf{1}_{\{ i\} } \rangle _p = p_i K(i, i)\).
By direct computation.
For a Markov kernel \(K\) reversible with respect to \(p\), \(\langle \varphi , K\varphi \rangle _p = \sum _i p_i \varphi (i)^2 - \frac{1}{2} \sum _{i, j} p_i K(i, j) \{ \varphi (i) - \varphi (j)\} ^2\). For a probability vector \(p\), \(p(\varphi )^2 \leq p(\varphi ^2)\). If, moreover, the Markov kernel \(K\) is reversible with respect to \(p\), then \(p(\varphi )^2 - \langle \varphi , K\varphi \rangle _p \leq p(\varphi ^2) - p(\varphi )^2\), that is, minus the lag-one autocovariance of \(\varphi \) is at most its variance.
The identity follows by expanding the squares. The inequality \(p(\varphi )^2 \leq p(\varphi ^2)\) is Lemma 7 applied to the Gibbs kernel and the identity kernel. The last inequality follows from the identity and from \(\{ \varphi (i) - \varphi (j)\} ^2 \leq 2 \{ \varphi (i) - p(\varphi )\} ^2 + 2 \{ \varphi (j) - p(\varphi )\} ^2\).