Trace powers, closed walks and independent edge moments

Repeatedly applying the matrix-product formula in [F1] gives $(W^k{)}_{ab}={\sum }_{i_1,\dots ,i_{k-1}}{W}_{a{i}_1}{W}_{i_1{i}_2}\cdots W_{i_{k-1}b}$ for $k\ge 2$. Taking $k=2\mathrm{\ell }$, setting $a=b=i_0$, and summing the diagonal entries yields the stated closed-walk expansion. The argument is pointwise and permits loops and repeated edges.

For a fixed walk, symmetry of $W$ groups its factors as ${\prod }_{e:m_e>0}{W}_e^{m_e}$. Each $m_e$ lies between $1$ and $2\mathrm{\ell }$. By [F7], all powers and absolute powers used here are measurable. By [F9] and [F8], the constant function $1$ is integrable. The moment hypothesis and [F4], [F8] imply that $\mathrm{\mid }{W}_e{\mathrm{\mid }}^{2\mathrm{\ell }}$ is integrable. Thus [F4] makes $1+\mathrm{\mid }{W}_e{\mathrm{\mid }}^{2\mathrm{\ell }}$ integrable, with integral $1+\mathbb{E}\mathrm{\mid }{W}_e{\mathrm{\mid }}^{2\mathrm{\ell }}<\mathrm{\infty }$. For $1\le m\le 2\mathrm{\ell }$, the bound $\mathrm{\mid }x{\mathrm{\mid }}^m\le 1+\mathrm{\mid }x{\mathrm{\mid }}^{2\mathrm{\ell }}$ follows by considering $\mathrm{\mid }x\mathrm{\mid }\le 1$ and $\mathrm{\mid }x\mathrm{\mid }>1$. Monotonicity in [F8] therefore gives ${\int }_{\mathrm{\Omega }}\mathrm{\mid }{W}_e{\mathrm{\mid }}^{m}d\mathbb{P}\le 1+\mathbb{E}\mathrm{\mid }{W}_e{\mathrm{\mid }}^{2\mathrm{\ell }}<\mathrm{\infty }$, so [F8] proves integrability of $W_e^m$, in particular of $W_e^{m_e}$.

Apply [F3] to the distinct independent entries and the Borel functions $x\mapsto x^{m_e}$ from [F7] to obtain the product formula and integrability of each walk product, using step 1.2. By [F4], integrate the finite sum of these integrable walk products over the underlying probability space; repeated $L^1$ linearity permits expectation to pass through the finite expansion in step 1.1.

If $W_e$ and $-W_e$ have the same law in the sense of [F6] and $m_e$ is odd, both are random variables by [F7], and both $m_e$th powers are integrable by step 1.2 because their absolute values agree. The change-of-variables formula in [F4] evaluates both expectations as the integral of the same Borel function $x\mapsto x^{m_e}$ against their common law. Since $(-W_e{)}^{m_e}=-W_e^{m_e}$, linearity in [F4] gives $\mathbb{E}[W_e^{m_e}]=\mathbb{E}[(-W_e{)}^{m_e}]=-\mathbb{E}[W_e^{m_e}]$. Thus that moment is zero. The product formula in step 2.1 then eliminates every walk with an odd multiplicity. For a comparison, the formula identifies precisely the moments to be compared; inequalities for those moments are additional hypotheses.

Fix an outcome. A real symmetric matrix is self-adjoint for the Euclidean inner product. By [F2] it has an orthonormal eigenbasis with real eigenvalues ${\lambda }_1,\dots ,{\lambda }_n$. Let $Q$ have those eigenvectors as columns. Then $Q^{-1}=Q^T$ and $Q^{-1}{W}^{2\mathrm{\ell }}Q=\mathrm{diag}({\lambda }_1^{2\mathrm{\ell }},\dots ,{\lambda }_n^{2\mathrm{\ell }})$, since both sides act on each coordinate vector by the same scalar. Thus [F5] and [F1] give $\mathrm{Tr}{W}^{2\mathrm{\ell }}={\sum }_r{\lambda }_r^{2\mathrm{\ell }}$ in the original coordinates. For $x={\sum }_r{c}_r{v}_r$ in the orthonormal eigenbasis, $\mathrm{\parallel }Wx{\mathrm{\parallel }}_2^2={\sum }_r\mathrm{\mid }{\lambda }_r{\mathrm{\mid }}^2\mathrm{\mid }{c}_r{\mathrm{\mid }}^2\le ({\max }_r\mathrm{\mid }{\lambda }_r\mathrm{\mid }{)}^2\mathrm{\parallel }x{\mathrm{\parallel }}_2^2$. This proves boundedness and the upper bound for the unit-ball supremum in [F2]; a unit eigenvector for a maximal absolute eigenvalue attains it. Hence the operator norm is ${\max }_r\mathrm{\mid }{\lambda }_r\mathrm{\mid }$. The same calculation in the complexified basis gives the identical complex operator norm. Every summand is nonnegative and at most the largest, while one attains it. Therefore $\mathrm{\parallel }W{\mathrm{\parallel }}^{2\mathrm{\ell }}\le \mathrm{Tr}{W}^{2\mathrm{\ell }}\le n\mathrm{\parallel }W{\mathrm{\parallel }}^{2\mathrm{\ell }}$. The preceding steps prove all assertions.