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Concept

Spectral radius

The largest absolute value among a matrix's eigenvalues. Powers of the matrix grow or shrink at that rate, so it decides whether a series of powers converges, and for a walk on a graph it sets how fast the chance of returning home decays.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Powers of a matrix that shrink in the end. Three curves of the norm of the n-th power of a two-by-two matrix against n on a logarithmic axis: one decays steadily, two rise to peaks of about 18 and 7 before decaying.

A geometric series whose ratio is a matrix

1 + r + r² + … adds to 1/(1 − r) when r is smaller than one. Put a matrix in place of r and the same formula holds, with the inverse matrix in place of the fraction — but what must be smaller than one is not the matrix's size. It is its largest eigenvalue. A matrix whose eigenvalues are 0.9 and 0.8 can stretch vectors ten times over before its powers begin to shrink, and the series still converges, after a detour the eigenvalues say nothing about.

analysis · Geometric series How often a walk comes home, on three lattices and a tree. Four curves of the logarithm of the return probability after 2n steps, for n up to 60: three lattices flattening, and the free group falling along a straight line.

How rarely a walk on a group comes home

Walk at random on the picture of a group, one generator at a time, and ask for the chance of standing at the start after 2n steps. On the line, the plane and three-dimensional space it falls like a power of n. On the tree that pictures the free group it falls by the factor √3/2 every step, exponentially. Kesten proved in 1959 that this is no accident of two examples: the chance falls exponentially exactly when the group's balls are mostly boundary, so a probabilistic rate and a geometric ratio are the same measurement.

algebra · Cayley graph

Named alongside it

The objects these essays reach for when they reach for this one.

AmenabilityCayley graphConvergenceEigenvalueFree groupGeometric seriesGrowth rateInverseMatrixRandom walkRecurrence

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