Critical Exponent

A critical exponent is an exponent that marks a threshold or governs asymptotic power-law behavior. Its precise meaning depends on the discipline.

In statistical mechanics and probability, critical exponents describe scaling near a phase transition or critical parameter. For example, the exponents and for self-avoiding walks describe the expected power-law factors in the asymptotic number of walks and their mean-square displacement, respectively (Madras and Slade 1993).

In hyperbolic geometry, the critical exponent of an infinite discrete group of isometries of a hyperbolic space, for a base point , is the threshold value such that converges for and diverges for . It is also the exponential growth rate of a group orbit (Dougall and Sharp 2016).

In nonlinear analysis and partial differential equations, a critical exponent is a threshold power separating qualitatively different regimes. For example, when , the target exponent is critical in the Sobolev embedding theorem (Brezis 2011).

In combinatorics on words, the critical exponent of a nonempty finite or right-infinite word is the supremum of the exponents of its nonempty finite factors. The exponent of a nonempty finite word is , where is its least period: the smallest positive integer for which whenever .

Let be the right-infinite fixed point beginning with 0 of the word morphism

It has factor complexity and critical exponent

where is the real root of . Currie (2026) proved that every right-infinite ternary word with factor complexity has critical exponent at least this value, confirming a conjecture of Shallit and Shur (2019).