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
govern the asymptotic number of walks and their mean-square displacement, respectively
(Madras and Slade 1993).
In hyperbolic geometry, the critical exponent 
of a discrete group 
of isometries,
for a base point 
,
is the boundary value such that ![sum_(gamma in Gamma)exp[-sd(o,gammao)]](/images/equations/CriticalExponent/Inline6.svg)
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 finite or right-infinite word is the supremum of the exponents of its finite factors.
The exponent of a nonempty finite word 
is 
, where 
is the least period of 
.
Let 
be the right-infinite fixed point beginning with 0 of the 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).
and Minimal Critical Exponent." Elec. J. Combin. 33,
No. 3, P3.31, 2026.