  We analyse the self-diffusiophoresis of a spherical particle animated by a
nonuniform chemical reaction at its boundary. We consider two models of solute
absorption, one with a specified distribution of interfacial solute flux, and
one where this flux is governed by first-order kinetics with a specified
distribution of rate constant. We employ a macroscale model where the
short-range interaction of the solute with the particle boundary is represented
by an effective slip condition. The solute transport is governed by an
advection-diffusion equation.
  We focus upon the singular limit of large P\'eclet numbers, $Pe\gg 1$. In the
fixed-flux model, the excess-solute concentration is confined to a narrow
boundary layer. The scaling pertinent to that limit allows to decouple the
problem governing the solute concentration from the flow field. The resulting
nonlinear boundary-layer problem is handled using a transformation to
stream-function coordinates and a subsequent application of Fourier transforms,
and is thereby reduced to a nonlinear integral equation governing the
interfacial concentration. Its solution provides the requisite approximation
for the particle velocity, which scales as $Pe^{-1/3}$.
  In the fixed-rate model, large P\'eclet numbers may be realized in different
limit processes. We consider the case of large swimmers or strong reaction,
where the Damk\"ohler number $Da$ is large as well, scaling as $Pe$. In that
double limit, where no boundary layer is formed, we obtain a closed-form
approximation for the particle velocity, expressed as a nonlinear functional of
the rate-constant distribution; this velocity scales as $Pe^{-2}$. Both the
fixed-flux and fixed-rate asymptotic predictions agree with the numerical
values provided by computational solutions of the nonlinear transport problem.
