  We consider a supercritical catalytic branching random walk (CBRW) on a
multidimensional lattice Z^d (d is positive integer). The main subject of study
is the behavior of particles cloud in space and time. For CBRW on an integer
line, Carmona and Hu (2014) examined the asymptotical behavior of the maximal
coordinate M_n of the particles at time n. They proved that M_n/n converges to
\mu almost surely (on a set of local non-degeneracy of CBRW), as n tends to
infinity, where \mu>0 is a certain constant. Under additional assumption of a
single catalyst in CBRW they also investigated the fluctuations of M_n with
respect to \mu n, as n grows to infinity. Bulinskaya (2018) extended the strong
limit theorem by Carmona and Hu having estimated the rate of the population
propagation for the front of a multidimensional CBRW. Now our aim is to analyze
fluctuations of the propagation front in CBRW on Z^d. We not only solve the
problem in a multidimensional setting but also, treating the case of an
arbitrary finite number of catalysts for d = 1, generalize the result by
Carmona and Hu with the help of other probabilistic-analytic methods.
  Keywords and phrases: catalytic branching random walk, supercritical regime,
spread of population, propagation front, fluctuations of front.
