  We consider the continuum version of the random field Ising model in one
dimension: this model arises naturally as weak disorder scaling limit of the
original Ising model. Like for the Ising model, a spin configuration is
conveniently described as a sequence of spin domains with alternating signs
(domain-wall structure). We show that for fixed centered external field and as
spin-spin couplings become large, the domain-wall structure scales to a
disorder dependent limit that coincides with the infinite disorder fixed point
process introduced by D. S. Fisher in the context of zero temperature quantum
Ising chains. In particular, our results establish a number of predictions that
one can find in [Fisher, Le Doussal and Monthus 2001]. The infinite disorder
fixed point process for centered external field is equivalently described in
terms of the process of suitably selected extrema of a Brownian trajectory
introduced and studied by J. Neveu and J. Pitman [Neveu and Pitman 1989]. This
characterization of the infinite disorder fixed point is one of the important
ingredients of our analysis.
