  In this paper, we study a disordered pinning model induced by a random walk
whose increments have a finite fourth moment and vanishing first and third
moments. It is known that this model is marginally relevant, and moreover, it
undergoes a phase transition in an intermediate disorder regime. We show that,
in the critical window, the point-to-point partition functions converge to a
unique limiting random measure, which we call the critical disordered pinning
measure. We also obtain an analogous result for a continuous counterpart to the
pinning model, which is closely related to two other models: one is a critical
stochastic Volterra equation that gives rise to a rough volatility model, and
the other is a critical stochastic heat equation with multiplicative noise that
is white in time and delta in space.
