We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β ⩾ 0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as dPβ,tdP0=Zβ,t(0)−1eβ∫t0δ0(xs)ds,dPβ,tdP0=Zβ,t(0)−1eβ∫0tδ0(xs)ds, (0.1) where Zβ,t(0)≡E0[eβ∫t0δ0(xs)ds].Zβ,t(0)≡E0[eβ∫0tδ0(xs)ds].. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper (Cranston and Molchanov, 2007), we showed that for dimensions d ⩾ 3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator Δ + βδ0, where Δ is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.