We make a new proposal to describe the very low temperature susceptibility of the doped Haldane gap compound Y$_2$BaNi$_{1-x}$Zn$_x$O$_5$. We propose a new mean field model relevant for this compound. The ground state of this mean field model is unconventional because antiferromagnetism coexists with random dimers. We present new susceptibility experiments at very low temperature. We obtain a Curie-Weiss susceptibility $chi(T) sim C / (Theta+T)$ as expected for antiferromagnetic correlations but we do not obtain a direct signature of antiferromagnetic long range order. We explain how to obtain the ``impurity susceptibility $chi_{imp}(T)$ by subtracting the Haldane gap contribution to the total susceptibility. In the temperature range [1 K, 300 K] the experimental data are well fitted by $T chi_{imp}(T) = C_{imp} (1 + T_{imp}/T )^{-gamma}$. In the temperature range [100 mK, 1 K] the experimental data are well fitted by $T chi_{imp}(T) = A ln{(T/T_c)}$, where $T_c$ increases with $x$. This fit suggests the existence of a finite Neel temperature which is however too small to be probed directly in our experiments. We also obtain a maximum in the temperature dependence of the ac-susceptibility $chi(T)$ which suggests the existence of antiferromagnetic correlations at very low temperature.
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