In this work we assess the ability of the derivativeexpansion of the FRG to quantitatively describe the long-distance physics of amodel without a priori knowledge of the relevant real-space coarse-grainedconfigurations, even when they are strongly nonuniform. To do so we investigatehow low orders of the derivative expansion describe the approach to the lower criticaldimension dlc of the Ising-like scalar φ 4 theory which is expected to becontrolled by the proliferation of localized excitations (instantons) [12–14].FRG is well suited for this task since even the lowest orders of approximationsare known to capture nontrivial nonperturbative phenomena [15, 16], and it hasan advantage of being a general approach, unlike specialized approaches onemust usually resort to capturing such behavior, like the droplet theory in thecase of the Ising model just above dlc, where enclosed domains of onespinorientation in the other (the droplets) must be introduced explicitly[12–14]. Additionally, the spatial dimension d is a parameter of the theorythat we can vary. We show that the convergence of the fixed-point effectivepotential describing the critical point of the model is nonuniform in the fieldwhen the dimension d approaches dlc, with the emergence of a boundary layeraround the minimum of the potential. < At the lowest nontrivialapproximation level (known as LPA’) this allows us to make analyticalpredictions for the value of the lower critical dimension dlc and for thebehavior of the critical temperature as d → dlc, which are both found in fairagreement with the known exact results. We check the stability of the resultsupon increasing the approximation order by studying the second order of the derivativeexpansion (∂2). The ∂2 investigation is still in progress. As of yet, less hasbeen found analytically due to the complexity coming from the nontrivial fieldrenormalization function z (ϕ) (z (ϕ) = 1 in LPA’). Numerical results above thedlc show strong indications that the boundary layer persists, tentatively withthe same behavior of the critical temperature in the dlc limit. However,predictions of the value of the lower critical dimension require matching theboundary layer solution with the rest of the field domain, which we have not achievedat the present moment. We also consider for the same Ising-like scalar φ 4model the ordered phase in the limit where d → dlc. We focus on the way theeffective potential in the FRG becomes convex as the proper spatialfluctuations, which now involve nonuniform configurations with a domain wallseparating two phases, are included in the calculation.