The so-called Lattice Discrete Particle Model (LDPM) naturally accounts for material heterogeneity by random particle placement and size, which is also constrained by a grading curve. This approach captures most microstructural effects of concrete very well, when compared to the continuum framework, however introducing higher order spatial variability enables to control and interpret the response scatter. This paper addresses the effects of various choices of spatially variable material property fields, such as random field described by power spectral functions or gradient based fields, and particle placement schemes, such as those derived from governing random or gradient based fields, in order to account for inherent variability and production processes of several classical concrete tests. These are e.g. cylinder and cube compression test, and unnotched three point bending test. As a consequence, the lattice models become sensitive to a particular choice of spatially variable material property fields and particular particle placement concept, which is no longer independent and random, and the scattering of the response can thus be associated with the physical meaning of an auto-correlation length and particular forms of the spectral function. In particular, the non-monotonous relationship between statistical characteristics of the response, such as the coefficient of variation of the load capacity, and spatial correlation structures, such as power spectral parameters, clearly support the hypotheses on causal relationship between spatial variability, auto-correlation length of the random fields, type of spectral function and meso/micro-structure of the material. By imposing correlated spatial variability the consistency and realism of the LDPM stochastic framework may dramatically increase if objective physical reference for the governing random field and correlation length is established. Since this represents a rather extreme case of high-dimensional problem, simple solutions are not to be expected any time soon. Until then, the presented original framework may serve for the interpretation of different sources and magnitudes of experimental scattering observed in various classical experiments and the illustration of the important implications towards enhanced realism in the reliability based assessment of concrete structures and infrastructure, including fastening systems.