## 1 Introduction  
Random media is a popular topic in modern material science and physics. In this paper, we focus on stationary two-dimensional processes in dispersed composites and discuss their effective properties within the framework of constructive homogenization theory.  
To illustrate the central concept of random media, we consider steady heat conduction. This formalism can also be applied to electric, magnetic, dielectric, diffusion, and other processes governed by Laplace's equation. Furthermore, the same homogenization concept could be extended to elasticity, piezoelectricity, and viscous fluids. These subjects will be discussed in a separate paper.  
The effective properties of random composite materials can be estimated on the basis of the representative volume element (RVE). [PERSON] [1] introduced the theoretical principles of RVE as \"... a sample that (a) is structurally entirely typical of the entire mixture on average, and (b) contains a sufficient number of inclusions for the apparent overall moduli to be effectively independent of the surface values of traction and displacement\". [PERSON]'s principles are frequently considered as a definition of RVE. Usually, numerical computations of effective constants are supported by a statistical approach when random inclusions are selected within a cell, and the effective constants are calculated based on the numerically given properties of constituents; see [2; 3; 4; 5] and the works cited there. Authors typically imply, though do not explicitly state, that they consider a solely uniform, non-overlapping distribution. Sometimes, the manipulations employed along the way are not grounded in homogenization theory. Sometimes, random locations may not be accurately simulated by computational rules [6] similar to those used in pseudo-random number simulations. This naive statistical approach is prevalent in the engineering literature concerning random composites.  
This situation must be rectified according to the mathematical principles established in homogenization theory [7; 8; 9]. The rules of asymptotic analysis [10; 11; 12; 13] and the upper and lower bounds should be obeyed [14; 15]. This paper aims to bridge the gap between rigorous mathematical theory and simplified engineering formulas for effective constants. It presents a strategy for studying random composites based on Image Analysis, partial differential equations (PDEs), computational simulations, and statistical processing.  
It should be noted that homogenization theory justifies the existence of effective constants under certain natural conditions and identifies the boundary value problem that must be solved to determine them. The existence and uniqueness of the solution to such a problem are essential results of the mathematical problems addressed. Then, the effective conductivity tensor \(\lambda_{e}\) should be computed. The effective tensors can be derived from a periodic boundary value problem with \(N\) inclusions per periodicity cell \(Q\), representing the composite under consideration displayed in Figure 1.  
The process of generating random arrangements of \(N\) inclusions within a two-dimensional domain \(Q\) gives examples of a microstructure. Two well-established approaches, such as random walks (RW) and Random Sequential Addition (RSA), are frequently employed to achieve this goal. Each method has well-defined procedural characteristics emergent from specific scenarios for randomness generation.  
Figure 1: The sample \(Q^{\prime}\) enclosing a part of the doubly periodic cell represented by the square \(Q\).  
Various mathematical methods are applied to samples with microstructure to determine the effective properties of random composites. These methods can be divided into three types: pure numerical, e.g., the Finite Element Method (FEM), empirical formulas, and analytical techniques. These methods will be discussed in the following subsections.