GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s51>GLYPH<s48>GLYPH<s48> GLYPH<s52>GLYPH<s48>GLYPH<s48> GLYPH<s53>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s51>GLYPH<s48>GLYPH<s48> GLYPH<s52>GLYPH<s48>GLYPH<s48> GLYPH<s53>GLYPH<s48>GLYPH<s48> GLYPH<s76>GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s76>GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49>GLYPH<s50>GLYPH<s48> GLYPH<s42> GLYPH<s113> GLYPH<s113> FIG. 1: Disorder distribution of the susceptibility maxima of two lattices with linear sizes L = 120 and L = 200 of the RBTrIM. The running averages over the samples is shown by the thick solid lines. the process and care is necessary in setting up a proper protocol of the recursion. In spite of the fact that the WL method has produced very accurate results in sev- eral models, it is fair to say that there is not a safe way to access possible systematic deviations in the general case. This has been pointed out and critiqued in a recent re- view by Janke [72]. For the 2d Ising model (where exact results are available for judgement) the WL method has been shown to converge very rapidly. Furthermore, from our experience and especially from our recent study of the 2d RBSqIM [59], for which the exact phase diagram is known by duality relations, our high-level WL imple- mentation has produced excellent results, enabling us to discriminate between competing theoretical predictions on that model. Since the RBTrIM is expected to have a similar “entropy structure” to the corresponding square model, we anticipate and it will be verified in the se- quel, that, our WL scheme produces sufficiently accurate estimates enabling us, also in this case, to distinguish between competing theoretical predictions, as we have already done in the corresponding model on the square lattice. Using this scheme we performed extensive simulations for several lattice sizes in the range L = 20 − 200, over large ensembles { 1 , · · · , q, · · · , Q } of random realizations ( Q = 500). Let us note here that for the pure model we simulated for each lattice size, 200 independent runs (WL random walks). It is well known that, extensive disorder averaging is necessary when studying random systems, where usually broad distributions are expected leading to a strong violation of self-averaging [73, 74]. A measure from the scaling theory of disordered systems, whose limiting behavior is directly related to the issue of self-averaging [73, 74] may be defined with the help of the relative variance of the sample-to-sample fluctuations of any relevant singular extensive thermodynamic prop- erty Z as follows: R$_{Z}$ = ([ Z $^{2}$]$_{av}$ − [ Z ] 2 $_{av}$) / [ Z ] 2 $_{av}$. Figure 1 In this Section we proceed to investigate the critical behavior of the pure TrIM defined in Eq. (1). Our aim is to observe the exact critical behavior of the model and also to estimate the whole set of critical exponents, pay- ing particular attention to the FSS behavior of the crit- ical specific heat. As mentioned above, the numerical data shown below in Figs. 2 - 4 have been estimated as averages over 200 independent runs together with the corresponding error bars. Figure 2(a) gives the shift behavior of the pseudocrit- ical temperatures corresponding to the peaks of the fol- Closely related to the above issue of self-averaging in disordered systems is the manner of averaging over the disorder. This non-trivial process may be performed in two distinct ways when identifying the finite-size anoma- lies, such as the peaks of the magnetic susceptibility. The first way corresponds to the average over disorder real- izations ([ . . . ]$_{av}$) and then taking the maxima ([ . . . ] ∗ $_{av}$), or taking the maxima in each individual realization first, and then taking the average ([ . . . ∗ ]$_{av}$). In the present paper we have undertaken our FSS analysis using both ways of averaging and have found comparable results for the values of the critical exponents, as will be discussed in more detail below. Closing this brief outline, let us comment on the statistical errors of our numerical data. The statistical errors of our WL scheme on the observed average behavior, were found to be of relatively small magnitude (of the order of the symbol sizes) when com- pared to the relevant disorder-sampling errors (due to the finite number of simulated realizations). Thus, the error bars in most of our figures below concerning the average [ . . . ] ∗ av and used also in the corresponding fitting attempts, reflect the disorder-sampling errors and have been estimated using groups of 50 realizations via the jackknife method [67]. On the other hand for the case [ . . . ∗ ]$_{av}$ the error bars shown reflect the sample-to-sample fluctuations. presents evidence that the above number of random re- alizations is sufficient in order to obtain the true average behavior and not a typical one. In particular, we plot in this figure (for lattice sizes L = 120 and L = 200) the disorder distribution of the susceptibility maxima χ ∗ q and the corresponding running average, i.e. a series of averages of different subsets of the full data set - each of which is the average of the corresponding subset of a larger set of data points, over the samples for the simu- lated ensemble of Q = 500 disorder realizations. A first striking observation from this figure is the existence of very large variance of the values of χ ∗ $_{q}$, indicating the ex- pected violation of self-averaging for this quantity. This figure illustrates that the simulated number of random realizations is sufficient in order to probe correctly the average behavior of the system, since already for Q ≈ 300 the average value of χ ∗ q appears quite stable. 3 III. PURE MODEL