**8. Numerical computations in the combinatorial multiverse**

With the setup in Sections 6 and 7, we can now use Mathematica or MATLAB to generate instances of the combinatorial multiverse for small values of *m* and *W* and then compute the corresponding probability weights *PLL*, *PLH*, *PHL* and *PHH*. It is important to note that the matrices here can be treated as sparse, rather than as full matrices, which make the computations considerably faster.

In particular, in the case *m* ¼ 2 in Section 6 and with a randomly generated dynamics which is manifested by an adjacency matrix *A*, we can compute the power *A*<sup>4</sup> and read of the first row, which contains all the information we need about the paths from the state at *t* ¼ �2 with *S* ¼ 0. So what do we find?

In **Figure 3**, I have plotted the ratio *NLL=*ð Þ *NLH* þ *NHL* for the cases *m* ¼ 2 (light gray) and *m* ¼ 3 (dark gray) for values of *W* ranging from 3 to 30. What is actually displayed are the mean values of 1000 randomly generated matrices as above for each value of *W*. Although the picture clearly supports the claim that

**Figure 3.** *The ratio NLL=*ð Þ *NLH* þ *NHL as a function of W for the cases m* ¼ *2 (light gray) and m* ¼ *3 (dark gray) [4].*

#### *Combinatorial Cosmology DOI: http://dx.doi.org/10.5772/intechopen.90696*

lim *PLL PLH* þ *PHL*

*Probability, Combinatorics and Control*

almost certainty he must live in a universe with monotonic entropy.

preferable one. This alternative will be further studied in Section 9.

**8. Numerical computations in the combinatorial multiverse**

matrices, which make the computations considerably faster.

**Figure 3.**

**318**

and lim *PLL* <sup>þ</sup> *PHH*

equal zero when certain parameters tend to infinity in some well-defined way. However, it is worthwhile at this stage to note their implications for cosmology. The strong broken symmetry in Definition 5 actually means that a monotonic behavior of the entropy is far more probable than a non-monotonic one. In the case of a weak broken symmetry, this is not necessarily so; it could very well be that the most probable scenario would be high entropy at both ends. Thus, this is definitely a weaker statement, but it can nevertheless be argued that it can be used to explain the time asymmetry that we observe, referring to a kind of anthropic principle: it is an obvious observational fact that we live in a universe with low entropy at at least one end. If the statement in Definition 4 is fulfilled, then clearly among such scenarios, the monotonic ones (LH and HL) are the by far most probable ones. Thus, since universes with high entropy at both ends would seem to be quite uninhabitable, one can argue that given the existence of an observer, then with

Summing up, both limits above can be used to argue in favor of time asymmetry. Nevertheless, at least to the mind of the author, the strong broken symmetry is the

With the setup in Sections 6 and 7, we can now use Mathematica or MATLAB to generate instances of the combinatorial multiverse for small values of *m* and *W* and then compute the corresponding probability weights *PLL*, *PLH*, *PHL* and *PHH*. It is important to note that the matrices here can be treated as sparse, rather than as full

In particular, in the case *m* ¼ 2 in Section 6 and with a randomly generated dynamics which is manifested by an adjacency matrix *A*, we can compute the power *A*<sup>4</sup> and read of the first row, which contains all the information we need

In **Figure 3**, I have plotted the ratio *NLL=*ð Þ *NLH* þ *NHL* for the cases *m* ¼ 2 (light gray) and *m* ¼ 3 (dark gray) for values of *W* ranging from 3 to 30. What is actually displayed are the mean values of 1000 randomly generated matrices as above for

*The ratio NLL=*ð Þ *NLH* þ *NHL as a function of W for the cases m* ¼ *2 (light gray) and m* ¼ *3 (dark gray) [4].*

about the paths from the state at *t* ¼ �2 with *S* ¼ 0. So what do we find?

each value of *W*. Although the picture clearly supports the claim that

*PLH* þ *PHL*

(13)

*NLL=*ð Þ! *NLH* þ *NHL* 0 when *W* ! ∞, there is not really enough support for a firm prediction about the more precise asymptotic behavior for large *W*. Having said this, the behavior seems to be rather close to a relationship of the form *ρ* � 1*=W*.

It should be possible, although perhaps not so easy, to prove exact limit theorems to confirm these kinds of predictions. The problem is that we use a large number of instances to model something much more complicated, namely, the full quantum mechanical development of the multiverse. For very special unlikely choices of these instances, the ratio *NLL=*ð Þ *NLH* þ *NHL* may behave quite differently.
