**Acknowledgements**

Most of the computations presented here have be done using Mathematica and/ or MATLAB. Many of them have been carried out on an ordinary mac, but the heavier ones have been processed using MATLAB at PDC, the center for parallel computing at the Royal Institute of Technology in Stockholm.

The author would like to thank PDC for the possibility to use their resources and also the friendly staff for helping to implement the computations in an economical way.

**Author details**

*Combinatorial Cosmology*

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

Department of Mathematics, University of Stockholm, Stockholm, Sweden

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: matamm@math.su.se

provided the original work is properly cited.

Martin Tamm

**321**

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

Given a real number *μ*≥0, we can then consider the probability measure which to each path *U* assigns the (un-normalized) probability exp f g *μξ* and replace the

With this definition, it is now again possible to compute the probability weights

**Conjecture 1**. If *μ*>0, then we have a strong broken time symmetry in the limit

Clearly, there is a large gap between the extremely simplified dynamics in this paper and more realistic dynamics based on, say, ordinary Newtonian physics or quantum mechanics. This is, for better or for worse, both the strength and the weakness of the combinatorial method presented here: extreme simplification may be the price we have to pay in order to see the forest in spite of all the trees.

In any case, the few simple examples in this paper should only be considered as a

Most of the computations presented here have be done using Mathematica and/

The author would like to thank PDC for the possibility to use their resources and also the friendly staff for helping to implement the computations in an economical

or MATLAB. Many of them have been carried out on an ordinary mac, but the heavier ones have been processed using MATLAB at PDC, the center for parallel

computing at the Royal Institute of Technology in Stockholm.

first step toward more realistic models. And in fact, when the object of study is something as enormously large as the multiverse, one should not expect a single method of attack to give all the answers. Rather, it can be expected that future developments will have to combine computer computations, heuristics, and exact

*PLL*, *PLH*, *PHL* and *PHH*, and we can note that for *μ* ¼ 0, these will be exactly the same as in the case without weights in Section 8. Thus, this model is really a

⋯*aq*2*m*�1*j:* (16)

X *q*1

generalization of the previous theory.

*Probability, Combinatorics and Control*

mathematical methods in completely new ways.

X *q*2

*m* ! ∞ (for a suitable fixed choice of *p*, *K*, and *W* with *K* ≪*W*).

⋯X *<sup>q</sup>*2*m*�<sup>1</sup> *e μξaiq*<sup>1</sup> *aq*1*q*<sup>2</sup>

sum in (14) by

**10. Conclusions**

**Acknowledgements**

way.

**320**
