A Class of Models with the Potential to Represent Fundamental Physics
  1. Introduction
  2. Basic Form of Models
  3. Typical Behaviors
  4. Limiting Behavior and Emergent Geometry
  5. The Updating Process for String Substitution Systems
  6. The Updating Process in Our Models
  7. Equivalence and Computation in Our Models
  8. Potential Relation to Physics
  9. Additional Material
  10. References
  11. Index

6.8 Large-Scale Structure of Causal Graphs

In section 5 we used the cone volume Ct to probe the large-scale structure of causal graphs generated by string substitution systems. Now we use Ct to probe the large-scale structure of causal graphs generated by our models.

Consider for example the rule

{{x, y}, {x, z}} -> {{x, y}, {x, w}, {y, w}, {z, w}}

We found in section 4 that after a few steps, the volumes Vr of balls in the hypergraphs generated by this rule grow roughly like r2.6, suggesting that in the limit the hypergraphs behave like a finite-dimensional space, with dimension 2.6.

The pictures below show the log differences in Vr and Ct for this rule after 15 steps of evolution:

With[{w = ResourceFunction[ "WolframModel"][{{x, y}, {x, z}} -> {{x, z}, {x, w}, {y, w}, {z, w}}, {{0, 0}, {0, 0}}, 15]}, ListLinePlot[ResourceFunction["LogDifferences"][#], Frame -> True, PlotStyle -> {Hue[0.985, 0.845, 0.638], Hue[0.05, 0.949, 0.955], Hue[0.089, 0.75, 0.873], Hue[0.06, 1., 0.8], Hue[0.12, 1., 0.9], Hue[0.08, 1., 1.], Hue[0.987, 0.673, 0.733], Hue[ 0.04, 0.68, 0.94], Hue[0.995, 0.989, 0.824], Hue[ 0.991, 0.4, 0.9]}] & /@ {ResourceFunction["RaggedMeanAround"][ Values@ResourceFunction["HypergraphNeighborhoodVolumes"][ w["FinalState"]]], ResourceFunction["GraphNeighborhoodVolumes"][ w["CausalGraph"], {1}][[1]]}]

The linear increase in this plot implies exponential growth in Ct and indeed we find that for this rule:

This exponential growthcompared with the polynomial growth of Vrimplies that expansion according to this rule is in a sense sufficiently rapid that there is increasing causal disconnection between different parts of the system.

The other three 22 42 globular-hypergraph-generating rules shown in the previous subsection show similar exponential growth in Ct, at least over the number of steps of evolution tested.

A rule such as

{{x, y, y}, {x, z, u}} -> {{u, v, v}, {v, z, y}, {x, y, v}}

whose hypergraph and causal graph (after 500 steps) are respectively

With[{w = ResourceFunction[ "WolframModel"][{{x, y, y}, {x, z, u}} -> {{u, v, v}, {v, z, y}, {x, y, v}}, {{0, 0, 0}, {0, 0, 0}}, 500]}, {ResourceFunction["WolframModelPlot"][w["FinalState"]], w["CausalGraph"]}]

gives the following for the log differences of Vr and Ct after 10,000 steps:

With[{w = ResourceFunction[ "WolframModel"][{{x, y, y}, {x, z, u}} -> {{u, v, v}, {v, z, y}, {x, y, v}}, {{0, 0, 0}, {0, 0, 0}}, 10000]}, ListLinePlot[ResourceFunction["LogDifferences"][#], Frame -> True, PlotRange -> {0, 2.5}, PlotStyle -> ResourceFunction["WolframPhysicsProjectStyleData"][ "GenericLinePlot", "PlotStyles"]] & /@ {ResourceFunction[ "RaggedMeanAround"][ Values@ResourceFunction["HypergraphNeighborhoodVolumes"][ w["FinalState"]]], ResourceFunction["GraphNeighborhoodVolumes"][ w["CausalGraph"], {1}][[1]]}]

This implies that for this rule the hypergraphs it generates and its causal graph both effectively limit to finite-dimensional spaces, with the hypergraphs having dimension perhaps slightly over 2, and the causal graph having dimension 2.

Consider now the rule:

{{x, y, x}, {x, z, u}} -> {{u, v, u}, {v, u, z}, {x, y, v}}

The hypergraph and causal graph (after 1500 steps) for this rule are respectively:

With[{w = ResourceFunction[ "WolframModel"][{{x, y, x}, {x, z, u}} -> {{u, v, u}, {v, u, z}, {x, y, v}}, {{0, 0, 0}, {0, 0, 0}}, 1500]}, {ResourceFunction["WolframModelPlot"][w["FinalState"]], w["CausalGraph"]}]

The log differences of Vr and Ct after 10,000 steps are then:

With[{w = ResourceFunction[ "WolframModel"][{{x, y, x}, {x, z, u}} -> {{u, v, u}, {v, u, z}, {x, y, v}}, {{0, 0, 0}, {0, 0, 0}}, 10000]}, ListLinePlot[ResourceFunction["LogDifferences"][#], Frame -> True, PlotRange -> {0, 3.5}, PlotStyle -> ResourceFunction["WolframPhysicsProjectStyleData"][ "GenericLinePlot", "PlotStyles"]] & /@ {ResourceFunction[ "RaggedMeanAround"][ Values@ResourceFunction["HypergraphNeighborhoodVolumes"][ w["FinalState"]]], ResourceFunction["RaggedMeanAround"][ Values@ResourceFunction["GraphNeighborhoodVolumes"][ w["CausalGraph"]]]}]

Both suggest limiting spaces with dimension 2, but with a certain amount of (negative) curvature.