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### Approximation of the effective resistance on Cayley graph

Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...
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### Why do we care about simplicity of the spectrum in Oseledets' theorem?

Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana). The simplicity of the spectrum has been studied ...
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### Unusual generalization of the law of large numbers

I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...
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### What is the relationship between complex time singularities and UV fixed points?

In this paper it is described how the turbulent kinetic energy spectrum and the flatness (a measure for intermittency) are governed by the position of the (dominant) singularities of the solutions of ...
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### Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diﬀusing Particles. The picture below shows the idea how permutations and inversion numbers reflect ...
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### The condition of maximality in branching rules of $SO$ group representations

Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...
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### Children's drawings and Seiberg-Witten curves

This physics (bear with me for a while) paper seems to say something about Gal \bar Q/Q: Children's Drawings From Seiberg-Witten Curves, hep-th/061108. Let's ...
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$). A vector $u\in ... 0answers 272 views ### Bosonic String Theory I would like clarification of 26 dimensional Bosonic String Theory. A definition would be, that this is free bosons compactified on a torus and orbifolded by a 2-point reflection group (or ... 0answers 110 views ### Geometric interpretation of table with permutations and inversions Let$T(n,k)$is the number of permutations of numbers$1, ..., n$and each of the permutations has$k$inversions. We can consider a table for$T(n,k)$for some$n$and$k$. For eg.$n=1,...,6$,$k=1,....
To make it simple, take the spin foam formalism of ($SU(2)$) 3D gravity. My question is about the choice of the data that will replace the (smoothly defined) fields $e$ (the triad) and $\omega$ (the ...