The Action is defined as S(q(t)):=\int_{[t_{1},t_{2}]}L(q,q',t)dt. For example, in absence of potential, the Lagrangian equals L=\frac{1}{2}\cdot|\gamma|^{2}= \frac{1}{2} m(\dot{r}^{2} + r^{2} \dot{\phi}^{2} ).

So the \partial S=0\Rightarrow EL equations are \partial_{t\dot{\phi}}L=\partial_{\phi}L and  \partial_{t\dot{r}}L=\partial_{r}L. For our example,m\cdot r^{2}\ddot{\phi} =0 and 2m \ddot{r}=m\cdot r\dot{\phi}\Rightarrow with solutions re^{i\phi}=(at+b)+i(ct+d).

Intuitively: Action is the distance in configuration space like the absolute metric is the distance in Euclidean space; it gives the shortest path between states.


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Concentration Inequalities: A Nonasymptotic Theory of Independence

Seriously good book and very related to Spin Glasses. Paperback is 60$.…/…/ref=sr_1_1…

“This is a truly wonderful book, that manages to convey the richness of concentration inequalities while maintaining
clarity and focus. Connections to entropy, influences, convex geometry and isoperimetric inequalities are highlighted.
The book should be useful for novices as well as seasoned experts. A remarkable achievement.” by Yuval Peres

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Convergence in distribution of conditional expectations

Convergence in distribution of conditional expectations

Main theorem

Let Q^{N} be measure s.t. P^{N}<<Q^{N} and under it, the X_{N},Y_{N} are independent. Let L^{N}(X_{N},Y_{N})=\frac{dP^{N}}{dQ^{N}}. Suppose Q^{N}\stackrel{weakly}{\to}Q , the distribution for (X,Y,L(X,Y)). Then a)P<<Q and $latex L(X,Y)=\frac{dP}{dQ}$ and b)for F\in C_{b},


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Gelfand–Naimark theorem and Probability

summarize connections with probability

In random matrix theory

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Buffon’s noodle problem

I will summarize steps in

Proof of Buffon’s noodle problem

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Large deviations and entropy

Details on

1)relation of microstate and rate functions

2)Kullback–Leibler divergence

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Distributions with identically distributed factors

Cramer’s theorem

Raikov’s theorem

Proof of Raikov’s theorem

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