Some papers on the confluence of probability and geometry I would like to get hold of some early papers on the confluence of Probability and Geometry. I am of course aware of the celebrated book, "Probability on Compact Lie Groups" but would like to understand the works of probability on Riemannian manifolds.
Can you kindly mention some old papers relating to the beginning of this connection between probability and geometry? Thanks a lot.
 A: Some early papers on probability on Riemannian manifolds include:


*

*Brownian
motion and random walks on manifolds, Nicolas Varopoulos (1984)

*The
Central Limit Problem for Geodesic Random Walks, Erik Jørgensen
(1975)

*Random Walk on a Sphere
and on a Riemannian Manifold, P.H. Roberts and H.D. Ursell (1960).
A: Some early papers concerning SDEs on manifolds are:


*

*Itô, Kiyoshi (1950). Stochastic differential equations in a
differentiable manifold. Nagoya Mathematical Journal. 1: 35–47.

*Itô, Kiyoshi (1962). The Brownian motion and tensor fields on
Riemannian manifold. Proc. Int. Congr. Math., Stockholm 2.


and see also,


*

*Molchanov S. A. (1975). Diffusion Processes and Riemannian
Geometry. Russian Mathematical Surveys. Vol. 30, 1-63.

*Hsu E. P. (1987). Brownian motion and Riemannian geometry. in
Geometry of Random Motion, edited by R. Durrett and M. Pinsky,
Contemporary Math., American Mathematical Society, 73,
95--104.


In addition, here are some highly cited books on stochastic differential geometry


*

*Malliavin P. (1978). Géometrie différentielle stochastique. Presses
de l'Université de Montréal, Montréal.

*Elworthy K. D. (1982). Stochastic differential equations on
manifolds, Cambridge University Press, Cambridge.

*M. Emery (1989). Stochastic calculus in manifolds, Springer,
Berlin.

*E. P. Hsu (2002). Stochastic analysis on manifolds, Graduate
Studies in Mathematics. Providence, RI: American Mathematical
Society, vol. 38.


On the topic of sampling on manifolds, see the following expository paper and the references given in their discussion of related literature:


*

*Diaconis, P., Holmes, S. and Shahshahani, M. (2013). Sampling from a
manifold. Advances in Modern Statistical Theory and Applications:
A Festschrift in Honor of Morris L. Eaton. Institute of Mathematical
Statistics. 102-125.

A: One very good introduction is this by Daniel W. Stroock. Is very good both as an introduction and as presentation of current research (at the time of the writing, that is 2000). Also it has plenty of historical discussions.
A: Hugo Duminil-Copin, Stanislav Smirnov 
The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$

We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt{2}}$. This value has been derived nonrigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self-avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to $\text{SLE}(8/3)$.

Let $f(z) = u(x,y) + iv(x,y)$ be a holomorphic function.  The derivative should be the same regardless of how we should take the derivative.
$$ \lim_{\Delta x \to 0} \frac{f(z + \Delta x)-f(z)}{\Delta x}
= \lim_{\Delta x \to 0} \frac{f(z + \Delta y)-f(z)}{i\Delta y} $$
Reading off the real and imaginary parts you obtain two different derivatives:
$$ \frac{\partial u}{\partial x} + i\, \frac{\partial v}{\partial x}  = -i\,\Big(\frac{\partial u}{\partial y}+ i\,\frac{\partial v}{\partial y} \big)$$
We have put a complex structure on the flat plane $\mathbb{C}$.  Why stop there?  Let's set:
$ dz = e^{i\theta} dx$ then:
$$ \frac{df}{dz} = e^{-i\theta} \lim_{\Delta x \to 0}\frac{f(z + e^{i\theta}\Delta x) - f(z)}{\Delta x}
= e^{-i\theta} \bigg( \cos \theta \frac{\partial }{\partial x} +  \sin \theta\; i\,\frac{\partial }{\partial y} \bigg) \bigg( u + i\, v\bigg)$$
If you let $\theta = 0$ or $\theta = \pi/2$ or $\theta = \pi/3$ this is:
$$ \bigg( \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}\bigg)  = \bigg(
-\frac{\partial v}{\partial y} + i \,\frac{\partial u}{\partial y}\bigg)
= e^{-i\pi/3}\bigg((
\frac{1}{2}\frac{\partial u}{\partial x} - 
\frac{\sqrt{3}}{2} \frac{\partial v}{\partial y}) + i 
(\frac{1}{2}\frac{\partial v}{\partial x} - \frac{\sqrt{3}}{2} \frac{\partial u}{\partial y}
)\bigg)
 $$
The algebra looks almost right.

Over a square lattice, all you have are approximations.  How do you know that the discrete derivative tends to the continuous complex derivative when $\Delta z \to 0$ ?  This might help us understand the limiting behavior of:


*

*Brownian motion as limit of random walk

*the limiting behavior of self-avoiding random walk

*loop-erased random walk (which I always mistake for self-avoiding)



The variable that Stanislav Smirnov and Hugo Duminil-Coupin define are not holomorphic, but parafermionic, they only obey half the Cauchy-Riemann equations.  
The geometric significance is still a little bit open, I think.  Overall geometric approaches to the Cauchy-Riemann equations are a bit under-valued in our modern time.
