The Brownian motion as scaling limit of a random walk in discrete time

I start with an infinite set of random variables $X_n$, with $n: 1,2, \ldots$ and probability density function proportional to \begin{equation*} \exp -\frac{1}{2} \sum_{n =0}^{+\infty} (x_{n+1} - x_n)^2 \end{equation*} with $x_0 =0$. Random paths of $X_n$ can be generated by \begin{equation*} X_{n+1} = X_n + G_n \end{equation*} starting at $X_0 =0$ and with $G_n$ independent standard Gaussian random variables. The $X_n$ form a random walk in discrete time ("the lattice"). The two-point correlation function is \begin{equation}\label{eq:20180123} \mathbb{E} [ X_m X_n ] = n \quad\text{if}\quad m \ge n \end{equation} Now I move to the real line: I imagine that the distance between the lattice points is $a$. Therefore, the ratio $t/a$ is the number of lattice points between $0$ and $t$, approximated as an integer. I define $B_t = a^h X_{t/a}$ for $a$ very small. The factor $a^h$ is an extra scaling to make everything work, see below. I calculate the limit \begin{equation*} \mathbb{E} [ B_s B_t] = \lim_{a \to 0}\mathbb{E} [ a^h X_{s/a} \ a^h X_{t/a}] \end{equation*} Using \eqref{eq:20180123}, this is equal to $\lim_{a \to 0} a^{2 h} \frac{t}{a}$ for $s \ge t$. This limit exists if $h = 1/2$ and is then equal to $t$. $B_t$ is the standard Brownian motion: it is a Gaussian process and has the correct two-point correlation function.

From the definition of the Brownian motion as scaling limit, one can also prove the following formula. Suppose $b > 0$, then \begin{equation}\label{eq:20180122} \mathbb{E}[B_{b t_1}\cdots B_{b t_n}] = b^{n/2} \mathbb{E}[B_{t_1}\cdots B_{t_n}] \end{equation} Indeed, the left hand side is equal to \begin{equation*} \lim_{a \to 0}\mathbb{E} \left[ a^{1/2} X_{b t_1/a}\ \cdots\ a^{1/2} X_{bt_n/a}\right] \end{equation*} Write $a = b a'$, then the limit is equal to \begin{equation*} \lim_{a' \to 0}\mathbb{E} \left[ (ba')^{1/2} X_{t_1/a'}\ \cdots \ (ba')^{1/2} X_{t_n/a'}\right] \end{equation*} This is equal to the right hand side of \eqref{eq:20180122}.

This is all quite similar to the scaling limit discussed in conformal field theory (CFT), see for example [1]. $h$ plays the role of the scaling dimension, formula \eqref{eq:20180122} is the analogue of scale invariance in CFT. For my job I use the Brownian motion and more general stochastic processes. As a hobby I wanted to do something different and study CFT. These topics are related after all: CFT seems to be some kind of two-dimensional generalization of stochastic processes.

References and comments

[1] Conformal field theory and statistical mechanics (Lecture - 01) by John Cardy

Extended solutions for the biadjoint scalar field

Chris D. White and I recently published a paper with new solutions for the biadjoint scalar field: link to Arxiv, link to Physics Letter B.

Girsanov for dummies

In quantitative finance, one uses change of numéraire, also known as Girsanov's theorem. In this post I explain this concept in a very simple situation. In practice, I find that this "Girsanov's theorem for dummies" version is good enough for most calculations.

Covariant Taylor Series

I recently saw for the first time formulas that are covariant versions of Taylor series. Because they are not easy to find on the internet, I write some down here. Suppose $x_0$ and $x_1$ are two points on a manifold, then the covariant Taylor series are formulas like \begin{align*} f(x_1) &= f(x_0) + f_{;\mu}(x_0)\, \eta^{\mu} + \dfrac{1}{2}f_{;\mu\nu}(x_0)\, \eta^{\mu}\eta^{\nu} + O(\eta)^3\\ T_{\mu}(x_1) &= T_{\mu}(x_0) + T_{\mu;\alpha}(x_0)\, \eta^{\alpha} + \dfrac{1}{2}\left( T_{\mu;\alpha\beta}(x_0)+\dfrac{1}{3} R^{\sigma}_{\ \ \alpha\beta\mu}(x_0)\, T_{\sigma}(x_0)\right) \eta^{\alpha}\eta^{\beta} + O(\eta)^3 \end{align*} The semi-colon denotes the covariant derivative with the Levi-Civita connection. The vector $\eta^{\mu}$ is defined as follows: take a geodesic $\gamma(t)$ such that $\gamma(0) = x_0$ and $\gamma(1) = x_1$, then $\eta^{\mu} = \dot\gamma^{\mu}(0)$. The higher coefficients in the series expansion become more and more complicated formulas involving the Riemann tensor and its covariant derivatives. The formulas can be proved using normal coordinates.

More information can be found in "The Background Field Method and the Ultraviolet Structure of the Supersymmetric Nonlinear Sigma Model", by Alvarez-Gaume, Freedman, Mukhi, 1981

Wu-Yang monopole: numerical calculation

I have been reading the paper by Wu and Yang [1] in which they find the famous Wu-Yang monopole. In the paper there are solutions for three types of monopoles: one has an analytical form, which is the one most often quoted, but there are also two other monopoles with numerical solution only. In this post I use Python/numpy to perform numerical analysis on the latter solution. I use the same notation as in [1].
Wu and Yang obtain the following system of ordinary differential equations \begin{align} \frac{d\Phi}{d \xi} &= \psi\label{eq:20170625a}\\ \frac{d\psi}{d \xi} &= \psi + \Phi(\Phi^2-1)\label{eq:20170626a} \end{align} Here $\xi$ is given by $r = e^{\xi}$, with $r$ the distance to the origin. The right-hand side of \eqref{eq:20170625a}-\eqref{eq:20170626a} defines the vector field ($d\Phi/d\xi, d\psi/d\xi)$ in the $(\Phi, \psi)$ plane. Its integral curves can be seen in the next figure

The integral curves of the vector field defined by \eqref{eq:20170625a}-\eqref{eq:20170626a}.
The stationary points are marked in red.

I calculate the integral curve from the point $(\Phi,\psi) = (0,0)$ to $(1,0)$ using the numpy function solve_bvp [2].

The integral curves of the vector field defined by \eqref{eq:20170625a}-\eqref{eq:20170626a}.
The integral curve from the stationary point $(0,0)$ to $(1,0)$ is added in red.

$\Phi(\xi)$ can be seen in the next graph. One sees that $\Phi(\xi) \to 0$ for $\xi \to -\infty$ and $\Phi(\xi) \to 1$ for $\xi \to +\infty$

In the rest of this post I reproduce part of Table 1 in [1].

Variance of Markov Chain Monte Carlo

In a previous post, I discussed the bias of Markov Chain Monte Carlo (MCMC) simulation. In this post I will discuss the variance. Please see the previous post for information about the notation that I use.
If \begin{equation*} S =\frac{1}{N} \sum_{t=1}^N f(X_t) \end{equation*} then for large $N$, the variance of $S$ is

Bias in Markov Chain Monte Carlo

Markov Chain Monte Carlo (MCMC) simulation can be used to calculate sums \begin{equation}\label{eq:20170427a} I = \sum_a \pi_a f(a) \end{equation} One finds a Markov process $X_t$ with stationary distribution $\pi_a$, then the sum \eqref{eq:20170427a} is approximated by \begin{equation*} S =\frac{1}{N} \sum_{t=1}^N f(X_t) \end{equation*} One can prove that under certain assumptions, \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \sum_{t=1}^N f(X_t) = \sum_a \pi_a f(a) \end{equation*} This is Birkhoff's ergodic theorem. In this post I illustrate the behaviour of $ES$ for large $N$.

A calculation on moduli stabilization

In section 21.6 "Moduli stabilization and the landscape" in Zwiebach's string theory book, I read the sentence "Deriving the potential $V(R)$ associated with $R$ is a straightforward but technical calculation in general relativity". At this point I did not understand what the calculation was. I vaguely remembered a paper by Witten about instabilities in Kaluza-Klein spacetimes related to instantons. A calculation with instantons is indeed technical, but perhaps straightforward for experts. I found more information in a paper by Denef [1]. The calculation has nothing to do with instantons, but is indeed a straightforward calculation in differential geometry. In the rest of this blog post I set out the calculation in the form of a new exercise for Zwiebach's book.

Comment about particle on a circle

The wave function of a particle on a circle is a solution of the Schrödinger equation \begin{equation}\label{eq:20170129a} i \frac{\partial \psi}{\partial t} = - \frac{1}{2 m} \frac{\partial^2 \psi}{\partial x^2} \end{equation} with $x \in [0 , 2 \pi]$ and $\hbar = 1$. When \eqref{eq:20170129a} is solved in physics books, it is usually imposed that the wave function should be periodic [1]. I used to be puzzled why one has to impose the periodicity. After all, I thought, only the probability density function $|\psi|^2$ has physical meaning, so one could as well impose that \begin{equation}\label{eq:20170129b} \psi(2 \pi) = e^{ i \alpha} \psi(0) \quad\text{with}\quad \alpha\in\mathbb{R} \end{equation}

A magnetostatic exercise in 10 dimensions

I calculate the electromagnetic field generated by electrical currents in 10 spacetime dimensions (9 space and 1 time). The set up is as follows: the current flows down the positive $x_1$-axis, hits the origin and then spreads out isotropically in the $x_2 x_3 x_4$ subspace, see figure 1 and 2. I wanted to calculate this because in string theory a similar calculation is needed to obtain the Kalb-Ramond field generated by a string ending on a $D3$-brane [1]

A calculation in magnetostatics

I wanted to calculate the magnetic field generated by a current which flows down the positive $z$-axis, hits the origin and then spreads out radially over the $xy$ plane, see figure 1.

Lorentz invariance of string theory in the light-cone gauge

On page 261 in his book [1] Zwiebach writes ''There is much at stake in this calculation. It is in fact, one of the most important calculations in string theory. [...] The calculation is long and uses many of our previously derived results''. Then Zwiebach states the result \begin{align} \left[ M^{- I}, M^{- J}\right] = &- \frac{1}{\alpha'\ { p^+ }^2} \sum_{m=1}^{\infty} \left(\alpha^I_{-m} \alpha^J_m - \alpha^J_{-m} \alpha^I_m \right)\nonumber\\ &\times \left\{ m \left[1 - \dfrac{1}{24} (D-2) \right] + \dfrac{1}{m} \left[ \dfrac{1}{24} (D-2) + a \right] \right\}\label{eq:20170103} \end{align} This is the commutator of two Lorentz transformations in the light-cone gauge. The commutator should be zero for string theory to be Lorentz invariant. The calculation of the commutator is indeed very tedious and after scribling too many pages I gave up. I googled for a quick way to obtain \eqref{eq:20170103}. Here are the more interesting results that I found.

On Killing spinors in general dimensions

The following property is true in four spacetime dimensions [1] [2]

If the electromagnetic field $F_{ab}$ satisfies Maxwell's equations \begin{equation*} \nabla_{[a}F_{bc]}= 0 \quad\text{and}\quad \nabla^a F_{ab} =0 \end{equation*} and there is a spinor $\psi$ such that \begin{equation}\label{eq:20161115b} (\nabla_{\mu} + i \sqrt{4 \pi} \not F \gamma_{\mu} ) \psi = 0 \end{equation} and $i \bar \psi \gamma^{\mu} \psi$ is time-like

then the Einstein equations are satisfied as well: \begin{equation*} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu} \end{equation*}

This property (property A from now on) can for example be used to obtain the metric and electromagnetic field of the Israel-Wilson-Perjés (IWP) black holes. Because I wanted to generalize the IWP black holes to higher dimensions, I wanted to find the generalization of property A in higher dimensions.

Extremal black holes via Killing spinors

The usual way to obtain the metric and electromagnetic field of a charged black hole in general relativity is to make a spherically symmetric ansatz, insert this ansatz into the Einstein-Maxwell equations and then solve the resulting set of non-linear ordinary differential equations. In this post I explain an alternative method that uses Killing spinors. This method can be used for extremal black holes. These are black holes with charge equal to the mass.

The isotropic harmonic oscillator

While studying Lie-algebras I read that the three-dimensional harmonic oscillator has an $SU(3)$ symmetry. I found this very unexpected; I thought it was ''obvious'' that the symmetry is only $SO(3)$.

$F_4$ tensor products

The product of two irreducible representations of a simple Lie algebra can be decomposed into irreducible components. There are various techniques to calculate this decomposition, see for example chapter XIV in [1]. However, the decomposition can also be calculated by brute force.

Weight diagrams of $G_2$

This post contains pictures of weight diagrams of irreducible representations of the Lie algebra $G_2$.

The 7-dimensional representation of $G_2$

The exceptional Lie algebra $G_2$ has an irreducible representation of dimension 7. In this post I calculate the matrices of this irrep.

Commutation relations in $G_2$

In this post I calculate the structure constants of the exceptional Lie algebra $G_2$. I assume the reader is familiar with Lie algebras, for example at the level of chapter 9 in [1].

Decay rate of the muon

The muon is a heavy cousin of the electron and decays into an electron and two neutrinos \begin{equation*} \mu \to e + \nu_{\mu} + \bar{\nu}_e \end{equation*} The decay rate of the muon is calculated in section 10.2 in Griffiths [1]. To calculate the decay rate $\Gamma$ one needs to calculate a 6-dimensional integral coming from 3 particles times 3 momentum integrals with momentum conservation. The calculation of this integral in Griffiths is quite lengthy and I do not have much insight about it. I have questions like

• Could I have calculated the integral in a different order than the one in Griffiths?
• Could one still calculate the result analytically if more particles were produced in the decay?
• Is there a faster way to obtain the result?

I therefore decided to calculate $\Gamma$ in a different way. My calculation is motivated by [2]. I set the mass of the electron $e$, of the muon neutrino $\nu_{\mu}$ and of the electron anti neutrino $\bar{\nu}_e$ to zero.