Research Diary June2018

Thursday-June 14-2018

1. A survey on KPZ and universality class by Ivan Corwin: arXiv 1106.1596. Another notes by Ivan Corwin: Exact solvability of some SPDEs. The stochastic heat equation (SHE) with multiplicative noise looks like (in differential form; formally, no exact meaning at the beginning)
\[\left\{ \begin{array}{l} {\partial _t}z = \frac{1}{2}{\partial _{xx}}z + z\xi \\ z(0,x) = {z_0}(x) \end{array} \right.\] where $z:\mathbb R_{+}\times \mathbb R \to \mathbb R$ and $z_0$ is (possibly random) initial data which is independent of the white noise $\xi$ (will be described in detail later). The only thing to keep in mind right now is $\xi$ can be thought of as a random distribution. (SHE) makes sense by the following notion of solution.
 
Definition. A mild solution to the SHE for all $t>0$ and $x\in \mathbb R$ is $z(\cdot,\cdot)\in L^2(\Omega,\mathcal{F},\mathbb P)$ satisfies the Duhamel formulation
\[z(t,x)=\int_{\mathbb R}p(t,x-y)z_0(y)dy+\int_0^t\int_{\mathbb R}p(t-s,x-y)z(s,y)\xi(s,y)dyds \] where $p(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-x^2/2t}$ is the heat kernel or fundamental solution to $\partial_tp=\partial_{xx}p$ with $p(0,x)=\delta_{x=0}$. Also, we must have a priori that $\int_0^t\int_{\mathbb R}p^2(t-s,x-y){\mathbb E}[z^2(s,y)]dyds<\infty$ for the Ito integrals (stochastic integrals) to make sense and finite.

Friday-June 15-2018

1. Progressively measurable stochastic process (vs adapted process) (look into link)

2. Martin Hairer's regularity structures (Link): Decomposition of heat kernel (Lemma 5.5, Theorem 5.12). "Canonical model" (Proposition 8.27). Renormalization is to "correct" Picard iteration at each step. To see how renormalization works, consider the term $({\mathcal D}{\mathcal I}(\Xi))^2$: $\Xi$ is the white noise, thought of as derivative of Brownian motion; hence, the action ${\mathcal D}{\mathcal I}$ on $\Xi$ basically recovers Brownian motion. To correct this, we have \[M(({\mathcal D}{\mathcal I}(\Xi))^2)=({\mathcal D}{\mathcal I}(\Xi))^2-\frac{c}{\varepsilon} \] where $M=I-cL-\cdots$ where $L$ represents the action of "removing" Brownian motion.

3. Problem about feeble fish. In homogenization theory, the underlying vector field is set to be random. Roughly speaking, consider a (random) vector field $V:(\text{space},\text{time})\to \mathbb R$. Consider the following dynamics: \[dX=Vdt,\quad \text{(or in discrete settings)}\quad X_{n+1}=X_n+V(X_n,n).\] We would expect in the long time, $X\sim \mathbb E(V) \cdot t$, or equivalently, $\frac{X}{t}\to \mathbb E(V)$ where the expectation is respect to the underlying probability measure. The main question is under what condition this would hold true.

Always to think about the purpose of defining an object.

We first work in the discrete settings. To even talk about expectation $\mathbb E$, we must have stationarity in both space and time. That's the first condition. We will consider its "ergodicity" (the relationship between time and space average). Let $W:\mathbb Z^2\to \mathbb Z^2$ is a deterministic velocity (with compact support to ensure convergence later), and $\xi_{ij}$ be i.i.d random variables (taking only $0$, $1$ or any finite set of values). One way to construct the random velocity $V$ is as follows \[V(x,t)=\sum\limits_{ij}\xi_{ij}V(x+i,t+j).\] Consider the (canonical) space-time shift $T_{ij}:\mathbb Z^2\to \mathbb Z^2$ \[T_{ij}(x,t)=(x+i,t+j).\] These shifts are measure-preserving $V(x,t,T_{ij}\omega)=V(x+i,t+j,\omega).$ The claim is that $V$ is strongly mixing with respect to these shifts. And we indeed get that $X_n/n\to \mathbb E(V)$ (link). The main question is can we reduce the strong mixing condition to just being ergodic, or is there a counterexample? "Ergodic" means that \[\frac{1}{N}\sum\limits_{n=1}^{N} V(T_{ij}^{n}(x,t)) \to \int Vd\mathbb P\] where $\mathbb P$ is the underlying probability measure. Do we have \[\frac{1}{N}\sum\limits_{n=1}^{N} V(X_n,n)\to\int Vd\mathbb P\;\;\] as $X_n$ follows the dynamics $X_{n+1}=X_n $$\;+ \;V(X_n,n)$? This is a type of questions in Random Walks in Random Environment.

4. How to solve stochastic heat equation: link

5. Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions

Sunday-June 17-2018

1. The Birkhoff-von Neumann theorem states that the set of $n \times n$ doubly stochastic matrices (non-negative matrices whose every row and every column sums to $1$) is convex, and that the extremal points of the set are (all) the permutation matrices.

2. The Hoffman-Wielandt inequality states that for any two $n \times n$ normal matrices $A$, $B$, with sets of eigenvalues $\lambda_1(X),\; \lambda_2 (X),\;\ldots,\; \lambda_n(X)$ for $X \in \{A, B\}$, there exists a permutation $\sigma \in S_n$ such that \[ \sum_{i=1}^n |\lambda_i(A) - \lambda_{\sigma(i)}(B)|^2 \leq \text{Tr}\left((A-B)(A-B)^*\right). \] Proof. (kind of) Express $\text{Tr}\left((A-B)(A-B)^*\right)$ as a function on doubly stochastic matrices. Then use Birkhoff-von Neumann theorem to achieve the minimum and maximum.

Thursday-June 21-2018

1. Bipartite biregular model $(m,n,d_1,d_2)$, $md_1=nd_2$. Eigenvalues of the adjency matrix $A$ are $n-m$ zeros and $\pm \sigma_i$ (singular values of $X$). Models for $X^TX$: Wishart.

2. Extremal eigenvalues of graphs. Graphs matrices:

  • adjacency matrix $A_{ij}=\lambda_{ij}$
  • Laplacian matrix $L=I-D^{-1}A$ where $D=\text{diag}(\deg(v_i))$, note that a graph defines a Markov chain: $D^{-1}A$ is Markov matrix for graph.

3. Cauchy-Binet formula: Let $A$ be an $m\times n$ matrix and $B$ be an $n\times m$ matrix. Let $1\le j_1,i_2,\ldots,j_m\le n$. Let $A_{j_1i_2\ldots j_m}$ denote the $m\times m$ matrix consisting of columns $j_1,i_2,\ldots,j_m$ of $A$. Let $B_{j_1i_2\ldots j_m}$ denote the $m\times m$ matrix consisting of rows $j_1,i_2,\ldots,j_m$ of $B$. Then \[\det(AB)=\sum\limits_{1\le j_1<j_2<\ldots<j_m\le n}\det(A_{j_1j_2\ldots j_m})\det(B_{j_1j_2\ldots j_m}) \] Using this we can prove the Jacobi-Trudi identity. Let $h_m(x)=h_m(x_1,x_2,\ldots)$ denote the complete homogeneous symmetric polynomial. The Jacobi-Trudi states that \[s_{\lambda}=\det(h_{\lambda_i-i+j})_{i,j=1}^{l(\lambda)} \] where $s_{\lambda}(x_1,\ldots,x_n)$ is Schur polynomial defined as \[s_{\lambda}(x_1,\ldots,x_n) = \frac{\det(x_i^{\lambda_j+n-j})}{\prod\limits_{i<j}(x_i-x_j)}\] The proof proceeds in three steps

  • Prove that if $f(u)=\sum\limits_{m=0}^{\infty} f_mu^m$, then using Cauchy-Binet, \[f(x_1)f(x_2)\ldots f(x_n) =\sum\limits_{\lambda:l(\lambda)\le n} \det(f_{\lambda_i-i+j})_{i,j=1}^{l(\lambda)}s_{\lambda}(x_1,\ldots,x_n). \]
  • Let $f_m=h_m(y_1,\ldots,y_n)$ to show that \[\prod\limits_{i,j}\frac{1}{1-x_iy_j} = \sum\limits_{\lambda}\det(h_{\lambda_i-i+j}(y_1,\ldots,y_n))_{i,j=1}^{l(\lambda)}s_{\lambda}(x_1,\ldots,x_n) .\]
  • Finish by using the Cauchy identity \[\sum\limits_{\lambda}s_{\lambda}(x)s_{\lambda}(y)=\prod\limits_{i,j}\frac{1}{1-x_iy_j}. \] Reference: Symmetric Functions and Hall Polynomials by MacDonald.

4. The $\lambda_k$ eigenvalue tells how a graph can be separated into $k$ connected components.

Friday-June 22-2018

1. Ihara-Bass formula: For non-backtracking matrix $B$ and adjacency matrix $A$ of a graph $(G,E,V)$, we have for every $z\in \mathbb C$ \[\det(B-zI)=(1-z^2)^{|E|-|D|}\det(z^2I-zA+D)\] where $D=\text{diag}(\det(v_i)-1)$.

Tuesday-June 26-2018

1. The goal is to express an arbitrary function $f(H)$ of a Hermitian matrix $H$ as a superposition of Green functions $G(z) = (H - z)^{-1}$. Note that $G$ has the following representation in terms of eigenvalues and eigenvectors \[G(z)= \sum\limits_i \frac{u_iu_i^*}{\lambda_i-z}.\]

  • If $f$ is holomorphic in an open domain containing the spectrum of $H$, then we have \[ f(H) = - \frac{1}{2 \pi i} \oint_{\Gamma} f(z) G(z) \, dz\,,\] where $\Gamma$ is a positively oriented contour encircling the spectrum of $H$.
  • In many applications, $f$ is not holomorphic. (For example, one may want to take $f$ to be a (smoothed) cutoff function.) A more general and flexible functional calculus is provided by the Helffer-Sjostrand formula. Let $n \in \mathbb N$ and $f \in \cal C^{n+1}(\mathbb R)$. We define the almost analytic extension of $f$ of degree $n$ through \[\tilde f_n(x + i y) \;:=\; \sum_{k = 0}^n \frac{1}{k !} (i y)^k f^{(k)}(x)\,.\] Let $\chi \in \cal C^\infty_c({\mathbb C};[0,1])$ be a smooth cutoff function. For any $\lambda \in \mathbb R$ satisfying $\chi(\lambda) = 1$ we have \[f(\lambda) = \frac{1}{\pi} \int_{\mathbb C} \frac{\bar \partial (\tilde f_n(z) \chi(z))}{\lambda - z} \, d^2z\,,\] where $d^2 z$ denotes the Lebesgue measure on $\mathbb C$ and $\bar \partial:= \frac{1}{2} (\partial_x + i \partial_y)$ is the antiholomorphic derivative. Note that we have the Green's formula in the complex form as \[\int_{\partial D}g(z)dz=\int_{\partial D}g(z)(dx+idy)=\int_{D}\left(\partial_x(ig(z))-\partial_y(g(z))\right)d^2z=2i\int_{D}{\bar \partial}g(z)d^2z.\] for some nice domain $D$. As a consequence, we find \[f(H) = \frac{1}{\pi} \int_{\mathbb C}\bar \partial (\tilde f_n(z) \chi(z)) \, G(z) \, d^2 z\,,\] provided that $\chi = 1$ on the spectrum of $H$. The good choice of $n$ depends on the applications, but in many cases $n = 0$ or $n = 1$ works well.

Wednesday-June 27-2018

1. For $k\ge 1$, let $f:\mathbb R^k\to \mathbb R$ and $g:\mathbb R\to \mathbb R$ be functions such that \[f(x_1,x_2,\ldots,x_k)=g(x_1+x_2+\ldots+x_k).\] We have that \[{\left. {\frac{\partial }{{\partial {x_1}}}\frac{\partial }{{\partial {x_2}}} \ldots \frac{\partial }{{\partial {x_k}}}f({x_1},{x_2}, \ldots ,{x_k})} \right|_{{x_1} = {y_1}, \ldots ,{x_k} = {y_k}}} = {\left. {\frac{{{\partial ^k}}}{{\partial {t^k}}}g(t)} \right|_{t = {y_1} + \ldots + {y_k}}}.\] In particular, we get that \[{\left. {\frac{\partial }{{\partial {x_1}}}\frac{\partial }{{\partial {x_2}}} \ldots \frac{\partial }{{\partial {x_k}}}\mathbb E{e^{({x_1} + \ldots + {x_k})Z}}} \right|_{{x_1} = \ldots = {x_k} = 0}} = {\left. {\frac{{{\partial ^k}}}{{\partial {t^k}}}\mathbb E e^{tZ}} \right|_{t = 0}}.\]

2. Moments and cumulants

Thursday-June 28-2018

1. Hook formula for dimension of Young diagrams (Bruce Sagan).

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