On the radius of self-repellent fractional Brownian motion
Le Chen
le.chen@auburn.eduDepartment of Mathematics and Statistics
Auburn University
Seminars on Analysis and Stochastic Analysis (SASA)
Auburn, Alabama, Aug. 28, 2024
SpringerLink
Sefika Kuzgun
Carl Mueller
Panqiu Xia
DMS-Probability: No. 2246850
(2023-2026)
No. 959981
(2022-2027)
Self Avoid Random walk on $\mathbb{Z}^2$
Number of Self Avoid Random walks
on $\mathbb{Z}^d$
\[ c_n ∼ A μ^n n^{\gamma -1} \]
| $d$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $γ$ | $1$ | $\frac{43}{32}$ | $1.16...$ | $1-$ | $1$ |
Connective constant $μ$
Conjectured (for square lattice with \(d=2\)): \(μ = 2.638 158 530 31(3)\).
Connective constant $μ$
Conjectured (for square lattice with \(d=2\)): \(μ = 2.638 158 530 31(3)\).
Only known case: \(μ = \sqrt{2+\sqrt{2}}\) for hexagonal lattice.
Paper link
Mean-square displacement
(end-to-end)
\[ \mathbb{E}\left(W(n)^2\right) ∼ D n^{2ν} \]
| $d$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $ν$ | $1$ | $3/4$ | $0.588...$ | $1/2 -$ | $1/2$ |
Flory Exponent $\nu$
\( \mathbb{E}\left(W(n)^2\right) ∼ D n^{2ν} \)
\[ ν_F = \frac{3}{d+2} \]| $d$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $ν$ | $1$ | $3/4$ | $0.588...$ | $1/2 -$ | $1/2$ |
| $ν_F$ | $1$ | $3/4$ | $3/5$ | $1/2$ | $1/2$ |
Universality
Universality
Both the critical exponents $γ$ and $ν$ are universal.Critical exponents $γ$ and $ν$ depend on
- Dimension
- Noise structure (BM or fBM)
But they do not depend on
- Lattice structure (square, triangular, or hexagonal...)
- Discrete Lattice or continuous space
- Strict self avoiding or self repellent
- Definitions of displacements
- ......
Models with the same critical exponents are in the same
Universality class
Fractional Brownian Motion (fBM)
Let $\left\{ B^H_t\right\}_{t\ge0}$ be a fBM with Hurst index $H \in (0,1)^d$, taking values in $\mathbb{R}^d$. That is, $B^H_t=\left(B^{H_1,1}_t,\dots,B^{H_d,d}_t\right)$ where $\left(B^{H_i,i}_\cdot\right)_{i=1}^{d}$ are independent one-dimensional fBMs with Hurst index $H$. Thus, each $B^{H_i,i}_\cdot$ is a centered Gaussian process with covariance
\begin{equation*} \mathbb{E}\left[B^{H_i,i}_s B^{H_i,i}_t\right]=\frac{1}{2}\left(t^{2H_i}+s^{2H_i}-|t-s|^{2H_i}\right). \end{equation*}self-repellent Random Walks/fBMs
\begin{align*} Q_n^\lambda(ω) = \frac{1}{Z_n(λ)} \prod_{0\le s< t\le n } \left(1- λ v_{st}(ω)\right), \quad λ\in(0,1] \end{align*}
Alternative/more common penalty
\begin{align*} Q_n^β(ω) = \frac{1}{Z_n(β)} \prod_{0\le s< t\le n } e^{- β v_{st}(ω)}, \quad β>0. \end{align*}Continuous time
Let $d W^T$ be the Wiener measure and the Edwards model:
\begin{align*} dμ^T = \frac{1}{Z^T} e^{- β J} d W^T \end{align*}where $J$ refers to the local time
\begin{align*} J(ω) = \int_0^T\int_0^T δ\left(ω(t)-ω(s)\right) ds dt. \end{align*}No square-integrable when
\[ d H \ge 1 \]Occupation time instead of local time
We define the occupation time as follows: \begin{align*} L_T(y) & := \left|\left\{t\in[0,T]: B^H_t\in\mathbf{O}_1(y)\right\}\right| \\ & = \int_0^T \mathbf{1}_{\mathbf{O}_1(y)} \left(B^H_t\right) d t, \end{align*} where $|S|$ denotes the Lebesgue measure of the set $S$, and $\mathbf{O}_r(y)$ is the open ball in $\mathbb{R}^d$, centered at $y$, of radius $r > 0$.
Radius of gyration
\begin{gather*} R_T := \left[\frac{1}{T}\int_{0}^{T}\left|B_t^H-\overline{B}_T^H\right|^2 d t\right]^{1/2} \quad \text{with} \\ \overline{B}_T^H := \frac{1}{T}\int_{0}^{T}B_t^H d t. \end{gather*}Conjecture for self repellent fractional BM
\[ ν = \frac{2(1+H)}{d+2} \]
Theorem (L. Kuzgun, Mueller, Xia 24)
Let $B^H$ be a $1$-d fBm with $H\in (0,1)$. Then, $∀ \beta > 0$, $∃ T_{\beta} \geq e$, $C_*$, $C^*$, $C> 0$ s.t. $∀ T \geq T_{\beta}$: \begin{gather*} \mathbb{Q}_T \left( C_* \beta^{1/3} T^{\frac{2(1+H)}{3}} \leq R_T \leq C^* \beta^{1/3} T^{\frac{2(1+H)}{3}} \right) \\ \ge 1 - 2 \exp\left(- C \beta^{2/3} T^{\frac{2(2-H)}{3}}\right). \end{gather*}Theorem (L. Kuzgun, Mueller, Xia 24)
Let $B^H$ be a $1$-d fBm with $H\in (0,1)$. Then, $∀ \beta > 0$, $∃ T_{\beta} \geq e$, $C_*$, $C^*$, $C> 0$ s.t. $∀ T \geq T_{\beta}$: \begin{gather*} \mathbb{Q}_T \left( C_* \beta^{1/3} T^{\frac{2(1+H)}{3}} \leq R_T \leq C^* \beta^{1/3} T^{\frac{2(1+H)}{3}} \right) \\ \ge 1 - 2 \exp\left(- C \beta^{2/3} T^{\frac{2(2-H)}{3}}\right). \end{gather*}
- $d=1$: Solved
- $d=2,3,4$: Completely open including $H=1/2$
- $d \ge 5$:
- $H=1/2$: Solved, using the Lace expansion
- $H \ne 1/2$: Open, no Markov property
Ideas in the proof
Aim: \begin{align}\label{E_:stg-0} \mathbb{Q}_T \left(a \leq R_T \leq b\right) \geq 1 - 2 \exp (- c), \end{align}
It suffices to show that \begin{align*} \mathbb{Q}_T (R_T \leq a) & = \frac{\mathbb{E}^{\mathbb{P}_T} \left[\mathbf{1}_{\{R_T \leq a\}} \mathcal{E}_T\right]}{Z_T} \leq \exp (-c), \\ \mathbb{Q}_T (R_T \geq b) & = \frac{\mathbb{E}^{\mathbb{P}_T} \left[\mathbf{1}_{\{R_T \geq b\}} \mathcal{E}_T\right]}{Z_T} \leq \exp (-c). \end{align*}
Hence, we need to show that \begin{gather*} Z_T \geq \exp (- c), \\ \mathbb{E}^{\mathbb{P}_T} \left[\mathbf{1}_{\{R_T \leq a\}} \mathcal{E}_T\right] \leq \exp (- 2c),\\ \mathbb{E}^{\mathbb{P}_T} \left[\mathbf{1}_{\{R_T \geq b\}} \mathcal{E}_T\right] \leq \exp (- 2 c). \end{gather*} Recall \begin{align*} \mathcal{E}_T := \exp\left(-\beta\int_{\mathbb{R}^d}L_T(z)^2 d z\right) \end{align*}
Girsanov formula for fBm
Large deviation for Gaussian process
Lots of computations