# Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model

## Abstract

**:**

## 1. Introduction

## 2. Stochastic Differential Equation for the ASM with the Time-Scale Separation between Energy Injection and Avalanche Relaxation

#### 2.1. Stochastic Differential Equation for the Coarse-Grained ASM

#### 2.2. Co-Variance of Random Forces with Time-Scale Separation

## 3. Functional Integral Formulation and Feynman Diagram Technique for the ASM

## 4. Multiplicative UV-Renormalization of the ASM

**Theorem**

**1.**

**Proof of Theorem**

**1.**

## 5. Calculation of an Infinite Number of Renormalization Constants in the ASM

**Theorem**

**2.**

**Proof of Theorem**

**2.**

## 6. RG-Equations for the ASM

## 7. Plane of Fixed Points in the ASM

- If there were no time-scale separation in the ASM (zero correlation time ${t}_{c}\left(k\right)=0$ for all wave numbers k), i.e., for the “white noise” model of energy injection uncorrelated in space and time, viz.,$$\u2329f\left(x\right)f\left({x}^{\prime}\right)\u232a\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}2\Gamma {\delta}^{d}(\mathbf{r}-{\mathbf{r}}^{\prime})\delta (t-{t}^{\prime}),$$
- In the opposite case of the “frozen” configuration of stochastic force, the first, trivial solution ${\varrho}_{*}=0$ comes into play, and therefore ${J}_{nm}=-(n-1)\u03f5{\delta}_{nm}<0$. Therefore, there are no IR-stable fixed points in the ASM in this case.

## 8. On the Critical Scaling in the ASM

## 9. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ACM | Abelian sandpile model |

IR | Infra-red (momentum scale) |

RG | Renormalization group |

SOC | Self-organized criticality |

UV | Ultra-violet (momentum scale) |

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**Figure 1.**The one-loop order diagram expansions for the simplest response function $\langle \delta E/\delta f\rangle $ and the pairwise correlation function $\u2329EE\u232a$.

Q | E | ${\mathit{E}}^{\prime}$ | ${\mathit{\nu}}_{\mathit{n}0}$ | ${\mathit{\nu}}_{\mathit{n}},\mathit{\varrho}$ | $\mathfrak{A},{\mathfrak{A}}_{0}$ | ${\mathit{\varrho}}_{0}$ |
---|---|---|---|---|---|---|

${d}^{k}$ | $-\u03f5$ | $d+\u03f5$ | $(n-1)\u03f5$ | 0 | $-2$ | $2\kappa $ |

${d}^{\omega}$ | 0 | 0 | 0 | 0 | 1 | 0 |

d | $-\u03f5$ | $d+\u03f5$ | $(n-1)\u03f5$ | 0 | 0 | $2\kappa $ |

**Table 2.**The critical dimensions of the fields E and ${E}^{\prime}$ for the different assumptions on energy injection in the ASM [14].

Assumption on Random Force | ${\mathit{\u03f5}}_{\mathbf{real}}$ | $\mathit{\Delta}\left[\mathit{E}\right]$ | $\mathit{\Delta}\left[{\mathit{E}}^{\prime}\right]$ | IR-Stability |
---|---|---|---|---|

Uncorrelated in space | $3-\kappa -d/2$ | $d/2+3(\kappa -1)$ | $d/2+3(1-\kappa )$ | Domain-wise |

“White noise” | $1-d/2+\kappa $ | $d/2+\kappa -1$ | $d/2+1-\kappa $ | No |

“Frozen” configuration | $4-d$ | $2\kappa -4+d$ | $4-2\kappa $ | No |

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**MDPI and ACS Style**

Volchenkov, D.
Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model. *Dynamics* **2024**, *4*, 40-56.
https://doi.org/10.3390/dynamics4010003

**AMA Style**

Volchenkov D.
Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model. *Dynamics*. 2024; 4(1):40-56.
https://doi.org/10.3390/dynamics4010003

**Chicago/Turabian Style**

Volchenkov, Dimitri.
2024. "Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model" *Dynamics* 4, no. 1: 40-56.
https://doi.org/10.3390/dynamics4010003