# The Continuous‐Time Lace Expansion

@article{Brydges2021TheCL, title={The Continuous‐Time Lace Expansion}, author={David Brydges and Tyler Helmuth and Mark Holmes}, journal={Communications on Pure and Applied Mathematics}, year={2021}, volume={74} }

We derive a continuous‐time lace expansion for a broad class of self‐interacting continuous‐time random walks. Our expansion applies when the self‐interaction is a sufficiently nice function of the local time of a continuous‐time random walk. As a special case we obtain a continuous‐time lace expansion for a class of spin systems that admit continuous‐time random walk representations.

## 13 Citations

### A simple convergence proof for the lace expansion

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2022

We use the lace expansion to give a simple proof that the critical two-point function for weakly self-avoiding walk on $\mathbb{Z}^d$ has decay $|x|^{-(d-2)}$ in dimensions $d>4$.

### The near-critical two-point function and the torus plateau for weakly self-avoiding walk in high dimensions

- Mathematics
- 2020

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice Z d in dimensions d > 4, in the vicinity of the critical…

### Self-Avoiding Walk and Supersymmetry

- MathematicsIntroduction to a Renormalisation Group Method
- 2019

Following a brief discussion of the critical behaviour of the standard self-avoiding walk, we introduce the continuous-time weakly self-avoiding walk (also called the lattice Edwards model). We…

### The near-critical two-point function for weakly self-avoiding walk in high dimensions

- Mathematics
- 2020

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the…

### Asymptotic behaviour of the lattice Green function

- MathematicsLatin American Journal of Probability and Mathematical Statistics
- 2022

The lattice Green function, i.e., the resolvent of the discrete Laplace operator, is fundamental in probability theory and mathematical physics. We derive its long-distance behaviour via a detailed…

### The geometry of random walk isomorphism theorems

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2021

The classical random walk isomorphism theorems relate the local times of a continuous-time random walk to the square of a Gaussian free field. A Gaussian free field is a spin system that takes values…

### The scaling limit of the weakly self-avoiding walk on a high-dimensional torus

- Mathematics
- 2021

How long does a self-avoiding walk on a discrete d -dimensional torus have to be before it begins to behave diﬀerently from a self-avoiding walk on Z d ? We consider a version of this question for…

### Kotani’s Theorem and the Lace Expansion

- Mathematics
- 2020

In 1991, Shinichi Kotani proved a theorem giving a sufficient condition to conclude that a function f(x) on Z decays like |x| for large x, assuming that its Fourier transform f̂(k) is such that…

### Limit Theorems for Random Walk Local Time, Bootstrap Percolation and Permutation Statistics

- Mathematics
- 2019

Author(s): Lee, Sangchul | Advisor(s): Biskup, Marek | Abstract: Limit theorems are established in three different contexts. The first one concerns exceptional points of the simple random walk in…

### Correct Bounds on the Ising Lace-Expansion Coefficients

- MathematicsCommunications in Mathematical Physics
- 2022

The lace expansion for the Ising two-point function was successfully derived in Sakai (Commun. Math. Phys., 272 (2007): 283--344). It is an identity that involves an alternating series of the…

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