arXiv subjects

Peter Wildemann

Publications and source records attributed to Peter Wildemann.

3 recordsLinked to original sources

arXiv2024

Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs

We consider the graphical representations of the Ising model on tree-like graphs. We construct a class of graphs on which the loop $\mathrm{O}(1)$ model and the single random current exhibit a non-unique phase transition with respect to the inverse temperature, highlighting the non-monotonicity of both models. It follows from the construction that there exist infinite graphs $\mathbb{G}\subseteq \mathbb{G}'$ such that the uniform even subgraph of $\mathbb{G}'$ percolates and the uniform even subgraph of $\mathbb{G}$ does not. We also show that on the wired $d$-regular tree, the phase transitions of the loop $\mathrm{O}(1)$, the single random current, and the random-cluster models are all unique and coincide.

math.PR↗

arXiv2024

Probabilistic Definition of the Schwarzian Field Theory

We provide mathematical foundations for the Schwarzian Field Theory as a finite Borel measure on $\mathrm{Diff}^1(\mathbb{T})/\mathrm{PSL}(2,\mathbb{R})$, a quotient of the space of circle reparametrisations. The measure is defined by a natural change of variables formula, which we show uniquely characterises it. We further compute its partition function (total mass) from this change of variable formula. The existence of the measure then follows from an explicit construction involving a nonlinear transformation of a Brownian Bridge, proposed by Belokurov--Shavgulidze. In two companion papers by Losev, the predicted exact cross-ratio correlation functions for non-crossing Wilson lines and the large deviations are derived from this measure.

math.PR↗

arXiv2023

$\mathbb{H}^{2|2}$-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

We explore the supercritical phase of the vertex-reinforced jump process (VRJP) and the $\mathbb{H}^{2|2}$-model on rooted regular trees. The VRJP is a random walk, which is more likely to jump to vertices on which it has previously spent a lot of time. The $\mathbb{H}^{2|2}$-model is a supersymmetric lattice spin model, originally introduced as a toy model for the Anderson transition. On infinite rooted regular trees, the VRJP undergoes a recurrence/transience transition controlled by an inverse temperature parameter $\beta > 0$. Approaching the critical point from the transient regime, $\beta \searrow \beta_{\mathrm{c}}$, we show that the expected total time spent at the starting vertex diverges as $\sim \exp(c/\sqrt{\beta - \beta_{\mathrm{c}}})$. Moreover, on large finite trees we show that the VRJP exhibits an additional intermediate regime for parameter values $\beta_{\mathrm{c}} < \beta < \beta_{\mathrm{c}}^{\mathrm{erg}}$. In this regime, despite being transient in infinite volume, the VRJP on finite trees spends an unusually long time at the starting vertex with high probability. We provide analogous results for correlation functions of the $\mathbb{H}^{2|2}$-model. Our proofs rely on the application of branching random walk methods to a horospherical marginal of the $\mathbb{H}^{2|2}$-model.

math.PR↗