arXiv subjects

Ilya Losev

Publications and source records attributed to Ilya Losev.

6 recordsLinked to original sources

arXiv2026

Schwarzian Field Theory at High Temperatures

In this work we study the high-temperature limit of the Schwarzian Field Theory probability measure. We show that on large scales this limit concentrates on jump processes, while its small-scale behaviour is governed by a process which we call Schwarzian Field Theory on the real line. This Schwarzian Theory on the real line can be viewed as the infinite-volume version of the Schwarzian Field Theory. In addition to showing this local convergence, we also provide a systematic treatment of the Schwarzian Theory on the real line. This includes the calculation of the correlation functions and a corresponding uniqueness theorem.

math.PR↗

arXiv2025

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↗

arXiv2025

How long are the arms in DBM?

Diffusion Limited Aggregation and its generalization, Dielectric Breakdown model play an important role in physics, approximating a range of natural phenomena. Yet little is known about them, with the famous Kesten's estimate on the DLAs growth being perhaps the most important result. Using a different approach we prove a generalisation of this result for the DBM in $\mathbb{Z}^2$ and $\mathbb{Z}^3$. The obtained estimate depends on the DBM parameter, and matches with the best known results for DLA. In particular, since our methods are different from Kesten's, our argument provides a new proof for Kesten's result both in $\mathbb{Z}^2$ and $\mathbb{Z}^3$.

math.PR↗

arXiv2024

Large Deviations of the Schwarzian Field Theory

We prove a large deviations principle for the probabilistic Schwarzian Field Theory at low temperatures. We demonstrate that the good rate function is equal to the action of the Schwarzian Field Theory, and we find its minimisers. In addition, we define an analogue of the H\"{o}lder condition on the functional space $\mathrm{Diff}^1(\mathbb{T})/\mathrm{PSL}(2,\mathbb{R})$ in terms of cross-ratio observables, characterise them in terms of the usual H\"{o}lder property on the space of continuous functions, and deduce the corresponding compact embedding theorem. We also show that the Schwarzian measure concentrates on functions satisfying the defined condition.

math.PR↗

arXiv2024

Probabilistic Correlation Functions of the Schwarzian Field Theory

We study correlation functions of the probabilistic Schwarzian Field Theory. We compute cross-ratio correlation functions exactly in the case when the corresponding Wilson lines do not intersect, confirming predictions made in the physics literature via limit of the conformal bootstrap and the DOZZ formula. Moreover, we prove that these correlation functions characterise the measure uniquely. We use them to define and compute the stress-energy tensor correlation functions, and demonstrate, in particular, that these agree with the results obtained earlier by formal differentiation of the partition function.

math.PR↗

arXiv2020

Jacobi matrices with lacunary spectrum

We find asymptotics of entries of Jacobi matrices with lacunary spectral data under some additional growth conditions. We also prove the inverse results. In addition, we study connections between Jacobi matrices, canonical systems and de Branges spaces for lacunary spectral data

math.CV↗