arXiv subjects

Christian Webb

Publications and source records attributed to Christian Webb.

At least 19 recordsLinked to original sources

arXiv2026

Exploring Expert Perspectives on Wearable-Triggered LLM Conversational Support for Daily Stress Management

Wearable devices increasingly support stress detection, while LLMs enable conversational mental health support. However, designing systems that meaningfully connect wearable-triggered stress events with generative dialogue remains underexplored, particularly from a design perspective. We present EmBot, a functional mobile application that combines wearable-triggered stress detection with LLM-based conversational support for daily stress management. We used EmBot as a design probe in semi-structured interviews with 15 mental health experts to examine their perspectives and surface early design tensions and considerations that arise from wearable-triggered conversational support, informing the future design of such systems for daily stress management and mental health support.

cs.HC↗

arXiv2025

Near-critical Ising, sine-Gordon at the free fermion point, and bosonization

In this article, we study the continuous correlations of the near-critical Ising model in two dimensions with plus boundary conditions, and prove that doubled correlation functions of primary fields (spin, disorder, fermions, energy) in the Ising model are given by correlation functions of the sine-Gordon model at the free fermion point. This is an instance of bosonization. The main ideas involve analyticity of correlation functions in a mass parameter in finite volume and proving that in a perturbative regime, the Taylor coefficients of the correlation functions match due to known bosonization results for the critical Ising model in terms of the Gaussian free field. The main techniques on the Ising side involve construction and precise estimates of certain massive holomorphic functions while on the sine-Gordon side, we control an iterated Mayer expansion with techniques going back to Brydges and Kennedy.

math-ph↗

arXiv2025

Twisted Dirac operators and fractional correlations of the massless sine-Gordon model at the free fermion point

For the massless sine-Gordon model at the free fermion point, in infinite volume, we define the fractional (charge or vertex operator) correlation functions from the probabilistic path integral and prove that they are given by renormalized determinants of massive twisted Dirac operators. The fractional correlation functions are the moments of the imaginary multiplicative chaos, a random generalized function that we construct with respect to the infinite-volume massless sine-Gordon measure. The renormalized determinants are the tau functions of Sato--Miwa--Jimbo as identified by Palmer. The construction and a priori control of the imaginary multiplicative chaos combines methods from stochastic analysis (of singular SPDE flavor) for short-scale regularity with qualitative input from integrability for large-scale control. The exact identification of the correlation functions with the renormalized determinants relies on finite-volume approximation, regularity estimates for the mass perturbation, and analytic continuation in the coupling constant. The combination of existing results for tau functions with our identification implies various predictions for the sine-Gordon model such as that the fractional two-point functions are expressed as Fredholm determinants and satisfy certain PDEs as predicted by Bernard--LeClair. Using asymptotics of Fredholm determinants of Basor--Tracy and mixing of the massless sine-Gordon model at the free fermion point, which we prove, we further derive the exact formula for the one-point function predicted by Lukyanov--Zamolodchikov (at the free fermion point).

math.PR↗

arXiv2025

Operator product expansions of derivative fields in the sine-Gordon model

In this article, we initiate the study of operator product expansions (OPEs) for the sine-Gordon model. For simplicity, we focus on the model below the first threshold of collapse ($\beta<4\pi$) and on the singular terms in OPEs of derivative-type fields $\partial \varphi$ and $\bar\partial\varphi$. We prove that compared to corresponding free field OPEs, the sine-Gordon OPEs develop logarithmic singularities and generate Wick ordered exponentials. Our approach for proving the OPEs relies heavily on Onsager-type inequalities and associated moment bounds for GFF correlation functions involving Wick ordered exponentials of the free field.

math-ph↗

arXiv2023

Bosonization of primary fields for the critical Ising model on multiply connected planar domains

We prove bosonization identities for the scaling limits of the critical Ising correlations in finitely-connected planar domains, expressing those in terms of correlations of the compactified Gaussian free field. This, in particular, yields explicit expressions for the Ising correlations in terms of domain's period matrix, Green's function, harmonic measures of boundary components and arcs, or alternatively, Abelian differentials on the Schottky double. Our proof is based on a limiting version of a classical identity due to D.~Hejhal and J.~Fay relating Szeg\H{o} kernels and Abelian differentials on Riemann surfaces, and a systematic use of operator product expansions both for the Ising and the bosonic correlations.

math-ph↗

arXiv2022

Multiplicative chaos measures from thick points of log-correlated fields

We prove that multiplicative chaos measures can be constructed from extreme level sets or thick points of the underlying logarithmically correlated field. We develop a method which covers the whole subcritical phase and only requires asymptotics of suitable exponential moments for the field. As an application, we establish these estimates hold for the logarithm of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix (CUE), using known asymptotics for Toeplitz determinant with (merging) Fisher-Hartwig singularities. Hence, this proves a conjecture of Fyodorov and Keating concerning the fluctuations of the volume of thick points of the CUE characteristic polynomial.

math.PR↗

arXiv2021

Asymptotics of Hankel determinants with a multi-cut regular potential and Fisher-Hartwig singularities

We obtain large $N$ asymptotics for $N \times N$ Hankel determinants corresponding to non-negative symbols with Fisher-Hartwig (FH) singularities in the multi-cut regime. Our result includes the explicit computation of the multiplicative constant. More precisely, we consider symbols of the form $\omega e^{f-NV}$, where $V$ is a real-analytic potential whose equilibrium measure $\mu_V$ is supported on several intervals, $f$ is analytic in a neighborhood of $\textrm{supp}(\mu_V)$, and $\omega$ is a function with any number of jump- and root-type singularities in the interior of $\textrm{supp}(\mu_V)$. While the special cases $\omega\equiv1$ and $\omega e^f\equiv1$ have been considered previously in the literature, we also prove new results for these special cases. No prior asymptotics were available in the literature for symbols with FH singularities in the multi-cut setting. As an application of our results, we discuss a connection between the spectral fluctuations of random Hermitian matrices in the multi-cut regime and the Gaussian free field on the Riemann surface associated to $\mu_V$. As a second application, we obtain new rigidity estimates for random Hermitian matrices in the multi-cut regime.

math-ph↗

arXiv2020

The Coleman correspondence at the free fermion point

We prove that the truncated correlation functions of the charge and gradient fields associated with the massless sine-Gordon model on $\mathbb{R}^2$ with $\beta=4\pi$ exist for all coupling constants and are equal to those of the chiral densities and vector current of free massive Dirac fermions. This is an instance of Coleman's prediction that the massless sine-Gordon model and the massive Thirring model are equivalent (in the above sense of correlation functions). Our main novelty is that we prove this correspondence starting from the Euclidean path integral in the non-perturative regime of the infinite volume models. We use this correspondence to show that the correlation functions of the massless sine-Gordon model with $\beta=4\pi$ decay exponentially and that the corresponding probabilistic field is localized.

math-ph↗

arXiv2019

Discretization of the maximum for the derivatives of a random model of $\log|\zeta|$ on the critical line

In this short note, we study the derivatives of all orders for the random field $$ X_T(h) = \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], $$ where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. We show that the maximum of $X_{T}$, and more generally the maximum of its $j$-th derivative, varies on a $(\log T)^{-\frac{1}{2}(j+2)}$ scale, which improves and extends the main result in Arguin & Ouimet (2019) and makes further progress towards the open problem of the tightness of the recentered maximum of $X_{T}$. Our proof is also much simpler and shorter.

math.PR↗

arXiv2019

How much can the eigenvalues of a random Hermitian matrix fluctuate?

The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -- or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices -- the Gaussian Unitary Ensemble being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher-Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.

math.PR↗

arXiv2018

Decompositions of log-correlated fields with applications

In this article we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a H\"older continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields whose covariance kernel has a logarithmic singularity on the diagonal -- or log-correlated Gaussian fields. The theorem states that any log-correlated Gaussian field $X$ can be decomposed locally into a sum of a H\"older continuous function and an independent almost $\star$-scale invariant field (a special class of stationary log-correlated fields with 'cone-like' white noise representations). This decomposition holds whenever the term $g$ in the covariance kernel $C_X(x,y)=\log(1/|x-y|)+g(x,y)$ has locally $H^{d+\varepsilon}$ Sobolev smoothness. We use these decompositions to extend several results that have been known basically only for $\star$-scale invariant fields to general log-correlated fields. These include the existence of critical multiplicative chaos, analytic continuation of the subcritical chaos in the so-called inverse temperature parameter $\beta$, as well as generalised Onsager-type covariance inequalities which play a role in the study of imaginary multiplicative chaos.

math.PR↗

arXiv2018

Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model

In this article we study imaginary Gaussian multiplicative chaos -- namely a family of random generalized functions which can formally be written as $e^{i X(x)}$, where $X$ is a log-correlated real-valued Gaussian field on $\mathbb{R}^d$, i.e. it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions do these objects live in, along with their basic stochastic properties, such as moment and tail estimates. After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the so called critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, i.e. the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution. The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.

math.PR↗

arXiv2018

Multiplicative chaos and the characteristic polynomial of the CUE: the $L^1$-phase

In this note we prove that suitable positive powers of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix converge in law, as the size of the matrix tends to infinity, to a Gaussian multiplicative chaos measure once correctly normalized. We prove this in the whole $L^1$- or subcritical phase of the chaos measure.

math.PR↗

arXiv2017

Quantitative Normal Approximation of Linear Statistics of $\beta$-ensembles

We present a new approach, inspired by Stein's method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions. The method relies on a general normal approximation result of independent interest which is valid for a large class of Gibbs-type distributions. In the context of $\beta$-ensembles, this leads to a multi-dimensional CLT for a sequence of linear statistics which are approximate eigenfunctions of the infinitesimal generator of Dyson Brownian motion once the various error terms are controlled using the rigidity results of Bourgade, Erd\H{o}s and Yau.

math.PR↗

arXiv2017

On the moments of the characteristic polynomial of a Ginibre random matrix

In this article we study the large $N$ asymptotics of complex moments of the absolute value of the characteristic polynomial of a $N\times N$ complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large $N$ asymptotics of $\mathbb{E}|\det(G_N-z)|^{\gamma}$, where $G_N$ is a $N\times N$ matrix whose entries are i.i.d and distributed as $N^{-1/2}Z$, $Z$ being a standard complex Gaussian, $\mathrm{Re}(\gamma)>-2$, and $|z|<1$. This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher-Hartwig type of singularity: $\det(\int_\mathbb{C} w^{i}\overline{w}^j |w-z|^\gamma e^{-N|w|^{2}}d^2 w)_{i,j=0}^{N-1}$. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight $|w-z|^\gamma e^{-N|w|^2}d^2 w$ along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher-Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two-dimensional set and a Fisher-Hartwig singularity, have been previously unknown.

math.PR↗

arXiv2017

Random Hermitian Matrices and Gaussian Multiplicative Chaos

We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called $L^2$-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher-Hartwig singularities. Using Riemann-Hilbert methods, we prove a rather general Fisher-Hartwig formula for one-cut regular unitary invariant ensembles.

math.PR↗

arXiv2016

The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line

We prove that if $\omega$ is uniformly distributed on $[0,1]$, then as $T\to\infty$, $t\mapsto \zeta(i\omega T+it+1/2)$ converges to a non-trivial random generalized function, which in turn is identified as a product of a very well behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let $\delta_T$ approach zero slowly enough as $T\to\infty$, then $t\mapsto \zeta(1/2+i\delta_T t+i\omega T)$ is asymptotically a product of a divergent scalar quantity suggested by Selberg's central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This essentially says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.

math.PR↗

arXiv2016

Multiplicative chaos measures for a random model of the Riemann zeta function

We prove convergence of a stochastic approximation of powers of the Riemann $\zeta$ function to a non-Gaussian multiplicative chaos measure, and prove that this measure is a non-trivial multifractal random measure. The results cover both the subcritical and critical chaos. A basic ingredient of the proof is a 'good' Gaussian approximation of the induced random fields that is potentially of independent interest.

math.PR↗