@unpublished{64816,
abstract = {{We study a block mean-field Ising model with $N$ spins split into $s_N$ blocks, with Curie-Weiss interaction within blocks and nearest-neighbor coupling between blocks. While previous models deal with the block magnetization for a fixed number of blocks, we study the the simultaneous limit $N\to\infty$ and $s_N\to\infty$. The model interpolates between Curie-Weiss model for $s_N=1$, multi-species mean field for fixed $s_N=s$, and the 1D Ising model for each spin in its own block at $s_N=N$.
Under mild growth conditions on $s_N$, we prove a law of large numbers and a multivariate CLT with covariance given by the lattice Green's function. For instance, the high temperature CLT essentially covers the optimal range up to $s_N=o(N/(\log N)^c)$ and the low temperature regime is new even for fixed number of blocks $s > 2$. In addition to the standard competition between entropy and energy, a new obstacle in the proofs is a curse of dimensionality as $s_N \to \infty$.}},
author = {{Jalowy, Jonas and Lammers, Isabel and Löwe, Matthias}},
booktitle = {{arXiv:2603.01994}},
title = {{{The infinite block spin Ising model}}},
year = {{2026}},
}
@unpublished{59664,
abstract = {{Given a sequence of polynomials $(P_n)_{n \in \mathbb{N}}$ with only
nonpositive zeros, the aim of this article is to present a user-friendly
approach for determining the limiting zero distribution of $P_n$ as
$\mathrm{deg}\, P_n \to \infty$. The method is based on establishing an
equivalence between the existence of a limiting empirical zero distribution
$\mu$ and the existence of an exponential profile $g$ associated with the
coefficients of the polynomials $(P_n)_{n \in \mathbb{N}}$. The exponential
profile $g$, which can be roughly described by $[z^k]P_n(z) \approx \exp(n
g(k/n))$, offers a direct route to computing the Cauchy transform $G$ of $\mu$:
the functions $t \mapsto tG(t)$ and $\alpha \mapsto \exp(-g'(\alpha))$ are
mutual inverses. This relationship, in various forms, has previously appeared
in the literature, most notably in the paper [Van Assche, Fano and Ortolani,
SIAM J. Math. Anal., 1987].
As a first contribution, we present a self-contained probabilistic proof of
this equivalence by representing the polynomials as generating functions of
sums of independent Bernoulli random variables. This probabilistic framework
naturally lends itself to tools from large deviation theory, such as the
exponential change of measure. The resulting theorems generalize and unify a
range of previously known results, which were traditionally established through
analytic or combinatorial methods.
Secondly, using the profile-based approach, we investigate how the
exponential profile and the limiting zero distribution behave under certain
operations on polynomials, including finite free convolutions, Hadamard
products, and repeated differentiation. In particular, our approach yields new
proofs of the convergence results `$\boxplus_n \to \boxplus$' and `$\boxtimes_n
\to \boxtimes$', extending them to cases where the distributions are not
necessarily compactly supported.}},
author = {{Jalowy, Jonas and Kabluchko, Zakhar and Marynych, Alexander}},
booktitle = {{arXiv:2504.11593}},
title = {{{Zeros and exponential profiles of polynomials I: Limit distributions, finite free convolutions and repeated differentiation}}},
year = {{2025}},
}
@article{59213,
abstract = {{Abstract
We compare a mean-field Gibbs distribution on a finite state space on N spins to that of an explicit simple mixture of product measures. This illustrates the situation beyond the so-called increasing propagation of chaos introduced by Ben Arous and Zeitouni [3], where marginal distributions of size
$$k=o(N)$$
k
=
o
(
N
)
are compared to product measures.}},
author = {{Jalowy, Jonas and Kabluchko, Zakhar and Löwe, Matthias}},
issn = {{1385-0172}},
journal = {{Mathematical Physics, Analysis and Geometry}},
number = {{1}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Propagation of Chaos and Residual Dependence in Gibbs Measures on Finite Sets}}},
doi = {{10.1007/s11040-025-09503-5}},
volume = {{28}},
year = {{2025}},
}
@article{59665,
author = {{Erbar, Matthias and Huesmann, Martin and Jalowy, Jonas and Müller, Bastian}},
issn = {{0022-1236}},
journal = {{Journal of Functional Analysis}},
number = {{4}},
publisher = {{Elsevier BV}},
title = {{{Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy}}},
doi = {{10.1016/j.jfa.2025.110974}},
volume = {{289}},
year = {{2025}},
}
@unpublished{60293,
abstract = {{In this work, we present a complete characterization of the covariance
structure of number statistics in boxes for hyperuniform point processes. Under
a standard integrability assumption, the covariance depends solely on the
overlap of the faces of the box. Beyond this assumption, a novel interpolating
covariance structure emerges. This enables us to identify a limiting Gaussian
'coarse-grained' process, counting the number of points in large boxes as a
function of the box position. Depending on the integrability assumption, this
process may be continuous or discontinuous, e.g. in d=1 it is given by an
increment process of a fractional Brownian motion.}},
author = {{Jalowy, Jonas and Stange, Hanna}},
booktitle = {{arXiv:2506.13661}},
title = {{{Box-Covariances of Hyperuniform Point Processes}}},
year = {{2025}},
}
@unpublished{63394,
abstract = {{We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are well understood, but the probabilities of rare events remain largely unexplored. Large deviation type results have been obtained only in extreme cases, when either a vanishingly small proportion of eigenvalues are real or almost all eigenvalues are real. Here, in both the strong and weak asymmetry regimes, we derive the probabilities of rare events in the moderate-to-large deviation regime, thereby providing a natural connection between the previously known regime of Gaussian fluctuations and the large deviation regime. Our results are new even for the classical real Ginibre ensemble.}},
author = {{Byun, Sung-Soo and Jalowy, Jonas and Lee, Yong-Woo and Schehr, Grégory}},
booktitle = {{arXiv:2511.09191}},
title = {{{Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices}}},
year = {{2025}},
}
@unpublished{63393,
abstract = {{We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane as $n$ tends to infinity. We describe this limit distribution $μ_t$ as a function of the time parameter $t$ of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution $μ_t$ satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for $μ_t$ is available.}},
author = {{Höfert, Antonia and Jalowy, Jonas and Kabluchko, Zakhar}},
booktitle = {{arXiv:2512.17808}},
title = {{{Zeros of polynomial powers under the heat flow}}},
year = {{2025}},
}
@article{59188,
author = {{Jalowy, Jonas and Kabluchko, Zakhar and Löwe, Matthias and Marynych, Alexander}},
issn = {{1083-6489}},
journal = {{Electronic Journal of Probability}},
number = {{none}},
publisher = {{Institute of Mathematical Statistics}},
title = {{{When does the chaos in the Curie-Weiss model stop to propagate?}}},
doi = {{10.1214/23-ejp1039}},
volume = {{28}},
year = {{2023}},
}
@unpublished{59209,
abstract = {{We start with a random polynomial $P^{N}$ of degree $N$ with independent
coefficients and consider a new polynomial $P_{t}^{N}$ obtained by repeated
applications of a fraction differential operator of the form $z^{a}%
(d/dz)^{b},$ where $a$ and $b$ are real numbers. When $b>0,$ we compute the
limiting root distribution $\mu_{t}$ of $P_{t}^{N}$ as $N\rightarrow\infty.$ We
show that $\mu_{t}$ is the push-forward of the limiting root distribution of
$P^{N}$ under a transport map $T_{t}$. The map $T_{t}$ is defined by flowing
along the characteristic curves of the PDE satisfied by the log potential of
$\mu_{t}.$ In the special case of repeated differentiation, our results may be
interpreted as saying that the roots evolve radially \textit{with constant
speed} until they hit the origin, at which point, they cease to exist. For
general $a$ and $b,$ the transport map $T_{t}$ has a free probability
interpretation as multiplication of an $R$-diagonal operator by an $R$-diagonal
"transport operator." As an application, we obtain a push-forward
characterization of the free self-convolution semigroup $\oplus$ of radial
measures on $\mathbb{C}$.
We also consider the case $b<0,$ which includes the case of repeated
integration. More complicated behavior of the roots can occur in this case.}},
author = {{Hall, Brian C. and Ho, Ching-Wei and Jalowy, Jonas and Kabluchko, Zakhar}},
booktitle = {{arXiv:2312.14883}},
title = {{{Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators}}},
year = {{2023}},
}
@unpublished{59187,
abstract = {{We investigate the evolution of the empirical distribution of the complex
roots of high-degree random polynomials, when the polynomial undergoes the heat
flow. In one prominent example of Weyl polynomials, the limiting zero
distribution evolves from the circular law into the elliptic law until it
collapses to the Wigner semicircle law, as was recently conjectured for
characteristic polynomials of random matrices by Hall and Ho, 2022. Moreover,
for a general family of random polynomials with independent coefficients and
isotropic limiting distribution of zeros, we determine the zero distribution of
the heat-evolved polynomials in terms of its logarithmic potential.
Furthermore, we explicitly identify two critical time thresholds, at which
singularities develop and at which the limiting distribution collapses to the
semicircle law. We completely characterize the limiting root distribution of
the heat-evolved polynomials before singularities develop as the push-forward
of the initial distribution under a transport map. Finally, we discuss the
results from the perspectives of partial differential equations (in particular
Hamilton-Jacobi equation and Burgers' equation), optimal transport, and free
probability. The theory is accompanied by explicit examples, simulations, and
conjectures.}},
author = {{Hall, Brian C. and Ho, Ching-Wei and Jalowy, Jonas and Kabluchko, Zakhar}},
booktitle = {{arXiv:2308.11685}},
title = {{{Zeros of random polynomials undergoing the heat flow}}},
year = {{2023}},
}
@unpublished{59211,
abstract = {{We develop a theory of optimal transport for stationary random measures with
a focus on stationary point processes and construct a family of distances on
the set of stationary random measures. These induce a natural notion of
interpolation between two stationary random measures along a shortest curve
connecting them. In the setting of stationary point processes we leverage this
transport distance to give a geometric interpretation for the evolution of
infinite particle systems with stationary distribution. Namely, we characterise
the evolution of infinitely many Brownian motions as the gradient flow of the
specific relative entropy w.r.t.~the Poisson point process. Further, we
establish displacement convexity of the specific relative entropy along optimal
interpolations of point processes and establish an stationary analogue of the
HWI inequality, relating specific entropy, transport distance, and a specific
relative Fisher information.}},
author = {{Erbar, Matthias and Huesmann, Martin and Jalowy, Jonas and Müller, Bastian}},
booktitle = {{arXiv:2304.11145}},
title = {{{Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy}}},
year = {{2023}},
}
@unpublished{59189,
abstract = {{We develop a theory of optimal transport for stationary random measures with
a focus on stationary point processes and construct a family of distances on
the set of stationary random measures. These induce a natural notion of
interpolation between two stationary random measures along a shortest curve
connecting them. In the setting of stationary point processes we leverage this
transport distance to give a geometric interpretation for the evolution of
infinite particle systems with stationary distribution. Namely, we characterise
the evolution of infinitely many Brownian motions as the gradient flow of the
specific relative entropy w.r.t.~the Poisson point process. Further, we
establish displacement convexity of the specific relative entropy along optimal
interpolations of point processes and establish an stationary analogue of the
HWI inequality, relating specific entropy, transport distance, and a specific
relative Fisher information.}},
author = {{Erbar, Matthias and Huesmann, Martin and Jalowy, Jonas and Müller, Bastian}},
booktitle = {{arXiv:2304.11145}},
title = {{{Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy}}},
year = {{2023}},
}
@article{59184,
author = {{Jalowy, Jonas}},
issn = {{0246-0203}},
journal = {{Annales de l'Institut Henri Poincaré, Probabilités et Statistiques}},
number = {{4}},
publisher = {{Institute of Mathematical Statistics}},
title = {{{The Wasserstein distance to the circular law}}},
doi = {{10.1214/22-aihp1317}},
volume = {{59}},
year = {{2023}},
}
@unpublished{59210,
abstract = {{We establish basic properties of the heat flow on entire holomorphic
functions that have order at most 2. We then look specifically at the action of
the heat flow on the Gaussian analytic function (GAF). We show that applying
the heat flow to a GAF and then rescaling and multiplying by an exponential of
a quadratic function gives another GAF. It follows that the zeros of the GAF
are invariant in distribution under the heat flow, up to a simple rescaling.
We then show that the zeros of the GAF evolve under the heat flow
approximately along straight lines, with an error whose distribution is
independent of the starting point. Finally, we connect the heat flow on the GAF
to the metaplectic representation of the double cover of the group
$SL(2;\mathbb{R}).$}},
author = {{Hall, Brian and Ho, Ching-Wei and Jalowy, Jonas and Kabluchko, Zakhar}},
booktitle = {{arXiv:2304.06665}},
title = {{{The heat flow, GAF, and SL(2;R)}}},
year = {{2023}},
}
@article{59186,
abstract = {{AbstractIn this note we study the block spin mean-field Potts model, in which the spins are divided into s blocks and can take $$q\ge 2$$
q
≥
2
different values (colors). Each block is allowed to contain a different proportion of vertices and behaves itself like a mean-field Ising/Potts model which also interacts with other blocks according to different temperatures. Of particular interest is the behavior of the magnetization, which counts the number of colors appearing in the distinct blocks. We prove central limit theorems for the magnetization in the generalized high-temperature regime and provide a moderate deviation principle for its fluctuations on lower scalings. More precisely, the magnetization concentrates around the uniform vector of all colors with an explicit, but singular, Gaussian distribution. In order to remove the singular component, we will also consider a rotated magnetization, which enables us to compare our results to various related models.}},
author = {{Jalowy, Jonas and Löwe, Matthias and Sambale, Holger}},
issn = {{0022-4715}},
journal = {{Journal of Statistical Physics}},
number = {{1}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Fluctuations of the Magnetization in the Block Potts Model}}},
doi = {{10.1007/s10955-022-02889-4}},
volume = {{187}},
year = {{2022}},
}
@article{59185,
author = {{Jalowy, Jonas}},
issn = {{1083-6489}},
journal = {{Electronic Journal of Probability}},
number = {{none}},
publisher = {{Institute of Mathematical Statistics}},
title = {{{Rate of convergence for products of independent non-Hermitian random matrices}}},
doi = {{10.1214/21-ejp625}},
volume = {{26}},
year = {{2021}},
}
@article{59214,
author = {{Jalowy, Jonas and Löwe, Matthias}},
issn = {{1083-589X}},
journal = {{Electronic Communications in Probability}},
number = {{none}},
publisher = {{Institute of Mathematical Statistics}},
title = {{{Reconstructing a recurrent random environment from a single trajectory of a Random Walk in Random Environment with errors}}},
doi = {{10.1214/21-ecp425}},
volume = {{26}},
year = {{2021}},
}
@article{59183,
abstract = {{ The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by [Formula: see text]. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials. }},
author = {{Götze, Friedrich and Jalowy, Jonas}},
issn = {{2010-3263}},
journal = {{Random Matrices: Theory and Applications}},
number = {{03}},
publisher = {{World Scientific Pub Co Pte Lt}},
title = {{{Rate of convergence to the Circular Law via smoothing inequalities for log-potentials}}},
doi = {{10.1142/s201032632150026x}},
volume = {{10}},
year = {{2020}},
}