arXiv subjects

Nikos Zygouras

Publications and source records attributed to Nikos Zygouras.

At least 19 recordsLinked to original sources

arXiv2026

Fractional moments of the Stochastic Heat Flow and 2D Directed Polymers

We estimate the fractional moments of the normalized mass assigned by the Critical 2D Stochastic Heat Flow to small balls. Our results also cover the discrete case corresponding to the 2D directed polymer model and provide estimates that are uniform in all parameters. One key takeaway of our results is that the vanishing of the fractional moments is completely governed by the divergence of the second moment. We use a quite robust method, by refining the change of measure argument and introducing a novel coarse-graining procedure, reducing the proof to essentially second moment estimates (in fact, we also provide sharp second moment estimates for directed polymers, of independent interest).

math.PR↗

arXiv2026

The Critical 2d Stochastic Heat Flow and Related Models

In these lecture notes, we review recent progress in the study of the stochastic heat equation and its discrete analogue, the directed polymer model, in spatial dimension 2. It was discovered that a phase transition emerges on an intermediate disorder scale, with Edwards-Wilkinson (Gaussian) fluctuations in the sub-critical regime. In the critical window, a unique scaling limit has been identified and named the critical 2d stochastic heat flow. This gives a meaning to the solution of the stochastic heat equation in the critical dimension 2, which lies beyond existing solution theories for singular SPDEs. We outline the proof ideas, introduce the key ingredients, and discuss related literature on disordered systems and singular SPDEs. A list of open questions is also provided.

math.PR↗

arXiv2026

On the moments of the mass of shrinking balls under the Critical $2d$ Stochastic Heat Flow

The Critical $2d$ Stochastic Heat Flow (SHF) is a measure valued stochastic process on $\mathbb{R}^2$ that defines a non-trivial solution to the two-dimensional stochastic heat equation with multiplicative space-time noise. Its one-time marginals are a.s. singular with respect to the Lebesgue measure, meaning that the mass they assign to shrinking balls decays to zero faster than their Lebesgue volume. In this work we explore the intermittency properties of the Critical 2d SHF by studying the asymptotics of the $h$-th moment of the mass that it assigns to shrinking balls of radius $ε$ and we determine that its ratio to the Lebesgue volume is of order $(\log\tfrac{1}ε)^{h\choose 2}$ up to possible lower order corrections.

math.PR↗

arXiv2026

From disordered systems to the Critical 2D Stochastic Heat Flow

We review our joint work on the scaling limits of disordered systems, linking the notion of disorder relevance/irrelevance to that of sub/super-criticality of singular SPDEs. This line of research culminated in the construction of the Critical 2D Stochastic Heat Flow (SHF), a universal process which provides a non-trivial solution to the Stochastic Heat Equation in dimension 2, a critical singular SPDE that lies beyond the reach of existing solution theories. The SHF also offers a rare example of a non-Gaussian scaling limit for a disordered system at its phase transition point in the critical dimension.

math.PR↗

arXiv2025

Singularity and regularity of the critical 2D Stochastic Heat Flow

The Critical 2D Stochastic Heat Flow (SHF) provides a natural candidate solution to the ill-posed 2D Stochastic Heat Equation with multiplicative space-time white noise. In this paper, we initiate the investigation of the spatial properties of the SHF. We prove that, as a random measure on $\mathbb{R}^2$, it is a.s. singular w.r.t. the Lebesgue measure. This is obtained by probing a "quasi-critical" regime and showing the asymptotic log-normality of the mass assigned to vanishing balls, as the disorder strength is sent to zero at a suitable rate, accompanied by similar results for critical 2D directed polymers. We also describe the regularity of the SHF, showing that it is a.s. Hölder $C^{-ε}$ for any $ε>0$, implying the absence of atoms, and we establish local convergence to zero in the long time limit.

math.PR↗

arXiv2024

Directed polymers in a random environment: a review of the phase transitions

The model of directed polymer in a random environment is a fundamental model of interaction between a simple random walk and ambient disorder. This interaction gives rise to complex phenomena and transitions from a central limit theory to novel statistical behaviours. Despite its intense study, there are still many aspects and phases which have not yet been identified. In this review we focus on the current status of our understanding of the transition between weak and strong disorder phases, give an account of some of the methods that the study of the model has motivated and highlight some open questions.

math.PR↗

arXiv2024

A multivariate extension of the Erdös-Taylor theorem

The Erdös-Taylor theorem [Acta Math. Acad. Sci. Hungar, 1960] states that if $\mathsf{L}_N$ is the local time at zero, up to time $2N$, of a two-dimensional simple, symmetric random walk, then $\tfracπ{\log N} \,\mathsf{L}_N$ converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if $\mathsf{L}_N^{(1,2)}=\sum_{n=1}^N 1_{\{S_n^{(1)}= S_n^{(2)}\}}$, then $\tfracπ{\log N}\, \mathsf{L}^{(1,2)}_N$ converges in distribution to an exponential random variable of parameter one. We prove that for every $h \geq 3$, the family $ \big\{ \fracπ{\log N} \,\mathsf{L}_N^{(i,j)} \big\}_{1\leq i<j\leq h}$, of logarithmically rescaled, two-body collision local times between $h$ independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdös-Taylor theorem. We also discuss connections to directed polymers in random environments.

math.PR↗

arXiv2023

The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos

The critical $2d$ Stochastic Heat Flow (SHF) is a stochastic process of random measures on ${\mathbb R}^2$, recently constructed in [CSZ23]. We show that this process falls outside the class of Gaussian Multiplicative Chaos (GMC), in the sense that it cannot be realised as the exponential of a (generalised) Gaussian field. We achieve this by deriving strict lower bounds on the moments of the SHF that are of independent interest.

math.PR↗

arXiv2023

The Allen-Cahn equation with weakly critical random initial datum

This work considers the two-dimensional Allen-Cahn equation $$ \partial_t u = \frac{1}{2}Δu + \mathfrak{m}\, u -u^3\;, \quad u(0,x)= η(x)\;, \qquad \forall (t,x) \in [0, \infty) \times \mathbb{R}^{2} \;, $$ where the initial condition $ η$ is a two-dimensional white noise, which lies in the scaling critical space of initial data to the equation. In a weak coupling scaling, we establish a Gaussian limit with nontrivial size of fluctuations, thus casting the nonlinearity as marginally relevant. The result builds on a precise analysis of the Wild expansion of the solution and an understanding of the underlying stochastic and combinatorial structure. This gives rise to a representation for the limiting variance in terms of Butcher series associated to the solution of an ordinary differential equation.

math.PR↗

arXiv2023

Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point

We consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters. We use the combinatorics of the Robinson-Schensted-Knuth correspondence and certain intertwining relations to express the transition kernel of this interacting particle system in terms of ensembles of weighted, non-intersecting lattice paths and, consequently, as a marginal of a determinantal point process. We next express the joint distribution of the particle positions as a Fredholm determinant, whose correlation kernel is given in terms of a boundary-value problem for a discrete heat equation. The solution to such a problem finally leads us to a representation of the correlation kernel in terms of random walk hitting probabilities, generalising the formulation of Matetski, Quastel and Remenik (Acta Math., 2021) to the case of both particle- and time-inhomogeneous rates. The solution to the boundary value problem in the fully inhomogeneous case appears with a finer structure than in the homogeneous case.

math.PR↗

arXiv2023

The Critical 2d Stochastic Heat Flow

We consider directed polymers in random environment in the critical dimension $d = 2$, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on $\mathbb{R}^2$ with logarithmic correlations, which we call the *Critical 2d Stochastic Heat Flow*. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise.

math.PR↗

arXiv2022

Some algebraic structures in KPZ universality

We review some algebraic and combinatorial structures that underlie models in the KPZ universality class.Emphasis is placed on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov. We present how these combinatorial constructions are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions, Young tableaux and Gelfand-Tsetlin patterns. We also present how fundamental representation theoretic concepts, such as the Cauchy identity, the Pieri rule and the branching rule, can be used, alongside RSK correspondences, and can be combined with probabilistic ideas, in order to construct integrable stochastic dynamics on two dimensional arrays of Gelfand-Tsetlin type, in ways that couple different one dimensional stochastic processes. For example, interacting particle systems, on the one hand, and processes related to eigenvalues of random matrices, on the other, thus illuminating the emergence of random matrix distributions in interacting stochastic processes. The goal of the notes is to expose some of the overarching principles, which have driven a significant number of developments in the field.

math.PR↗

arXiv2022

Moments of the 2d directed polymer in the subcritical regime and a generalisation of the Erdös-Taylor theorem

We compute the limit of the moments of the partition function $Z_{N}^{β_N} $ of the directed polymer in dimension $d=2$ in the subcritical regime, i.e. when the inverse temperature is scaled as $β_N \sim \hatβ \sqrt{\tfracπ{\log N}}$ for $\hatβ \in (0,1)$. In particular, we establish that for every $h \in \mathbb{R}$, $\lim_{N \to \infty } \mathbb{E}\big[\big(Z_{N}^{β_N}\big)^h\big]=\big(\frac{1}{1-\hatβ^2}\big)^{\frac{h(h-1)}{2}}.$ We also identify the limit of the moments of the averaged field $\tfrac{\sqrt{\log N}}{N} \sum_{x \in \mathbb{Z}^2} φ(\tfrac{x}{\sqrt{N}})\big(Z_{N}^{β_N}(x)-1 \big )$, for $φ\in C_c(\mathbb{R}^2)$, as those of a gaussian free field. As a byproduct, we identify the limiting probability distribution of the total pairwise collisions between $h$ independent, two dimensional random walks starting at the origin. In particular, we derive that $$ \fracπ{\log N}\sum_{1 \leq i<j\leq h} \mathsf{L}_N^{(i,j)}\xrightarrow[N \to \infty ]{(d)} Γ\big( \tfrac{h(h-1)}{2},1\big) \, ,$$ where ${\mathsf{L}}^{(i,j)}_N$ denotes the collision local time by time $N$ between copies $i,j$ and $Γ$ denotes the Gamma distribution. This generalises a classical result of Erdös-Taylor.

math.PR↗

arXiv2022

Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation

We study the combinatorial structure of the irreducible characters of the classical groups ${\rm GL}_n(\mathbb{C})$, ${\rm SO}_{2n+1}(\mathbb{C})$, ${\rm Sp}_{2n}(\mathbb{C})$, ${\rm SO}_{2n}(\mathbb{C})$ and the "non-classical" odd symplectic group ${\rm Sp}_{2n+1}(\mathbb{C})$, finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of ${\rm Sp}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$ and between characters of ${\rm SO}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$. We identify the first family as a one-parameter specialization of Koornwinder polynomials, for which we thus provide a novel combinatorial structure; on the other hand, the second family appears to be new. We next develop a method of Gelfand-Tsetlin pattern decomposition to establish identities between all these polynomials that, in the case of irreducible characters, can be viewed as branching rules. Through these formulas we connect orthogonal and symplectic characters, and more generally the interpolating polynomials, to LPP models with various symmetries, thus going beyond the link with classical Schur polynomials originally found by Baik and Rains (Duke Math. J., 2001). Taking the scaling limit of the LPP models, we finally provide an explanation of why the Tracy-Widom GOE and GSE distributions from random matrix theory admit formulations in terms of both Fredholm determinants and Fredholm Pfaffians.

math.CO↗

arXiv2021

Edwards-Wilkinson fluctuations for the directed polymer in the full $L^2$-regime for dimensions $d \geq 3$

We prove that in the full $L^2$-regime the partition function of the directed polymer model in dimensions $d\geq 3$, if centered, scaled and averaged with respect to a test function $φ\in C_c(\mathbb{R}^d)$, converges in distribution to a Gaussian random variable with explicit variance. Introducing a new idea in this context of a martingale difference representation, we also prove that the log-partition function, which can be viewed as a discretisation of the KPZ equation, exhibits the same fluctuations, when centered and averaged with respect to a test function. Thus, the two models fall within the Edwards-Wilkinson universality class in the full $L^2$-regime, a result that was only established, so far, for a strict subset of this regime in $d\geq 3$.

math.PR↗

arXiv2020

The geometric Burge correspondence and the partition function of polymer replicas

We construct a geometric lifting of the Burge correspondence as a composition of local birational maps on generic Young-diagram-shaped arrays. We establish its fundamental relation to the geometric Robinson-Schensted-Knuth correspondence and to the geometric Schützenberger involution. We also show a number of properties of the geometric Burge correspondence, specializing them to the case of symmetric input arrays. In particular, our construction shows that such a mapping is volume preserving in log-log variables. As an application, we consider a model of two polymer paths of given length constrained to have the same endpoint, known as polymer replica. We prove that the distribution of the polymer replica partition function in a log-gamma random environment is a Whittaker measure, and deduce the corresponding Whittaker integral identity. For a certain choice of the parameters, we notice a distributional identity between our model and the symmetric log-gamma polymer studied by O'Connell, Seppäläinen, and Zygouras (2014).

math.PR↗

arXiv2019

The Dickman subordinator, renewal theorems, and disordered systems

We consider the so-called Dickman subordinator, whose Levy measure has density 1/x restricted to the interval (0,1). The marginal density of this process, known as the Dickman function, appears in many areas of mathematics, from number theory to combinatorics. In this paper, we study renewal processes in the domain of attraction of the Dickman subordinator, for which we prove local renewal theorems. We then present applications to marginally relevant disordered systems, such as pinning and directed polymer models, and prove sharp second moment estimates on their partition functions.

math.PR↗

arXiv2019

The two-dimensional KPZ equation in the entire subcritical regime

We consider the KPZ equation in space dimension 2 driven by space-time white noise. We showed in previous work that if the noise is mollified in space on scale $ε$ and its strength is scaled as $\hatβ/ \sqrt{|\log ε|}$, then a transition occurs with explicit critical point $\hatβ_c = \sqrt{2π}$. Recently Chatterjee and Dunlap showed that the solution admits subsequential scaling limits as $ε\downarrow 0$, for sufficiently small $\hatβ$. We prove here that the limit exists in the entire subcritical regime $\hatβ\in (0, \hatβ_c)$ and we identify it as the solution of an additive Stochastic Heat Equation, establishing so-called Edwards-Wilkinson fluctuations. The same result holds for the directed polymer model in random environment in space dimension 2.

math.PR↗