Abstracts

Mini-courses:

Ivan Corwin (Columbia University) – A tail of two times for the KPZ equation.

What begins as a story about the decay of correlations for the KPZ equation a two different times quickly morphs into a vignette on integrable probability. Featured in the drama are two main characters. The first is an identity between the KPZ equation and the Airy point process. The second is the KPZ line ensemble and its Brownian Gibbs property. At the end of the day, we reveal that these are really estranged siblings, both springing from the Yang-Baxter equation.

Grégory Miermont (ENS Lyon) – Brownian Surfaces

I will review the topic of random maps and its connection to classical objects of probability theory, namely random walks and peeling processes, and branching processes arising in slice decompositions and layer decompositions. The first lecture will focus on the “local” model of Boltzmann maps and will comment on its relation to “non-local” models of maps, and will introduce peeling processes and some applications. The second and third lectures will focus on slice decompositions, which is, from the geometric point of  view, a convenient way to understand the Bouttier-di Francesco-Guitter bijection between maps and trees. We will show how this decomposition can be used to show convergence of random maps to Brownian surfaces in various topologies. Time permitting, I will comment on the use by Curien-Le Gall of layer decompositions to show robustness of these results when modifying locally the distances, and how it is used by Carrance to show convergence of Eulerian triangulations to the Brownian map.

Ron Rosenthal (Technion, Haifa) – Random topological structures
We will review the use of probability theory for the study of random topological structures. In the first lecture we will discuss the notion of high-dimensional graphs and some of their topological properties. The second lecture will be devoted to a high-dimensional version of the ErdősRényi model and the topological phase transitions it exhibits. Finally, in the third talk we will discuss a model for high-dimensional random walks and how it can be used to study topological properties. 

Invited talks:

Guillaume Barraquand (ENS Paris, France) – Diffusions in random environment

Consider the simple random walk on Z. What happens if transition probabilities are themselves random variables ? Using an integrable model, a random walk with Beta distributed transition probabilities, we will see that the extreme value behavior of many random walks in the same environment is governed by  scalings and statistics that arise in random matrix theory and the the Kardar-Parisi-Zhang universality class. Then we will see that the relevant continuous limit of the model is a stochastic flow introduced by Le Jan and Raimond. Several diffusions following this stochastic flow behave as Brownian motions with a local attractive interaction called sticky Brownian motions. This talk is based on joint works with Ivan Corwin and Mark Rychnovsky.

Omer Bobrowski (Technion, Israel)  – Topological phase transitions in random geometric complexes

Connectivity and percolation are two well studied phenomena in random graphs. In this talk we will discuss higher-dimensional analogues of connectivity and percolation that occur in random simplicial complexes. Simplicial complexes are a natural generalization of graphs, that consist of vertices, edges, triangles, tetrahedra, and higher dimensional simplexes. We will mainly focus on random geometric complexes. These complexes are generated by taking the vertices to be a random point process, and adding simplexes according to their geometric configuration. Our generalized notions of connectivity and percolation use the language of homology – an algebraic-topological structure representing cycles of different dimensions. In this talk we will discuss recent results analyzing phase transitions related to these topological phenomena.

Christina Goldschmidt (Oxford, UK) – The scaling limit of a critical random directed graph

We consider the random directed graph D(n, p) with vertex set {1, 2, . . . , n} in which each of the n(n − 1) possible directed edges is present independently with probability p. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at p = 1/n, with critical window p = 1/n + \lambda n^{-4/3} for \lambda \in \R. We show that, within this critical window, the strongly connected components of D(n, p), ranked in decreasing order of size and rescaled by n^{-1/3}, converge in distribution to a sequence of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. This is joint work with Robin Stephenson (Oxford).

Kostya Khanin (U. of Toronto, Canada) – On stationary solutions to the stochastic heat equation  

We shall discuss the problem of uniqueness of global solutions to the
random Hamilton-Jacobi equation. We shall formulate several conjectures
and present results supporting them. Finally, we discuss a new uniqueness result

for the Stochastic Heat equation in the regime of weak disorder.

Sayan Mukherjee (Duke, US) – Minimal Spanning Acycles and Giant Shadows

A classic result by Frieze is the expected weight of the minimum
spanning tree (MST) of particular random graph models when properly
normalized is Zeta(3). This type of result has been extended from
random graph models to models on simplicial complexes, where the
random object studied is the minimum spanning acycle. I will state
some results on minimal spanning acycles (MSA) for a variant of the
Linial-Meshulam model. I will discuss distributional properties of the
weights included in the MSA, both in the bulk and in the extremes. A
key observation will be the relation between the weights of faces
included in the MSA and the idea of the giant shadow. The giant shadow
in the Linial-Meshulam model is the analog of a giant component in the
Erdos-Renyi model. Time permitting I will discuss an on-line minimal
spanning tree algorithm that these results characterize. Joint work with Nichols Fraiman (UNC, CH) and Gugan Thoppe (Indian
Institute of Science)

Oren Louidor (Technion) – A Scaling limit for the Cover Time of the Binary Tree

We consider a continuous time random walk on the rooted binary tree of depth $n$ with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by $2^{n+1} n$ and then centered by $(\log 2) n – \log n$, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a Gumbel random variable with rate one, shifted randomly by the logarithm of the sum of the limits of the derivative martingales associated with two negatively correlated discrete Gaussian free fields on the infinite version of the tree. The existence of the limit and its overall form were conjectured in the literature. Our approach is quite different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field on the tree. Joint work with Aser Cortines and Santiago Saglietti.

 

Short talks:

Alessandra Occelli Stationary half-space last passage percolation We study stationary last passage percolation in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. We derive our results using a related integrable model having Pfaffian structure together with analytic continuation and steepest descent analysis.
Amit Weintroub On the angles between eigenvectors in the Ginibre ensemble A random matrix is said to be sampled from the Ginibre ensemble if all of its entries are i.i.d., complex normal random variables with mean zero. In this work, we study the asymptotic behavior of the angles between pairs of eigenvectors of such matrices. In particular, we compute the limiting distribution of the angle for fixed pairs of eigenvalues, obtain precise bounds on the typical behavior of the maximal angle with high probability and find the limiting distribution for the location of the eigenvalues which attain the maximal angle with high probability. The talk will present the main results and demonstrate some of the techniques used to reach them.
Biltu Dan Scaling limit of semiflexible polymers: a phase transition We consider a semiflexible polymer in the d-dimensional integer lattice which is a random interface model with a mixed gradient and Laplacian interaction. The strength of the two operators is governed by two parameters called lateral tension and bending rigidity, which might depend on the size of the graph. In this talk we show a phase transition in the scaling limit according to the strength of these parameters: we prove that the scaling limit is, respectively, the Gaussian free field, a “mixed” random distribution and the continuum membrane model in three different regimes. This is based on joint work with Alessandra Cipriani (TU Delft) and Rajat Subhra Hazra (ISI Kolkata).
Carlo Bellingeri Transport equations with low regularity geometric rough noise We consider the transport equation driven by a geometric rough path of any Hölder regularity. This type of equation has been intensively studied in recent years, providing a comprehensive solution theory when the driver has only two terms. We provide an intrinsic characterisation of the strong solution obtained through rough characteristics. A key point to formulate this condition is the existence of a path-wise Itô formula in the context of the solution of a rough differential equation. Some possible applications of this dynamic are related to when the transport term is a fractional Brownian motion of Hurst parameter H>1/4. Joint work with P. Friz, N. Tapia (WIAS, TU Berlin) and A. Djurdjevac (TU Berlin).
Dan Betea From Gumbel to Tracy–Widom (via cylinders and finite temperature) We study edge scaling of the finite temperature Plancherel measure (and process), a measure on partitions interpolating between the poissonized Plancherel and the uniform measures. In the KPZ-like edge scaling limit, we obtain Johansson’s finite temperature Tracy–Widom distribution, itself interpolating between two well-known extremal statistics distributions: Tracy–Widom GUE (governing the maxima of correlated variables) and Gumbel (governing maxima of iid variables). This can be viewed as a generalization of both the Baik–Deift–Johansson theorem (concerning longest increasing subsequences of random permutations) and the Erdos–Lehner theorem. Towards the former, we also exhibit a directed polymer (last passage percolation) model on a cylinder. Its length—asymptotically and upon convolution with an independent Gumbel random variable—exhibits the aforementioned finite temperature Tracy–Widom fluctuations. This is joint work with Jérémie Bouttier.
Dan Mikulincer Stability of functional inequalities in Gauss space We will discuss how several known functional inequalities in Gauss space arise from general principles in stochastic analysis.
This point of view will give rise to a unified framework from which one may study the stability of those inequalities.
Several results in this direction will be presented with further application to concentration of measure and normal approximations.
Fabian Gerle On the convergence of symmetric Feller processes Donsker’s classical invariance principle is usually stated in terms of
conditions on the increments.Athreya L\”ohr and Winter (2017) introduced a change of perspective by
associating stochastic processes with metric measure spaces (trees). The authors
showed that Gromov-Hausdorff vague convergence of the metric measure spaces
implies weak convergence in path space of the associated processes.This concept was extended by Croydon (2018) to resistance forms.Both results are tailored to essentially one-dimensional state spaces where the
resistance between points is finite. This is not the case, for example, for
2-dimensional Brownian motion.We propose a new approach based on occupation time functionals of the form
\[
G_A f(x):= \mathbb{E}_x\left[\int_0^{\tau_A} f(X_t) \mathrm dx \right],
\]
where $A$ is a Borel-set of the state space and $\tau_A$ denotes the first
hitting time of $A$.
This approach is applicable for general symmetric Feller processes which are not
necessarily associated with a metric on the state space.This is an ongoing project with Anita Winter.
Florian Bechtold Law of large numbers for particle systems in a mild formulation Consider a system of weakly interacting particles perturbed by Brownian motion. In the theory of propagation of chaos it is well known that under suitable assumptions the associated empirical measure converges to the solution of a McKean-Vlasov PDE (which can be seen as a law of large numbers). We present an alternative proof to this statement by showing that the empirical measure associated to particle systems of fixed size $n$ satisfies a certain stochastic differential equation, which can be shown to converge to the corresponding deterministic McKean-Vlasov PDE as $n \rightarrow \infty$. Towards this end, we prove two distinct bounds of the noise term exploiting rough path theory for the first and martingale techniques for the second one. This is joint work in progress with Fabio Coppini (LPSM, Université de Paris).
Geronimo Rojas On the rate of convergence of diffusion approximation on compact trees. In this talk we introduce a new distance for stochastic processes taking values on compact metric measure trees based on hitting times. We show how this new metric can be bounded in terms of the Gromov-Prokhorov distance and argue how this yields rates of convergence in the f.d.d. sense.

We complete the metric by adding a functional that captures tightness. We discuss how this complete metric can be used to derive rates for weak convergence in path-space.

As an application, we use this distance to derive a rate for weak convergence in path-space for SRW to BM on compact intervals.

Helena Kremp Multidimensional SDE with distributional drift and Lévy noise We solve multidimensional SDEs with distributional drift of rather general regularity driven by a symmetric, α-stable Lévy process for α∈(1,2]. To this end, we introduce the notion of a (singular) martingale problem and solve the backward generator PDE in the case when the Besov regularity of the drift is below the critical threshold for the classical PDE theory of (1−α)/2. Therefore, we employ the theory of paracontrolled distributions estabished by Gubinelli, Imkeller and Perkowski to solve singular PDEs. We prove existence and uniqueness of a solution to the martingale problem associated to the Lévy SDE with Besov drift. Our work extends a recent result of Cannizzaro and Chouk from the Brownian to the stable Lévy noise case. If time permits, we apply our theory to construct the Brox diffusion with α-stable Lévy noise dXt =ξ(Xt)dt+dLt, X0=x∈ R, where ξ is periodic white noise independent of L and α>7/4.
Keywords: multidimensional SDE, Besov drift, stable Lévy noise, paracontrolled distributions, singular martingale problem
joint work with: Nicolas Perkowski
Henri Elad Altman A wetting model in the continuum In this talk I will introduce a continuous version of wetting model consisting of the law of a Brownian meander tilted by its local time at a positive level h, with h small. I will prove that this measure converges, as h tends to 0, to the same weak limit as in the discrete setting. I will also discuss the corresponding Langevin dynamics, which is expected to converge to a Bessel SPDE admitting the law of a reflecting Brownian motion as invariant measure. This is based on joint work with Jean-Dominique Deuschel and Tal Orenshtein.
Immanuel Zachhuber Strichartz estimates for the Anderson Hamiltonian After recalling some recent developments related to the continuum Anderson Hamiltonian in 2- and 3- dimensions I will present some new results that combine tools from singular SPDEs, namely Paracontrolled Distributions, with a method due to Burq/Gerard/Tzvetkov. This gives rise to Strichartz-type estimates which allow to solve multiplicative stochastic NLS in a low-regularity regime.
Josué Nussbaumer The alpha-Ford algebraic measure trees We are interested in the infinite limit of the alpha-Ford model, which is a family of random cladograms, interpolating between the coalescent tree (or Yule tree) and the branching tree (or uniform tree). For this, we use the notion of algebraic measure trees, which are trees without edge length and equipped with a sampling measure. In the space of algebraic measure trees, the limit of the alpha-Ford model is well defined. We then describe some statistics on the limit trees, allowing for tests of hypotheses on real world phylogenies.
Luis Enrique Osorio Puentes A two level branching model for virus populations In this talk we present a two-level branching model for virus populations under cell division. We assume that the cells are carrying a virus population which evolve as a branching particle system with competition, while the cells split according to a Yule process thereby dividing their virus populations into two independently evolving sub-populations.
Then we assume that sizes of the virus populations are huge and characterize the fast branching rate and huge population density limit as the solution of a well-posed martingale problem. Finally we provide a Feynman-Kac duality relation to conclude uniqueness. Moreover, the duality relation allows for a further study of the long term behavior of the model.
Luisa Andreis Phase transitions in inhomogeneous random graphs and coagulation processes. Sparse inhomogeneous random graphs are a natural generalization of the well-known Erdos-Reényi random graph (ERRG), where vertices are characterized by a type and edges are independent but distributed according to the types of the vertices that they are connecting. These graphs undergo a phase transition in terms of the emergence of a giant component, exactly as the classical ERRG. In this talk we will present an approach, based on large deviations, to prove the existence of this phase transition and we will relate it to the phase transition (usually called gelation) in coagulation processes. This is a joint work with W. König, H. Langhammer and R. Patterson.
Lukas Wresch An Introduction to path by path uniqueness for stochastic differential equations In this short talk I will introduce the notion of path-by-path solutions and path-by-path uniqueness of stochastic differential equations, which is a slightly stronger notion than the standard pathwise uniqueness.
I will present recent results in this field and techniques on how the theorem have been established.
Mario Antonio Ayala Valenzuela Higher order fluctuation fields and orthogonal duality polynomials Inspired by the works in [1] and [2] we introduce what we call $k$-th-order fluctuation fields and study their scaling limits. This construction is done in the context of particle systems with the property of orthogonal self-duality. This type of duality provides us with a setting in which we were able to interpret these fields as some type of discrete analogue of powers of the well-known density fluctuation field. We show that the weak limit of the $k$-th order field satisfies a recursive martingale problem that informally corresponds to the SPDE associated with the $k$th-power of a generalized Ornstein-Uhlenbeck process.

[1] Assing, S. A limit theorem for quadratic fluctuations in symmetric simple exclusion. Stochastic Processes and their Applications 117, 766–790 (2007).

[2] Gonçalves, P. & Jara, M. Quadratic fluctuations of the symmetric simple exclusion. Latin American Journal of Probability and Mathematical Statistics 16, 605 (2019).

Martin Friesen Ergodicity and regularity for affine processes An affine process is characterized by its log-characteristic function which depends in an affine manner on the initial state of the process. On the canonical state space $\mathbb{R}_+^m \times \mathbb{R}^n$, this notion includes Ornstein-Uhlenbeck as well as continuous-state branching processes with immigration and, moreover, unifies them in a general framework.
The study of invariant measures and ergodicity for affine processes has recently received much attention in the literature.
In this talk, we first address the existence and uniqueness of invariant measures and then, based on the coupling technique,
prove convergence of transition probabilities with exponential rate towards the unique invariant measure in the Wasserstein distance. Afterward, we discuss the regularity (in anisotropic Besov spaces) of corresponding heat kernels and deduce from that the strong Feller property. By a combination of this regularity results and the coupling technique, we finally also prove convergence in total variation distance for affine processes.This talk is based on several joint works with: P. Jin, J. Kremer, and B. Rüdiger.
Matan Shalev The Diameter of Uniform Spanning Trees in High Dimensions A uniform spanning tree of a finite connected graph G is a spanning tree T_G of G drawn uniformly at random. We show that the diameter of T_G is of order \sqrt{|G|} for a large class of “high-dimensional” graphs such as expander graphs and tori of dimension above 5. Joint work with Asaf Nachmias and Peleg Michaeli
Maximilian Fels Extreme value theory of the scale-inhomogeneous 2d discrete Gaussian free field The extremes of strongly correlated fields, such as branching Brownian motion (BBM), the 2d discrete Gaussian free field (DGFF), the maximum of the randomized Riemann zeta function on the critical line or cover times of Brownian motion on the torus, have received a lot of attention in the last couple of years. In the cases of branching Brownian motion or the 2d discrete Gaussian free field, there exists by now a detailed description of their maximum value and their extreme value statistics. We introduce a model derived from the 2d DGFF by introducing a scale-dependent variance. In the context of the 2d DGFF, it is the analogue model to the time-inhomogeneous branching random walk (BRW) or variable-speed BBM in their respective contexts of BRW or BBM. Moreover, we discuss our recent results on its maximum value and extremal process.
Oren Yakir Recovering the lattice from its random perturbations Given a d-dimensional Euclidean lattice, we consider the point process obtained by adding an independent Gaussian vector to each of the lattice points. In my talk, I will explain a simple procedure that recovers the underlying lattice from a single realization of this random point process.
Quan Shi Interval partition evolutions with Poisson-Dirichlet stationary distributions We introduce interval partition evolutions for $\alpha\in (0, 1)$ and $\theta \ge 0$, in which the total sums of interval lengths evolve as squared Bessel processes of dimension $2\theta$. These diffusions arise as continuum limits of up-down Markov chains on Chinese restaurant processes and have pseudo-stationary distributions related to regenerative Poisson-Dirichlet interval partitions.
This talk is based on joint works in progress with Noah Forman, Soumik Pal, Douglas Rizzolo, and Matthias Winkel.
Sergey Berezin Berry-Esseen type estimates for linear statistics of unitary matrix ensembles We are going to talk about the Berry-Esseen type estimates, which give an upper bound for the convergence rate in the central limit theorem for linear statistics of several canonical unitary matrix ensembles in a scaling limit. To control the Kolmogorov-Smirnov distance we obtain the uniform asymptotic for the characteristic function with growing argument via the Riemann-Hilbert approach.

The talk is based on a joint work with A. Bufetov.

Shu Kanazawa A limit theorem for Betti numbers of homogeneous and spatially independent random simplicial complexes The Erdős–Rényi graph model has been extensively studied since the 1960s as a typical random graph model. Recently, the study of random simplicial complexes has drawn attention as a higher-dimensional generalization of random graphs. In this talk we introduce the class of homogeneous and spatially independent random simplicial complexes, including Linial–Meshulam complexes and random clique complexes as special cases, and give a limit theorem for their Betti numbers. This result includes the limit theorem for Betti numbers of Linial–Meshulam complexes, proved by Linial and Peled. Furthermore, applying our main result to the random clique complex model, we show that the Betti number of a random clique complex is asymptotically unimodal with respect to the underlying parameter. The unimodality comes from the competitive relationship between the effect of creating holes and that of filling them by higher-dimensional simplices.
One of the key elements in the proof of our main result is the concept of the local weak convergence of simplicial complexes. This concept is a generalization of the Benjamini–Schramm convergence of graphs. Inspired by the significant work by Linial and Peled, we prove the local weak limit theorem for homogeneous and spatially independent random simplicial complexes.
Stefan Junk Large deviations for directed polymers in the whole weak disorder phase In the directed polymer model we study a random process affected by a space-time random environment. The process is known to satisfy a large deviation principle, but not much is known about its rate function. In weak disorder, the rate function is known to agree with the rate function of simple random walk only under a sub-optimal L^2-boundedness assumption. We show that if the discrete-time model is replaced by a natural continuous-time model, then this result holds without assuming L^2-boundedness. Joint work with Ryoki Fukushima.
Sukrit Chakraborty Largest eigenvalue of the adjacency matrix of non-sparse inhomogeneous Erd\H{o}s-R\’enyi random graphs We consider an inhomogeneous non-sparse Erd\H{o}s-R\’enyi random graph on $N$ vertices with edge probabilities $p_N f(i/N,j/N)$ between each of the vertices $i$ and $j$, where $f$ is non-negative, bounded, Riemann integrable and symmetric. We shall discuss the convergence in distribution of the suitably scaled and centered largest eigenvalue of the adjacency matrix to the normal distribution.
This is based on a joint work with Dr. Arijit Chakrabarty and Dr. Rajat Subhra Hazra.
Tal Orenshtein Aging for the stationary KPZ equation We shall discuss aging properties for the Cole-Hopf solution to the stationary KPZ equation. These rely on an explicit covariance-to-variance formula we shall present and which may be of independent interest. The formula is obtained first for semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime using (only) the stationary structure. It is then lifted, after proper rescaling, to the continuum by a concentration of mass argument for functionals of bounded Malliavin derivative.
Based on a joint work with Gregorio Moreno-Flores (PUC Chile) and Jean-Dominique Deuschel (TU Berlin).
Tatsuya Mikami First Passage Percolation on a Crystal Lattice Percolation theory is a branch of probability theory which describes the the behavior of clusters in a random graph, and it has many applications to material science such as immersion in a porous stone. One of the most basic percolation model is the bond percolation model on the cubical lattice, where each edge is assumed to be open with the same probability and we consider the cluster obtained by open edges. The time evolution version of the bond percolation model is called the first passage percolation (FPP) model: each edge is assigned a random passage time, and consider the behavior of “percolation region” B(t), which consists of the vertices that can be arrived from the origin within the time t > 0. Cox and Durrett (1981) showed the “shape theorem” for the region, saying that the normalized region B(t)/t converges to some limit shape.
In this study, I consider a generalized model for the bond percolation model and FPP model formulated on a crystal lattice. In this talk, I will give a generalized version of the shape theorem. I will also give an observation about the relation between the structure of the crystal lattice and the limit shape.
vincenzo crescimanna Martingale: a machine learning description Learning a representation of the visible data is a key skill for any leaning system. In the talk would be described the relationship between the representation of the stochastic process and its associated martingale. In particular would be considered the case of Gaussian process and the possible applications in the forecasting scenario
Wolfgang Loehr Continuum trees and triangulations of the circle For constructing limit processes of tree-valued Markov chains, as the size of the trees tends to infinity, it is crucial to choose an appropriate state space of “continuum trees” for the limit. A classical approach is to use metric (measure) spaces with certain tree-like properties. In this setup, trees can be encoded by excursions on an interval. An alternative is to use what we call algebraic trees, which can be coded by triangulations of the circle. This concept of trees contains less structure, but the natural topology on the set of trees is also more stringent on some aspects of the “tree structure” such as the maximal degree.
(joint work with Anita Winter)
Yulia Meshkova On homogenization of hyperbolic systems The purposed talk concerns homogenization of periodic differential operators. There are no randomness / probability in the problem at all. We consider a hyperbolic system for an operator with periodic rapidly oscillating coefficients. The aim is to give an approximation for the solution as the period tends to zero and to obtain a quantitative error estimate. The main goal of the talk is to give an impression on the spectral approach to periodic homogenization problems. This method is based on the scaling transformation, the Floquet-Bloch theory, and the analytic perturbation theory. More details can be found at arXiv:1705.02531.
Zhenyao Sun Quasi-stationary distributions for subcritical superprocesses Suppose that X is a subcritical superprocess. Under some asymptotic conditions on the mean semigroup of X, we prove the Yaglom limit of X exists and identify all quasi-stationary distributions of X.