Publications
Articles in Refereed Journals

A. Zass, Gibbs point processes on path space: Existence, cluster expansion and uniqueness, Markov Processes and Related Fields, 28 (2022), pp. 329364.
Abstract
We study a class of infinitedimensional diffusions under Gibbsian interactions, in the context of marked point configurations: The starting points belong to R^d, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinitevolume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds. 
A. Bianchi, F. Collet, E. Magnanini, The GHS and other inequalities for the twostar model, ALEA. Latin American Journal of Probability and Mathematical Statistics, 19 (2022), pp. 16791695, DOI 10.30757/ALEA.v1964 .
Abstract
We consider the twostar model, a family of exponential random graphs indexed by two real parameters, h and ?, that rule respectively the total number of edges and the mutual dependence between them. Borrowing tools from statistical mechanics, we study different classes of correlation inequalities for edges, that naturally emerge while taking the partial derivatives of the (finite size) free energy. In particular, under a mild hypothesis on the parameters, we derive first and second order correlation inequalities and then prove the socalled GHS inequality. As a consequence, the average edge density turns out to be an increasing and concave function of the parameter h, at any fixed size of the graph 
A. Agazzi, L. Andreis, R.I.A. Patterson, D.R.M. Renger, Large deviations for Markov jump processes with uniformly diminishing rates, , published online: 20.07.2022 (2022), DOI 10.1016/j.spa.2022.06.017 .
Abstract
We prove a largedeviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further show that our assumptions on the decay of the jump rates are optimal. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of Mass action kinetics. 
N. Fountoulakis, T. Iyer, Condensation phenomena in preferential attachment trees with neighbourhood influence, Electronic Journal of Probability, 27 (2022), pp. 149, DOI 10.1214/22EJP787 .
Abstract
We introduce a model of evolving preferential attachment trees where vertices are assigned weights, and the evolution of a vertex depends not only on its own weight, but also on the weights of its neighbours. We study the distribution of edges with endpoints having certain weights, and the distribution of degrees of vertices having a given weight. We show that the former exhibits a condensation phenomenon under a certain critical condition, whereas the latter converges almost surely to a distribution that resembles a power law distribution. Moreover, in the absence of condensation, we prove almostsure setwise convergence of the related quantities. This generalises existing results on the BianconiBarabÃ¡si tree as well as on an evolving tree model introduced by the second author. 
P. Houdebert, A. Zass, An explicit Dobrushin uniqueness region for Gibbs point process with repulsive interactions, Journal of Applied Probability, (2022), pp. 115, DOI 10.1017/jpr.2021.70 .
Abstract
We present a uniqueness result for Gibbs point processes with interactions that come from a nonnegative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature ?. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interactions 
Z. Mokhtari, R.I.A. Patterson, F. Höfling, Spontaneous trail formation in populations of autochemotactic walkers, New Journal of Physics, 24 (2022), 013012, DOI 10.1088/13672630/ac43ec .
Abstract
We study the formation of trails in populations of selfpropelled agents that make oriented deposits of pheromones and also sense such deposits to which they then respond with gradual changes of their direction of motion. Based on extensive offlattice computer simulations aiming at the scale of insects, e.g., ants, we identify a number of emerging stationary patterns and obtain qualitatively the nonequilibrium state diagram of the model, spanned by the strength of the agentpheromone interaction and the number density of the population. In particular, we demonstrate the spontaneous formation of persistent, macroscopic trails, and highlight some behaviour that is consistent with a dynamic phase transition. This includes a characterisation of the mass of systemspanning trails as a potential order parameter. We also propose a dynamic model for a few macroscopic observables, including the subpopulation size of trailfollowing agents, which captures the early phase of trail formation. 
N. Perkowski, W. van Zuijlen, Quantitative heat kernel estimates for diffusions with distributional drift, Potential Analysis, published online on 27.01.2022 (2022), DOI 10.1007/s11118021099843 .
Abstract
We consider the stochastic differential equation on ℝ ^{d} given by d X _{t} = b(t,X_{t} ) d t + d B_{t}, where B is a Brownian motion and b is considered to be a distribution of regularity >  1/2. We show that the martingale solution of the SDE has a transition kernel Γ_{t} and prove upper and lower heat kernel bounds for Γ_{t} with explicit dependence on t and the norm of b. 
B. Jahnel, A. Tóbiás, Absence of percolation in graphs based on stationary point processes with degrees bounded by two, Random Structures and Algorithms, 62 (2022), pp. 240255, DOI 10.1002/rsa.21084 .
Abstract
We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edgedrawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional $k$nearest neighbor graph of a twodimensional homogeneous Poisson point process does not percolate for k=2. 
T. Orenshtein, Rough invariance principle for delayed regenerative processes, Electronic Communications in Probability, 26 (2021), pp. 37/137/13, DOI 10.1214/21ECP406 .
Abstract
We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the pvariation settings, where a rough Donsker Theorem is available under the second moment condition. The key renewal theorem is applied to obtain an optimal moment condition. 
C. Giardinà, C. Giberti, E. Magnanini, Approximating the cumulant generating function of triangles in the ErdösRényi random graph, Journal of Statistical Physics, 182 (2021), pp. 23/123/22, DOI 10.1007/s10955021027073 .
Abstract
We study the pressure of the “edgetriangle model”, which is equivalent to the cumulant generating function of triangles in the ErdösRényi random graph. The investigation involves a population dynamics method on finite graphs of increasing volume, as well as a discretization of the graphon variational problem arising in the infinite volume limit. As a result, we locate a curve in the parameter space where a onestep replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to t he one of an equibipartite graph. 
S. Jansen, W. König, B. Schmidt, F. Theil, Distribution of cracks in a chain of atoms at low temperature, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 22 (2021), pp. 41314172, DOI 10.1007/s00023021010767 .
Abstract
We consider a onedimensional classical manybody system with interaction potential of LennardJones type in the thermodynamic limit at low temperature 1/β ∈ (0, ∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of N exp(β e _{surf} /2) with e _{surf} > 0 a surface energy. 
O. Lopusanschi, T. Orenshtein, Ballistic random walks in random environment as rough paths: Convergence and area anomaly, ALEA. Latin American Journal of Probability and Mathematical Statistics, 18 (2021), pp. 945962, DOI 10.30757/ALEA.v1834 .
Abstract
Annealed functional CLT in the rough path topology is proved for the standard class of ballistic random walks in random environment. Moreover, the `area anomaly', i.e. a deterministic linear correction for the second level iterated integral of the rescaled path, is identified in terms of a stochastic area on a regeneration interval. The main theorem is formulated in more general settings, namely for any discrete process with uniformly bounded increments which admits a regeneration structure where the regeneration times have finite moments. Here the largest finite moment translates into the degree of regularity of the rough path topology. In particular, the convergence holds in the alphaHölder rough path topology for all alpha<1/2 whenever all moments are finite, which is the case for the class of ballistic random walks in random environment. The latter may be compared to a special class of random walks in Dirichlet environments for which the regularity alpha<1/2 is bounded away from 1/2, explicitly in terms of the corresponding trap parameter. 
K. Chouk, W. van Zuijlen, Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions, The Annals of Probability, 49 (2021), pp. 19171964, DOI 10.1214/20AOP1497 .
Abstract
In this paper we consider the Anderson Hamiltonian with white noise potential on the box [0,L]² with Dirichlet boundary conditions. We show that all the eigenvalues divided by log L converge as L → ∞ almost surely to the same deterministic constant, which is given by a variational formula. 
J.D. Deuschel, T. Orenshtein, N. Perkowski, Additive functionals as rough paths, The Annals of Probability, 49 (2021), pp. 14501479, DOI 10.1214/20AOP1488 .
Abstract
We consider additive functionals of stationary Markov processes and show that under KipnisVaradhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Lévy area that can be described in terms of the asymmetry (nonreversibility) of the underlying Markov process. We apply this abstract result to three model problems: First we study random walks with random conductances under the annealed law. If we consider the Itô rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a nonreversible OrnsteinUhlenbeck process, while the last example is a diffusion in a periodic environment. As a technical step we prove an estimate for the pvariation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path BurkholderDavisGundy inequality for local martingale rough paths of [FV08], [CF19] and [FZ18] to the case where only the integrator is a local martingale. 
R. Kraaij, F. Redig, W. van Zuijlen, A HamiltonJacobi point of view on meanfield GibbsnonGibbs transitions, Transactions of the American Mathematical Society, 374 (2021), pp. 52875329, DOI 10.1090/tran/8408 .
Abstract
We study the loss, recovery, and preservation of differentiability of timedependent large deviation rate functions. This study is motivated by meanfield GibbsnonGibbs transitions. The gradient of the ratefunction evolves according to a Hamiltonian flow. This Hamiltonian flow is used to analyze the regularity of the time dependent rate function, both for Glauber dynamics for the CurieWeiss model and Brownian dynamics in a potential. We hereby create a unifying framework for the treatment of meanfield GibbsnonGibbs transitions, based on Hamiltonian dynamics and viscosity solutions of HamiltonJacobi equations. 
M.A. Peletier, D.R.M. Renger, Fast reaction limits via Gammaconvergence of the flux rate functional, Journal of Dynamics and Differential Equations, published online in July 2021, DOI 10.1007/s10884021100242 .
Abstract
We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; `slow' rates are constant, and `fast' rates are scaled as 1/∈, and we prove the convergence in the fastreaction limit ∈ → 0. We establish a Γconvergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level2.5 rate function). The limiting system is again described by a functional, and characterizes both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γconvergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes. 
S. Stivanello, G. Bet, A. Bianchi, M. Lenci, E. Magnanini, Limit theorems for Lévy flights on a 1D Lévy random medium, Electronic Journal of Probability, 26 (2021), pp. 57/157/25, DOI 10.1214/21EJP626 .
Abstract
We study a random walk on a point process given by an ordered array of points ( ? k , k ? Z ) (?k,k?Z) on the real line. The distances ? k + 1 ? ? k ?k+1??k are i.i.d. random variables in the domain of attraction of a ?stable law, with ? ? ( 0 , 1 ) ? ( 1 , 2 ) ??(0,1)?(1,2). The random walk has i.i.d. jumps such that the transition probabilities between ? k ?k and ? ? ?? depend on ? ? k ??k and are given by the distribution of a Z Zvalued random variable in the domain of attraction of an ?stable law, with ? ? ( 0 , 1 ) ? ( 1 , 2 ) ??(0,1)?(1,2). Since the defining variables, for both the random walk and the point process, are heavytailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters ? and ?, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finitedimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology. 
L. Andreis, W. König, R.I.A. Patterson, A largedeviations principle for all the cluster sizes of a sparse ErdősRényi random graph, Random Structures and Algorithms, 59 (2021), pp. 522553, DOI 10.1002/rsa.21007 .
Abstract
A largedeviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a timedependent version of the ErdősRényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller ErdősRényi graphs are connected.
Contributions to Collected Editions

F. DEN Hollander, W. König, R. Soares Dos Santos, The parabolic Anderson model on a GaltonWatson tree, in: In and Out of Equilibrium 3: Celebrating Vladas Sidovaricius, M.E. Vares, R. Fernandez, L.R. Fontes, C.M. Newman, eds., 77 of Progress in Probability, Birkhäuser, 2021, pp. 591635, DOI 10.1007/9783030607548_25 .
Abstract
We study the longtime asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical GaltonWatson random tree with bounded degrees. We identify the secondorder contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally treelike random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence. 
W. König, Branching random walks in random environment, in: Probabilistic Structures in Evolution, E. Baake, A. Wakolbinger, eds., Probabilistic Structures in Evolution, EMS Series of Congress Reports, European Mathematical Society Publishing House, 2021, pp. 2341, DOI 10.4171/ECR/171/2 .
Preprints, Reports, Technical Reports

T. Matsuda, W. van Zuijlen, Anderson Hamiltonians with singular potentials, Preprint no. 2976, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2976 .
Abstract, PDF (732 kByte)
We construct random Schrödinger operators, called Anderson Hamiltonians, with Dirichlet and Neumann boundary conditions for a fairly general class of singular random potentials on bounded domains. Furthermore, we construct the integrated density of states of these Anderson Hamiltonians, and we relate the Lifschitz tails (the asymptotics of the left tails of the integrated density of states) to the left tails of the principal eigenvalues. 
Z. Benomar, Ch. Ghribi, E. Cali, A. Hinsen, B. Jahnel, J.P. Wary, Multiagent simulations for virus propagation in D2D 5G+ networks, Preprint no. 2953, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2953 .
Abstract, PDF (367 kByte)
In this paper we present results for an extended class of multiagent simulation models for malware propagation in devicetodevice 5G networks, first exhibited in citeAgentBasedAAMAS. The models allow to understand and analyze mobile malware spreading dynamics in highly dynamical networks and also to assess the effectiveness of a proposed counter measure policy for reversing attacks and securing the system. Our main simulation studies identify critical thresholds for maximal malware propagation and isolate two distinguished regimes for malware survival and extermination depending on a variety of parameters. We further predict via simulations the malware spreading velocities, depending on device density and speed, as well as the percentage of counter agents that have to be introduced into the network for malware elimination. We complement these findings and state also an associated theoretical study that highlights the key parameters of our agentbased model and exhibit certain linear relationships between them [1]. 
A. Quitmann, L. Taggi, Macroscopic loops in the $3d$ doubledimer model, Preprint no. 2944, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2944 .
Abstract, PDF (265 kByte)
The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of selfavoiding loops. Our first result is that in ℤ ^{d}, d>2, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer doubledimer model, namely the doubledimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from [Betz, Taggi] that a finite critical threshold of the monomer activity exists, below which a selfavoiding walk forced through the system is macroscopic. Our paper shows that, when d >2, such a critical threshold is strictly positive. In other words, the selfavoiding walk is macroscopic even in the presence of a positive density of monomers. 
D.R.M. Renger, U. Sharma, Untangling dissipative and Hamiltonian effects in bulk and boundary driven systems, Preprint no. 2936, WIAS, Berlin, 2022.
Abstract, PDF (330 kByte)
Using the theory of large deviations, macroscopic fluctuation theory provides a framework to understand the behaviour of nonequilibrium dynamics and steady states in emphdiffusive systems. We extend this framework to a minimal model of nonequilibrium emphnondiffusive system, specifically an open linear network on a finite graph. We explicitly calculate the dissipative bulk and boundary forces that drive the system towards the steady state, and nondissipative bulk and boundary forces that drives the system in orbits around the steady state. Using the fact that these forces are orthogonal in a certain sense, we provide a decomposition of the largedeviation cost into dissipative and nondissipative terms. We establish that the purely nondissipative force turns the dynamics into a Hamiltonian system. These theoretical findings are illustrated by numerical examples. 
A. Quitmann, L. Taggi, Macroscopic loops in the Bose gas, Spin O(N) and related models, Preprint no. 2915, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2915 .
Abstract, PDF (598 kByte)
We consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model, random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in ℤ^{d}, d ≥ 3, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate ℤ^{d} by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hardcore constraints. 
O. Collin, B. Jahnel, W. König, The free energy of a boxversion of the interacting Bose gas, Preprint no. 2914, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2914 .
Abstract, PDF (1441 kByte)
The interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous BoseEinstein condensation phase transition is sought for. We introduce a simplified version of the model with an organisation of the particles in deterministic boxes instead of Brownian cycles as the marks of a reference Poisson point process (for simplicity, in Z ^{d}, instead of R ^{d}). We derive an explicit and interpretable variational formula in the thermodynamic limit for the limiting free energy of the canonical ensemble for any value of the particle density. This formula features all relevant physical quantities of the model, like the microscopic and the macroscopic particle densities, together with their mutual and selfenergies and their entropies. The proof method comprises a twostep largedeviation approach for marked Poisson point processes and an explicit distinction into small and large marks. In the characteristic formula, each of the microscopic particles and the statistics of the macroscopic part of the configuration are seen explicitly; the latter receives the interpretation of the condensate. The formula enables us to prove a number of properties of the limiting free energy as a function of the particle density, like differentiability and explicit upper and lower bounds, and a qualitative picture below and above the critical threshold (if it is finite). This proves a modified saturation nature of the phase transition. However, we have not yet succeeded in proving the existence of this phase transition. 
L. Andreis, W. König, H. Langhammer, R.I.A. Patterson, A largedeviations principle for all the components in a sparse inhomogeneous random graph, Preprint no. 2898, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2898 .
Abstract, PDF (690 kByte)
We study an inhomogeneous sparse random graph, G_{N}, on [N] = { 1,...,N } as introduced in a seminal paper [BJR07] by Bollobás, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N → ∞ , we consider the sparse regime, where the average degree is O(1). We prove a largedeviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size ≈ N). In doing so, we derive explicit logarithmic asymptotics for the probability that G_{N} is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of G_{N}. In particular, we recover the criterion for the existence of the phase transition given in [BJR07]. 
D.R.M. Renger, Anisothermal chemical reactions: OnsagerMachlup and macroscopic fluctuation theory, Preprint no. 2893, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2893 .
Abstract, PDF (379 kByte)
We study a micro and macroscopic model for chemical reactions with feedback between reactions and temperature of the solute. The first result concerns the quasipotential as the largedeviation rate of the microscopic invariant measure. The second result is an application of modern OnsagerMachlup theory to the pathwise large deviations, in case the system is in detailed balance. The third result is an application of macroscopic fluctuation theory to the reaction flux large deviations, in case the system is in complex balance. 
D.R.M. Renger, S. Schindler, Gradient flows for bounded linear evolution equations, Preprint no. 2881, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2881 .
Abstract, PDF (198 kByte)
We study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real diagonalisable. The proof is constructive, from which we also derive geodesic lambdaconvexity. 
A. Zass, Gibbs point processes on path space: Existence, cluster expansion and uniqueness, Preprint no. 2859, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2859 .
Abstract, PDF (1749 kByte)
We study a class of infinitedimensional diffusions under Gibbsian interactions, in the context of marked point configurations: The starting points belong to R^d, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinitevolume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds. 
R.I.A. Patterson, D.R.M. Renger, U. Sharma, Variational structures beyond gradient flows: A macroscopic fluctuationtheory perspective, Preprint no. 2826, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2826 .
Abstract, PDF (522 kByte)
Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the largedeviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode nondissipative effects. Our main contribution is an abstract framework, which for a given fluxdensity cost and a quasipotential, provides a decomposition into dissipative and nondissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems  independent copies of jump processes, zerorange processes, chemicalreaction networks in complex balance and latticegas models. 
N. Nüsken, D.R.M. Renger, Stein variational gradient descent: Manyparticle and longtime asymptotics, Preprint no. 2819, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2819 .
Abstract, PDF (430 kByte)
Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: emphvariational inference and emphMarkov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and largedeviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the largedeviation functional governing the manyparticle limit for the empirical measure. Moreover, we identify the emphSteinFisher information (or emphkernelised Stein discrepancy) as its leading order contribution in the longtime and manyparticle regime in the sense of $Gamma$convergence, shedding some light on the finiteparticle properties of SVGD. Finally, we establish a comparison principle between the SteinFisher information and RKHSnorms that might be of independent interest.
Talks, Poster

A. Quitmann, Macroscopic loops in a random walk loop soup, Spring School on Random geometric graphs, March 28  April 1, 2022, Technische Universität Darmstadt, Fachbereich Mathematik.

A. Quitmann, Macroscopic loops in a random walk loop soup, Mathematics of Random Systems Summer School 2022, September 26  30, 2022, Centre for Doctoral Trainig, Mathematics of Random Systems, Oxford, UK, September 27, 2022.

A. Quitmann, Macroscopic loops in an interacting random walk loop soup, Focused Research: Graphical Representations of Spin Systems, September 5  7, 2022, University of Bristol, Heilbronn Institute for Mathematical Research, Bristol, UK, September 6, 2022.

A. Quitmann, Macroscopic loops in random walk loop soups, Workshop and Summerschool on Random Graphs (RandNET), August 22  30, 2022, Workshop Centre in the area of Stochastics (EURANDOM), Eindhoven, Netherlands, August 24, 2022.

A. Quitmann, Macroscopic loops in the Spin O(N) and related models (online talk), Percolation Today (Online Event), ETH Zürich, Italy, February 15, 2022.

T. Iyer, Preferential attachment trees with neighbourhood influence, Summer School: Mathematics of Large Networks, May 30  June 3, 2022, Erdős Center, Budapest, Hungary, May 31, 2022.

T. Iyer, Preferential attachment trees with neighbourhood influence, Probability Seminar, Universität zu Köln, Department Mathematik/Informatik, April 29, 2022.

D.R.M. Renger, Variational formulations beyond gradient flows (online talk), British Applied Mathematics Colloquium BAMC 2022 (HybridEvent), April 11  13, 2022, Loughborough University, Loughborough, UK, April 13, 2022.

M. Renger, Variational structures beyond gradient flows  Part II (online talk), Seminar on Variational Evolutionary Problems and Related Problems (Online Event), Technische Universität Dresden, Fakulät für Mathematik, January 19, 2022.

A. Zass, Existence of infinitevolume marked Gibbs point processes: a path space example, Third Italian Meeting on Probability and Mathematical Statistics, June 13  16, 2022, University of Bologna, Department of Mathematics, Department of Statistical Sciences of the Alma Mater Studiorum, Bologna, Italy, June 15, 2022.

A. Zass, Gibbs point process on path space: existence, cluster, cluster expansion and uniqueness, Oberseminar zur Stochastik, Otto von Guericke Universität Magdeburg, Fakulät für Mathematik, January 20, 2022.

A. Zass, Interacting diffusions as marked Gibbs point processes, Seminar of Stochastic Geometry, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic, October 11, 2022.

W. van Zuijlen, Anderson Hamiltonians with singular potentials, 15th OxfordBerlin Young Researchers Meeting on Applied Stochastic Analysis, May 12  14, 2022, WIAS Berlin, May 13, 2022.

E. Magnanini, Limit theorems for the edge density in exponential random graphs, Third Italian Meeting on Probability and Mathematical Statistics, June 13  16, 2022, University of Bologna, Department of Mathematics, Department of Statistical Sciences of the Alma Mater Studiorum, Bologna, Italy, June 15, 2022.

E. Magnanini, Limit theorems for the edge density in exponential random graphs, Workshop and Summerschool on Random Graphs (RandNET), August 22  30, 2022, Workshop Centre in the area of Stochastics (EURANDOM), Eindhoven, Netherlands, August 24, 2022.

A. Quitmann, Macroscopic loops in interacting random walk loop soups, Oberseminar AG Stochastik, Technische Universität Darmstatd, Fachbereich Mathematik, November 3, 2022.

A. Quitmann, Macroscopic loops in interacting random walk loop soups, Oberseminar Wahrscheinlichkeitstheorie, LudwigMaximiliansUniversität, Mathematisches Institut, November 14, 2022.

W. König, A largedeviations principle for all the components in a sparse inhomogeneous ErdősRényi, Workshop: Interacting Particle Systems and Hydrodynamic Limits, March 14  25, 2022, Université Montreal, Centre de Recherches de Mathématiques, Montreal, Canada, March 22, 2022.

W. König, Das interagierende Bosegas im Lichte der Wahrscheinlichkeitstheorie, SommerfeldTag 2022, WorkshopDialogKolloquium ''Wissenschaft vs. Wahrscheinlichkeit?¡`, ArnoldSommerfeldGesellschaft e.V., Leipzig, April 28, 2022.

W. König, The free energy of a box version of the interacting Bose gas, Quantum many body system and interacting particles: in honor of Herbert Spohn, June 20  24, 2022, Westfälische WilhelmsUniversität Münster, June 24, 2022.

A. Quitmann, Introduction to GFF, Multiplicative Chaos and Liouville Quantum Gravity, Minicourse for PhD students, June 14  July 7, 2021, Universitá di Roma la Sapienza, Dipartimento di Matematica, Rome, Italy.

T. Iyer, Degrees of fixed vertices and power law degree distributions in preferential attachment trees with neighbourhood influence, Probability Seminar, Università degli Studi di Firenze, Dipartimento di Matematica e Informatica ``Ulisse Dini'', Florence, Italy, November 17, 2021.

T. Orenshtein, A discussant for a talk by Jeremy QUASTEL (UBC) titled ``The KPZ fixed point'' (online talk), SPDEs & friends (Online Event), May 31  June 2, 2021, Technische Universität Berlin, May 31, 2021.

T. Orenshtein, Rough walks in random environment (online talk), BernoulliIMS 10th World Congress in Probability and Statistics (Online Event), July 19  23, 2021, Seoul National University, Korea (Republic of), July 22, 2021.

T. Orenshtein, Rough walks in random environment (online talk), ArgentinaBrasilPortugal Joint Probability Seminar (Online Event), Instituto Nacional de Matemática Pura e Aplicada (IMPA), Brazil, May 19, 2021.

T. Orenshtein, Rough walks in random environments (online talk), Chilean Probability Seminar (Online Event), Pontificia Universidad Católica de Chile, Facultad de Matemáticas, Santiago de Chile, Chile, December 1, 2021.

A. Zass, A marked Gibbs point process on path space: Existence and uniqueness, Oberseminar Stochastik, Universität zu Köln, Department Mathematik/Informatik, Abteilung Mathematik, November 4, 2021.

A. Zass, Gibbs point processes on path space: Existence, cluster expansion and uniqueness, AG Stochastische Geometrie, Karlsruher Institut für Technologie, Fakultät für Mathematik, December 10, 2021.

L. Andreis, Introduction to large deviations and random graphs (online minicourse), cycle of doctoral seminars (8hour course) for the Ph.D. program of Turin University, January 14  25, 2021, Università degli Studi di Torino, Dipartimento di Matematica, Italy.

H. Langhammer, A Largedeviations Approach to the Phase Transition in Inhomogeneous Random Graphs, Workshop ``Junior Female Researchers in Probability'', October 4  6, 2021, Stochastic Analysis in Interaction. BerlinOxford IRTG 2544, October 5, 2021.

E. Magnanini, Limit theorems for the edge density in exponential random graphs, Workshop ``Junior Female Researchers in Probability'', October 4  6, 2021, Stochastic Analysis in Interaction. BerlinOxford IRTG 2544, October 5, 2021.

E. Magnanini, Limit theorems for the edge density in exponential random graphs, Probability Seminar, Università degli Studi di Firenze, Dipartimento di Matematica e Informatica ``Ulisse Dini'', Italy, November 17, 2021.

W. König, A box version of the interacting Bose gas, Workshop on Randomness Unleashed Geometry, Topology, and Data, September 22  24, 2021, University of Groningen, Faculty of Science and Engineering, Groningen, Netherlands, September 23, 2021.

W. König, A box version of the interacting Bose gas, Stochastic Geometry Days, November 15  19, 2021, Dunkerque, France, November 18, 2021.

W. König, A grid version of the interacting Bose gas (online talk), Probability Seminar (Online Event), University of Bath, Department of Mathematical Sciences, UK, February 15, 2021.

W. König, A largedeviations principle for all the components in a sparse inhomogeneous ErdősRényi graph (online talk), UC San Diego Probability Seminar (Online Event), University of California, Department of Mathematics, San Diego, USA, October 14, 2021.

W. König, Cluster size distributions in a classical manybody system (online talk), Thematic Einstein Semester on Geometric and Topological Structure of Materials, Summer Semester 2021, Technische Universität Berlin, May 20, 2021.

W. König, Das interagierende Bosegas im Lichte der Wahrscheinlichkeitstheorie, Sommerfeld Seminar, ArnoldSommerfeldGesellschaft e.V., Leipzig, October 21, 2021.

R. Patterson, Decomposing large deviations rate functions into reversible and irreversible parts (online talk), The British Mathematical Colloquium (BMC) and the British Applied Mathematics Colloquium (BAMC) : BMCBAMC GLASGOW 2021, April 6  9, 2021, University of Glasgow (Online Event), UK, April 7, 2021.

W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability Meeting (Online Event), University of Oxford, Department of Statistics, UK, February 10, 2021.

W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability and Statistical Physics Seminar (Online Event), The University of Chicago, Department of Mathematics, Statistics, and Computer Science, USA, February 12, 2021.

W. van Zuijlen, Quantitative heat kernel estimates for diffusions with distributional drift (online talk), 14th OxfordBerlin Young Researchers Meeting on Applied Stochastic Analysis (Online Event), February 10  12, 2021, University of Oxford, Mathematical Institute, UK, February 12, 2021.

W. van Zuijlen, Total mass asymptotics of the parabolic Anderson model (online talk), SPASS  Probability, Stochastic Analysis and Statistics Seminar (Online Event), Pisa, Italy, June 8, 2021.
External Preprints

A. Bianchi, F. Collet, E. Magnanini, Limit theorems for exponential random graphs, Preprint no. arXiv:2105.06312, Cornell University Library, arXiv.org, 2021.
Abstract
We consider the edgetriangle model, a twoparameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the socalled replica symmetric regime, the limiting free energy exists together with a complete characterization of the phase diagram of the model. We borrow tools from statistical mechanics to obtain limit theorems for the edge density. First, we determine the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases. Then we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a meanfield approximation of the model. Some of our results can be extended with no substantial changes to more general classes of exponential random graphs 
A. Bianchi, F. Collet, E. Magnanini, The GHS and other inequalities for the twostar model, Preprint no. arXiv:2107.08889, Cornell University Library, arXiv.org, 2021.
Abstract
We consider the twostar model, a family of exponential random graphs indexed by two real parameters, h and ?, that rule respectively the total number of edges and the mutual dependence between them. Borrowing tools from statistical mechanics, we study different classes of correlation inequalities for edges, that naturally emerge while taking the partial derivatives of the (finite size) free energy. In particular, under a mild hypothesis on the parameters, we derive first and second order correlation inequalities and then prove the socalled GHS inequality. As a consequence, the average edge density turns out to be an increasing and concave function of the parameter h, at any fixed size of the graph 
N. Fountoulakis, T. Iyes, Condensation phenomena in preferential attachment trees with neighbourhood influence, Preprint no. arXiv:2101.027, Cornell University Library, arXiv.org, 2021.
Abstract
We introduce a model of evolving preferential attachment trees where vertices are assigned weights, and the evolution of a vertex depends not only on its own weight, but also on the weights of its neighbours. We study the distribution of edges with endpoints having certain weights, and the distribution of degrees of vertices having a given weight. We show that the former exhibits a condensation phenomenon under a certain critical condition, whereas the latter converges almost surely to a distribution that resembles a power law distribution. Moreover, in the absence of condensation, we prove almostsure setwise convergence of the related quantities. This generalises existing results on the BianconiBarabÃ¡si tree as well as on an evolving tree model introduced by the second author.
Research Groups
 Partial Differential Equations
 Laser Dynamics
 Numerical Mathematics and Scientific Computing
 Nonlinear Optimization and Inverse Problems
 Interacting Random Systems
 Stochastic Algorithms and Nonparametric Statistics
 Thermodynamic Modeling and Analysis of Phase Transitions
 Nonsmooth Variational Problems and Operator Equations