Publications
Articles in Refereed Journals

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. 122, 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. 125, 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. XXIII, 820, DOI 10.1007/9783030607548 .
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

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. 
Z. Mokhtari, R.I.A. Patterson, F. Höfling, Spontaneous trail formation in populations of autochemotactic walkers, Preprint no. 2900, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2900 .
Abstract, PDF (2776 kByte)
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. 
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. 
A. Agazzi, L. Andreis, R.I.A. Patterson, D.R.M. Renger, Large deviations for Markov jump processes with uniformly diminishing rates, Preprint no. 2816, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2816 .
Abstract, PDF (358 kByte)
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.
Talks, Poster

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, BernoulliIMS 10th World Congress in Probability and Statistics, July 19  23, 2021, Virtual Congress Organized by Bernoulli Society and Institute of Mathematical Statistics and Hosted by Seoul National University and The Korean Statistical Society, July 22, 2021.

T. Orenshtein, Rough walks in random environment (online talk), ArgentinaBrasilPortugal joint probability seminar (Online Event), Instituto Nacional de MatemÃ¡tica 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, Karlsruhe Institut für Technologie, Fakultät für Mathematik, December 10, 2021.

L. Andreis, Introduction to large deviations and random graphs (online talk), Minicourse , cycle of doctoral seminars (a short 8hour course) for the Ph D program of Turin University, January 14  25, 2021, Università degli Studi di Torino, Dipartimento di Matematica (Online Event), 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), April 7, 2021.

R.I.A. Patterson, Decomposing large deviations rate functions into reversible and irreversible parts (online talk), BMCBAMC Glasgow 2021 (Online Event), April 6  9, 2021, The British Mathematical Colloquium (BMC) and the British Applied Mathematics Colloquium (BAMC), 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 parabilic Anderson model (online talk), Seminarion de Probabilitá , Analysi Stochastica e Statistica (Online Event), Pisa, Italy, June 8, 2021.
External Preprints

A. Zass, Gibbs point processes on path space: existence, cluster expansion and uniqueness, Preprint no. arXiv:2106.14000, Cornell University Library, arXiv.org, 2021.
Abstract
We study a class of infinitedimensional diffusions under Gibbsian interactions, in the context of marked point configurations: the starting points belong to Rd, 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, 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