Papers on the arxiv

• The directed landscape seen from its trees
  with Mustazee Rahman, The directed landscape can be reconstructed from the shapes of its semi-infinite geodesics using the Busemann process. A novel coordinate system and differential distance analysis enable this reconstruction, revealing the geometric structure of geodesics.
• Upper tail large deviations of the directed landscape
  with Sayan Das, Duncan Dauvergne, An upper tail large deviation principle for the directed landscape is established, linking metrics to measures on countably many paths with a Kruzhkov entropy rate function. This principle is applied to derive a large deviation result for related stochastic processes.
• The directed landscape from Brownian motion
  with Duncan Dauvergne, An almost sure bijection constructs the directed landscape from independent Brownian motions, analogous to the RSK correspondence. The inverse map provides a natural coupling for Brownian last passage percolation, revealing the landscape’s structure.
• The geometry of coalescing random walks, the Brownian web distance and KPZ universality
  with Bálint Vető, Coalescing random walks form an infinite tree with a directed distance that scales to the Brownian web distance, exhibiting 0:1:2 exponents. This distance connects to KPZ universality, distinguishing it from the 1:2:3 exponents of the KPZ world.
• Eigenvectors of the square grid plus GUE
  with András Mészáros, Communications in Mathematical Physics, Volume 405, article number 2, 2024, Eigenvectors of the GUE-perturbed discrete torus transition from product structure to discrete Gaussian waves as perturbation increases. The phase transition point for this behavior is precisely determined.
• KPZ fluctuations in the planar stochastic heat equation
  with Jeremy Quastel, Alejandro Ramírez, A Wick-ordered stochastic heat equation with planar white noise is formulated using the Skorokhod integral, representing a random polymer’s free energy. The solution, expressed via the Feynman-Kac formula, defines a Gaussian multiplicative chaos with KPZ fluctuations.
• Palm measures for Dirac operators and the Sine beta process
  with Benedek Valkó, Stochastic Processes and their Applications, Volume 163, September 2023, Pages 106-135, The Palm measure of the Sine beta process is characterized as the eigenvalues of an associated Dirac operator with specific boundary conditions. This provides a new perspective on the statistical properties of the process.
• Infinite geodesics, competition interfaces and the second class particle in the scaling limit
  with Mustazee Rahman, Infinite geodesics in the directed landscape are constructed, proving their uniqueness and coalescence using Busemann functions. The second class particle in TASEP converges to a competition interface under KPZ scaling.
• Amenability of quadratic automaton groups
  with Gideon Amir, Omer Angel, Lower bounds on electrical resistance in Schreier graphs of quadratic automaton groups demonstrate their amenability. This uses a weighted Nash-Williams criterion, resolving an open question in group theory.
• Large deviations for the interchange process on the interval and incompressible flows
  with Michał Kotowski, Geometric and Functional Analysis, 32, 1357-1427 (2022), Large deviations of the interchange process are shown to be controlled by Dirichlet energy, linking permutations to incompressible Euler equations. This implies the Archimedean limit for relaxed sorting networks and enables their asymptotic counting.
• RSK in last passage percolation: a unified approach
  with Duncan Dauvergne, Mihai Nica, Probability Surveys 19 (2022): 65-112, A unified RSK correspondence based on the Pitman transform is presented, connecting ordinary, dual, and continuous RSK, and shown to be bijective and isometric. This enables non-computational proofs of properties like dual RSK mapping to Bernoulli measures.
• The scaling limit of the longest increasing subsequence
  with Duncan Dauvergne, A framework proves convergence of last passage models to the directed landscape via the Airy line ensemble and Airy sheet. In i.i.d. environments, this convergence extends to distances and geodesics in the directed landscape.
• Three-halves variation of geodesics in the directed landscape
  with Duncan Dauvergne, Sourav Sarkar, The Annals of Probability 50, no. 5 (2022): 1947-1985, Geodesics in the directed landscape exhibit 3/2-variation, with weight functions showing cubic variation. The small-scale limit around interior points is characterized by a Brownian-Bessel boundary condition in the directed landscape.
• The many faces of the stochastic zeta function
  with Benedek Valkó, Geometric and Functional Analysis 32, no. 5 (2022): 1160-1231, The stochastic zeta function, with zeros from the Sineβ process, is studied via a power series built from Brownian motion. Multiple characterizations and stochastic differential equations reveal its properties and related distributions.
• The heat and the landscape I
  Heat flows in a 1+1 dimensional stochastic environment converge to the directed landscape, with the O'Connell-Yor polymer and KPZ equation converging to the KPZ fixed point. Baik-Ben Arous-Peché statistics provide a general method for this convergence.
• Brownian absolute continuity of the KPZ fixed point with arbitrary initial condition
  with Sourav Sarkar, The Annals of Probability 2021, Vol. 49, No. 4, 1718-1737, The KPZ fixed point’s law, starting from any initial condition, is absolutely continuous with respect to Brownian motion on compact intervals. This includes the Airy1 process, confirming its Brownian-like behavior.
• Uniform convergence to the Airy line ensemble
  with Duncan Dauvergne, Mihai Nica, Ann. Inst. H. Poincaré Probab. Statist. 59(4): 2220-2256, November 2023, Classical integrable models of last passage percolation converge uniformly to the Airy line ensemble. This is shown via convergence of nonintersecting Bernoulli random walks in all feasible directions.
• Uniform point variance bounds in classical beta ensembles
  with Joseph Najnudel, Random Matrices: Theory and Applications, Vol. 10, No. 04, 2150033 (2021), Logarithmic bounds are derived for the variance of point counts in circular and Gaussian β ensembles. These bounds are optimal up to a constant depending on β.
• The bead process for beta ensembles
  with Joseph Najnudel, Probab. Theory Relat. Fields (2021), The bead process for general sine beta processes is constructed as an infinite Markov chain, generalizing the determinantal sine-kernel process. It is shown to be the bulk scaling limit of the Hermite beta corner process.
• Bulk properties of the Airy line ensemble
  with Duncan Dauvergne, The Annals of Probability 2021, Vol. 49, No. 4, 1738-1777, The Airy line ensemble is represented using independent Brownian bridges on a fine grid, enabling precise probabilistic analysis. A modulus of continuity is also established for its paths.
• The directed landscape
  with Duncan Dauvergne, Janosch Ortmann, Acta Mathematica, 229, 201-285, 2022, The directed landscape is constructed as the limit of Brownian last passage percolation, characterized via the Airy sheet and line ensemble. Last passage geodesics converge to random functions with Hölder-2/3- continuous paths.
• Entropy and expansion
  with Endre Csóka, Viktor Harangi, Ann. Inst. H. Poincaré Probab. Statist. 56(4): 2428-2444 (2020), A lower bound for the sum of joint entropies is derived using individual entropies and system expansion properties, generalizing entropy inequalities in invariant settings. This method yields bounds for independent sets in graphs.
• Operator limits of random matrices
  Proceedings of the International Congress of Mathematicians, Seoul 2014, Volume 4, 247-272, An introduction to operator limits of random matrices is provided, discussing connections to statistics, integrable systems, and orthogonal polynomials. Several open problems and conjectures are highlighted for non-experts.
• Tracy-Widom fluctuations in 2D random Schrödinger operators
  with Michał Kotowski, Communications in Mathematical Physics 370.3 (2019): 873-893, A random Schrödinger operator on the hexagonal lattice is mapped to the log-Gamma polymer, revealing Tracy-Widom fluctuations in its smallest eigenvalues. This leverages integrability properties of the polymer model.
• Circular support in random sorting networks
  with Duncan Dauvergne, Transactions of the American Mathematical Society 373.3 (2020): 1529-1553, Particle trajectories in random sorting networks are shown to be supported on π-Lipschitz paths in the global limit. The permutation matrix’s weak limit is supported on a specific set, revealing structural properties.
• Operator limit of the circular beta ensemble
  with Benedek Valkó, Annals of Probability 2020, Vol. 48, No. 3, 1286-1316, A precise coupling of finite circular beta ensembles with their limit process is provided via operator representations, with explicit bounds on operator distances. An estimate on β-dependence in the Sineβ process is also proven.
• The Local Limit of Random Sorting Networks
  with Omer Angel, Duncan Dauvergne, Alexander E. Holroyd, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, Vol. 55, No. 1, 2019, A local limit is established for the space-time locations of transpositions in random sorting networks as n approaches infinity. This limit reveals the asymptotic behavior of particle trajectories in the network.
• Brownian motion as limit of the interchange process
  with Mustazee Rahman, The empirical measure of rescaled particle trajectories in the interchange process on path graphs converges to the deterministic measure of stationary Brownian motion. This establishes a law of large numbers for particle trajectories.
• Geometry of Permutation Limits
  with Mustazee Rahman, Máté Vizer, Combinatorica 39 (2019), 933–960, The asymptotic behavior of random sorting networks is studied, proving that the Archimedean path is the unique minimal-energy path from identity to reverse permuton. This confirms conjectures about permutation limit geometry.
• Eigenvectors of the critical 1-dimensional random Schrödinger operator
  with Ben Rifkind, Geom. Funct. Anal. 28 (2018), no. 5, 1394–1419, Eigenvectors of the random Schrödinger operator with i.i.d. potential are shown to be delocalized at criticality. Their transfer matrix evolution converges to a stochastic differential equation, revealing their structure.
• The Sineβ operator
  with Benedek Valkó, Invent. Math. 209 (2017), no. 1, 275-327, The Sineβ process is shown to be the spectrum of a random differential operator involving hyperbolic Brownian motion. This provides a novel operator representation for the bulk limit of Gaussian β-ensembles.
• Dyson's spike for random Schrödinger operators and Novikov-Shubin invariants of groups
  with Michał Kotowski, Comm. Math. Phys. 352 (2017), no. 3, 905-933, Random Schrödinger operators exhibit a Dyson spike in the spectral measure near zero, with a logarithmic form. The limiting local eigenvalue distribution is identified, connecting to Novikov-Shubin invariants.
• Hölder continuity of the integrated density of states in the one-dimensional Anderson model
  with Eric Hart, Comm. Math. Phys. 355 (2017), no. 3, 839-863, The integrated density of states in the one-dimensional Anderson model is proven to be Hölder continuous with an exponent approaching 1 as disorder decreases. This improves prior results in the Anderson-Bernoulli setting.
• Spectral measures of factor of i.i.d. processes on vertex-transitive graphs
  with Ágnes Backhausz, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 4, 2260-2278, Spectral measures of factor of i.i.d. processes on vertex-transitive graphs are shown to be absolutely continuous with respect to the graph’s spectral measure. The sets of such measures and their limits are proven identical.
• A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes
  with Christian Sadel, Comm. Math. Phys. 343 (2016), no. 3, 881-919, A central limit theorem is derived for products of random matrices, with eigenvalues converging to a stochastic differential equation. This yields GOE statistics for the Anderson model on long boxes.
• Non-Liouville groups with return probability exponent at most 1/2
  with Michał Kotowski, Electron. Commun. Probab. 20 (2015), no. 12, A finitely generated group is constructed without the Liouville property, with random walk return probability bounded by e^(-n^(1/2)+o(1)). This uses permutational wreath products over tree-like Schreier graphs.
• Local algorithms for independent sets are half-optimal
  with Mustazee Rahman, Ann. Probab. 45 (2017), no. 3, 1543-1577, The largest density of factor of i.i.d. independent sets on d-regular trees is shown to be asymptotically (log d)/d, matching known constructions. This implies the same density for local algorithms on random d-regular graphs.
• Independence ratio and random eigenvectors in transitive graphs
  with Viktor Harangi, Annals of Probability 2015, Vol. 43, No. 5, 2810-2840, Lower bounds on the independence ratio of 3-regular transitive graphs are derived using random eigenvectors. These bounds are expressed in terms of the minimum eigenvalue of the adjacency matrix.
• Mean quantum percolation
  with Charles Bordenave, Arnab Sen, J. Eur. Math. Soc. 19, (2017), no. 12, 3679-3707, The spectral measure of bond percolation in the two-dimensional lattice is shown to have a non-trivial continuous part in the supercritical regime. Techniques lower bound the mass of the continuous spectral measure.
• The Liouville property for groups acting on rooted trees
  with Gideon Amir, Omer Angel, Nicolás Matte Bon, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, Vol. 52, No. 4, pp. 1763-1783, 2016, Every symmetric, finitely supported probability measure on bounded activity automaton groups has the Liouville property. A uniform entropy bound is provided for such measures.
• Universality of the Stochastic Airy Operator
  with Manjunath Krishnapur, Brian Rider, Communications on Pure and Applied Mathematics, Volume 69, Issue 1, 145-199, 2016, The top eigenvalues of beta ensembles converge to the Stochastic Airy operator after scaling. This universality extends to eigenvectors, confirming conjectures for random matrix models.
• Ramanujan graphings and correlation decay in local algorithms
  with Ágnes Backhausz, Balázs Szegedy, Random Structures & Algorithms, Volume 47, Issue 3, 424-435, 2015, An upper bound for correlation in random processes on large-girth d-regular graphs is derived, optimal for factor of i.i.d. processes. This applies to Ramanujan graphings, showing rapid correlation decay.
• Invariant Gaussian processes and independent sets on regular graphs of large girth
  with Endre Csóka, Balázs Gerencsér, Viktor Harangi, Random Structures & Algorithms, Volume 47, Issue 2, pages 284-303, 2015, Every 3-regular graph with large girth contains an independent set of size at least 0.4361n, improving prior bounds. Computer simulations suggest a bound of about 0.438n using invariant Gaussian processes.
• Random walks veering left
  with Raoul Normand, Electron. J. Probab 18.89 (2013): 1-25, Coupled random walks with a deterministic angle bias have a Hausdorff dimension computed for unusual behavior. This model relates to the spectral measure of certain random matrices.
• Speed exponents of random walks on groups
  with Gideon Amir, Int. Math. Res. Not. IMRN 2017, no. 9, 2567-2598, Finitely generated groups are constructed with random walk expected distances scaling as n^β for 3/4 ≤ β < 1. The speed can be set to match any prescribed function up to a constant factor.
• Kesten's theorem for Invariant Random Subgroups
  with Miklós Abert, Yair Glasner, Duke Math. J. 163, no. 3 (2014), 465-488, Nonamenable invariant random subgroups yield a strictly smaller spectral radius for random walks compared to the quotient group. This generalizes Kesten’s result for normal subgroups.
• The Ginibre ensemble and Gaussian analytic functions
  with Manjunath Krishnapur, International Mathematics Research Notices 2014.6 (2014): 1441-1464, The characteristic polynomial of the Ginibre ensemble evolves via a Pólya urn-like process as n changes. This yields a limiting random analytic function as a mixture of Gaussian analytic functions.
• The measurable Kesten theorem
  with Miklós Abert, Yair Glasner, Ann. Probab. 44 (2016), no. 3, 1601-1646, Explicit estimates relate spectral radius and short cycle density in finite d-regular Ramanujan graphs. Infinite d-regular Ramanujan unimodular random graphs are proven to be trees.
• Absolute continuity of the limiting eigenvalue distribution of the random Toeplitz matrix
  with Arnab Sen, Elect. Comm. in Probab. 16 (2011), 706-711, The limiting eigenvalue distribution of random symmetric Toeplitz matrices is shown to be absolutely continuous with a bounded density. Spectral averaging techniques from random Schrödinger operators are used in the proof.
• The top eigenvalue of the random Toeplitz matrix and the sine kernel
  with Arnab Sen, Annals of Probability 2013, Vol. 41, No. 6, 4050-4079, The top eigenvalue of a random symmetric Toeplitz matrix, scaled appropriately, converges to the square of the sine kernel’s 2→4 operator norm. This establishes a precise limit for the eigenvalue distribution.
• Limits of spiked random matrices II
  with Alex Bloemendal, Annals of Probability 2016, Vol. 44, No. 4, 2726-2769, Near-critical perturbations in spiked real Wishart matrices yield limiting distributions for top eigenvalues, resolving a conjecture. A new (2r+1)-diagonal matrix form is used to derive these limits.
• The scaling limit of the critical one-dimensional random Schrödinger operator
  with Evgenij Kritchevski, Benedek Valkó, Communications in Mathematical Physics 2012, Volume 314, Issue 3, pp 775-806, Eigenvectors of critical one-dimensional random Schrödinger operators are delocalized, with transfer matrices converging to a stochastic differential equation. This reveals the operator’s scaling limit behavior.
• Positive speed for high-degree automaton groups
  with Gideon Amir, Groups, Geometry, and Dynamics, 2014, Vol. 8, Issue 1, 23-38, Bounded, symmetric random walks on mother groups of degree at least 3 have positive speed. Resistance analysis in fractal mother graphs supports this result.
• The right tail exponent of the Tracy-Widom-beta distribution
  with Laure Dumaz, Ann. Inst. H. Poincaré Probab. Statist. Volume 49, Number 4 (2013), 915-1231, The Tracy-Widom β distribution’s right tail is shown to follow a specific exponential decay using the stochastic Airy operator. This provides precise asymptotics for large values.
• Patterns in Sinai's walk
  with Dimitris Cheliotis, Annals of Probability 2013, Vol. 41, No. 3B, 1900-1937, Sinai’s random walk in a random environment exhibits patterns on exponential time scales, characterized by a functional law of iterated logarithm. The rate function captures differences between one-sided and two-sided behavior.
• Limits of spiked random matrices I
  with Alex Bloemendal, Probability Theory and Related Fields 2013, 156, 3-4, pp 795-825, Top eigenvalues in rank-one spiked real Wishart matrices exhibit a phase transition with asymptotic distributions near criticality. This resolves conjectures for spiked Gaussian orthogonal ensembles.
• Random Schrödinger operators on long boxes, noise explosion and the GOE
  with Benedek Valkó, Transactions of the American Mathematical Society, 2014, Vol. 366, Issue 7, 3709-3728, Eigenvalues of random Schrödinger operators on long boxes converge to the Sine1 process under low disorder, resembling GOE statistics. This provides evidence for GOE behavior at localization transitions.
• Amenability of linear-activity automaton groups
  with Gideon Amir, Omer Angel, Journal of the European Mathematical Society, Volume 15, Issue 3, 2013, pp. 705-730, Linear-activity automaton groups are proven amenable using a random walk with zero asymptotic entropy on a mother group. This answers an open question by Nekrashevich and partially addresses Sidki’s query.
• Large gaps between random eigenvalues
  with Benedek Valkó, Annals of Probability 2010, Vol. 38, No. 3, 1263-1279, The probability of no eigenvalues in a fixed interval for β-ensembles is given by an exponential decay with a specific exponent. This quantifies large gaps in the limiting point process.
• The spectrum of the random environment and localization of noise
  with Dimitrios Cheliotis, Probab. Theory Related Fields 148 (2010), no. 1-2, 141-158, The analytic spectrum of random walk transition matrices on large-girth graphs converges to Gaussian noise after scaling. At d=2, noise localization occurs as the limit graph transitions to the integers.
• Continuum limits of random matrices and the Brownian carousel
  with Benedek Valkó, Invent. Math. 177 (2009), no. 3, 463-508, Eigenvalues of Gaussian unitary ensembles converge to the Sineβ process, described by a Brownian carousel in the hyperbolic plane. This provides a geometric interpretation of the continuum limit.
• On the girth of random Cayley graphs
  with Alex Gamburd, Shlomo Hoory, Mehrdad Shahshahani, Aner Shalev, Random Structures Algorithms 35 (2009), no. 1, 100-117, Random d-regular Cayley graphs of the symmetric group have girth at least (log_{d-1}|G|)^(1/2)/2. For simple algebraic groups, girth is at least log_{d-1}|G|/dim(G).
• Random Sorting Networks
  with Omer Angel, Alexander E. Holroyd, Dan Romik, Adv. Math. 215 (2007), no. 2, 839-868, The space-time process of swaps in random sorting networks converges to the product of semicircle law and Lebesgue measure. Particle trajectories are conjectured to converge to random sine curves.
• Complex determinantal processes and H1 noise
  with Brian Rider, Electron. J. Probab. 12 (2007), no. 45, 1238-1257, Invariant determinantal point processes with intensity ρ converge to H1 noise as ρ increases. This holds for the plane, sphere, and hyperbolic plane, with precise distributional limits.
• Beta ensembles, stochastic Airy spectrum, and a diffusion
  with José Ramírez, Brian Rider, J. Amer. Math. Soc. 24 (2011), no. 4, 919-944, The largest eigenvalues of β ensembles converge to the stochastic Airy operator’s low-lying eigenvalues. This extends the Tracy-Widom distribution to all β > 0 with precise tail estimates.
• The noise in the circular law and the Gaussian free field
  with Brian Rider, Int. Math. Res. Not. 2007, no. 2, Art. ID rnm006, 33 pp, Eigenvalues of n x n Gaussian matrices converge to a sum of H^1 and H^{1/2} noises on the unit disk and circle. This describes the circular law’s noise structure for C^1 functions.
• Determinantal Processes and Independence
  with J. Ben Hough, Manjunath Krishnapur, Yuval Peres, Probability Surveys 2006, Vol. 3, 206-229, Determinantal processes, arising in physics and combinatorics, have point counts expressible as sums of independent Bernoulli variables. This property facilitates probabilistic analysis in random matrix theory and related fields.
• Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process
  with Yuval Peres, Acta Math. 194 (2005), no. 1, 1-35, The zero set of a random Gaussian power series forms a determinantal process with joint intensity given by the Bergman kernel. The number of zeros in a disk is a sum of independent Bernoulli variables.
• Amenability via random walks
  with Laurent Bartholdi, Duke Math. J. 130 (2005), no. 1, 39–56, Amenability of groups is characterized using random walks, separating amenable groups from subexponentially growing groups. This resolves questions about group extensions and direct limits.
• Brownian beads
  Probab. Theory Related Fields 127 (2003), no. 3, 367-387, Half-plane Brownian motion’s past and future at cutpoints are independent after conformal transformation, forming a Poisson process. The path size as a function of local time is a stable subordinator.
• Dimension and randomness in groups acting on rooted trees
  with Miklós Abert, J. Amer. Math. Soc. 18 (2005), no. 1, 157-192, The asymptotic order of typical elements in p-adic automorphism groups of rooted trees is determined, addressing Turán’s question. A dimension theory for such groups relates to solvability and dense subgroups.
• Random walks that avoid their past convex hull
  with Omer Angel, Itai Benjamini, Electron. Comm. Probab. 8 (2003), 6-16, Planar random walks conditioned to avoid their past convex hull escape with positive limsup speed. Fluctuations from a limiting direction are observed to be on the order of n^(3/4).
• Fast graphs for the random walker
  Probab. Theory Related Fields 124 (2002), no. 1, 50-72, Lower bounds for the hitting time of random walks on weighted graphs are derived using volume and graph distance. These bounds are sharp and imply positive escape rates on infinite graphs.
• Anchored expansion and random walk
  Geom. Funct. Anal. 10 (2000), no. 6, 1588-1605, Graphs with anchored expansion contain subgraphs with high Cheeger constants, confirming a conjecture by Benjamini, Lyons, and Schramm. Anchored expansion implies positive lim inf speed for random walks.