About

Nina Holden is an Associate Professor at the Courant Institute of Mathematical Sciences at New York University. She completed her PhD in mathematics at the Massachusetts Institute of Technology in 2018 under the supervision of Scott Sheffield. Her research focuses on probability theory and mathematical physics, including topics such as Liouville quantum gravity, Schramm-Loewner evolutions, random planar maps, and statistical physics. Prior to her current position, she was a Junior Fellow and postdoctoral researcher at ETH Zurich, working in the group of Wendelin Werner. Her academic background also includes a Master in Mathematics from the University of Oslo and a visiting student experience at Oxford University. She is actively involved in teaching, with a scheduled course on Theory of Probability in Spring 2026.

Research topics

• Artificial Intelligence
• Computer Science
• Algorithm

Selected publications

• Circle packing and Riemann uniformization of random planar maps in an ergodic scale-free environment

  Open MIND · 2026-03-06

  preprint1st authorCorresponding

  We prove that embedded infinite planar maps in ergodic scale-free environments are close to their circle packing and Riemann uniformization embedding on a large scale, as long as suitable moment and connectivity conditions are satisfied. Ergodic scale-free environments were earlier considered by Gwynne, Miller and Sheffield (2018) in the context of the invariance principles for random walk, and they arise naturally in the study of random planar maps and Liouville quantum gravity.

  DOI

• Regularity of the Schramm–Loewner evolution: Up-to-constant variation and modulus of continuity

  The Annals of Probability · 2026-01-01

  preprintOpen access1st authorCorresponding

  We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm–Loewner evolution (SLE): variation regularity, modulus of continuity and law of the iterated logarithm. For the latter two, we consider the SLE with its natural parametrization. More precisely, denoting by d∈(0,2] the dimension of the curve, we show the following: 1. The optimal ψ-variation is ψ(x)=xd(loglogx−1)−(d−1) in the sense that η is a.s. of finite ψ-variation for this ψ and not for any function decaying more slowly as x↓0. 2. The optimal modulus of continuity is ω(s)=cs1/d(logs−1)1−1/d, that is, for some random c>0 we have |η(t)−η(s)|≤ω(t−s) a.s., while this does not hold for any function ω decaying faster as s↓0. 3. lim supt↓0|η(t)|(t1/d(loglogt−1)1−1/d)−1 is a.s. equal to a deterministic constant in (0,∞). We also show that the natural parametrization of SLE is given by the fine mesh limit of the ψ-variation. As part of our proof, we show that every stochastic process whose increments satisfy a particular moment condition attains a certain variation regularity.

  PublisherDOI

• Regularity of the Schramm–Loewner evolution: Up-to-constant variation and modulus of continuity

  The Annals of Probability · 2026-01-01

  articleOpen access1st authorCorresponding
  PublisherDOI

• Circle packing and Riemann uniformization of random planar maps in an ergodic scale-free environment

  ArXiv.org · 2026-03-06

  articleOpen access1st authorCorresponding

  We prove that embedded infinite planar maps in ergodic scale-free environments are close to their circle packing and Riemann uniformization embedding on a large scale, as long as suitable moment and connectivity conditions are satisfied. Ergodic scale-free environments were earlier considered by Gwynne, Miller and Sheffield (2018) in the context of the invariance principles for random walk, and they arise naturally in the study of random planar maps and Liouville quantum gravity.

  PublisherOA PDF

• Conformal welding of quantum disks and multiple SLE: the non-simple case

  Probability Theory and Related Fields · 2025-09-27

  article
  PublisherDOI

• Liouville quantum gravity: from random planar maps to conformal field theory

  ArXiv.org · 2025-10-18

  preprintOpen access1st authorCorresponding

  Originating in theoretical physics, Liouville quantum gravity (LQG) has been an important topic in probability theory and mathematical physics in the past two decades. In this proceeding, we review two aspects of this topic. The first is that LQG describes the random conformal geometry of the scaling limit of random planar maps. We highlight the convergence of random planar maps under discrete conformal embedding, where couplings between LQG and the Schramm-Loewner evolution (SLE) play a key role. The second aspect is the connection to conformal field theory (CFT). Here we highlight the interplay between Liouville CFT and the SLE/LQG coupling, the CFT description of 2D quantum gravity coupled with conformal matter, and applications to SLE and 2D statistical physics. We conclude with several open questions and future directions.

  PublisherOA PDFDOI

• What is the effect of measurable respiratory muscle training on respiratory muscle strength in mechanically ventilated adults in intensive care units? A systematic review and meta-analysis

  Australian Critical Care · 2025-09-23 · 2 citations

  reviewOpen access

  BACKGROUND: Respiratory muscle weakness, associated with mechanical ventilation during critical illness, is well established. Respiratory muscle strength training (RMST) including inspiratory muscle training (IMT) and expiratory muscle strength training (EMST) aims to address this weakness. The aim of this systematic review and meta-analysis was to assess the effectiveness of RMST, delivered using measurable load devices, to increase respiratory muscle strength in mechanically ventilated adults in the intensive care unit. METHODS: Conducted per Preferred Reporting Items for Systematic reviews and Meta-Analyses guidelines, the review included randomised controlled trials of intensive care unit patients aged ≥16 years, ventilated ≥24 h, receiving RMST (IMT or EMST) via measurable load devices before extubation, published from January 2000 to January 2024. Preoperative/postoperative training and cohorts with other causes of respiratory weakness were excluded. Searches covered electronic databases, clinical registers, reference lists, and SCOPUS. Meta-analyses and sensitivity and subgroup analyses were performed using Cochrane Review Manager (RevMan). Risk of bias (RoB2) and Grading of Recommendations, Assessment, Development, and Evaluation tools were applied to assess respiratory muscle strength. RESULTS: = 78%) were also observed. CONCLUSION: This systematic review supports the use of IMT delivered using measurable load devices, initiated during mechanical ventilation, in critical care patients, to increase MIP measures. Other potentially positive effects found in this review such as reduced weaning and mechanical ventilation durations in response to IMT need further confirmation. REGISTRATION: This protocol was registered with the International Prospective Register of Systematic Reviews (CRD42023431244).

  PublisherDOI

• Liouville quantum gravity weighted by conformal loop ensemble nesting statistics

  Probability and Mathematical Physics · 2024-07-18 · 1 citations

  articleOpen access1st authorCorresponding

  We study Liouville quantum gravity (LQG) surfaces whose law has been reweighted according to nesting statistics for a conformal loop ensemble (CLE) relative to n ∈ {0, 1, 2, . . .} marked points z1, . . ., zn.The idea is to consider a reweighting by B⊆[n] e σ B N B , where σB ∈ R and NB is the number of CLE loops surrounding the points zi for i ∈ B. This is made precise via an approximation procedure where as part of the proof we derive strong spatial independence results for CLE.The reweighting induces logarithmic singularities for the Liouville field at z1, . . ., zn with a magnitude depending explicitly on σ1, . . ., σn.We define the partition function of the surface and explain its relationship to Liouville conformal field theory (CFT) correlation functions and a potential relationship with a CFT for CLE.In the case of n ∈ {0, 1} points we derive an explicit formula for the partition function.Furthermore, we obtain a recursive formula for the partition functions where we express the n point partition function in terms of the partition function for disks with k < n marked points, and we use this to partially determine for which values of (σB : B ⊆ [n]) the partition function is finite.The recursive formula is derived via a continuum counterpart of the peeling process on planar maps, which was earlier studied in works of Miller, Sheffield, and Werner in the setting of n = 0 marked points.We also find an explicit formula for the generator of the Markov process describing the LQG boundary lengths in the continuum peeling process.Via an explicit calculation for this Markov process for n = 0 we give a new proof for the law of the conformal radius of the outermost CLE loop in the unit disk around 0, which was earlier established by Schramm, Sheffield, and Wilson.Finally, we state precise conjectures relating the LQG surfaces with marked points to random planar maps.

  PublisherDOI

• Conformal welding of quantum disks

  Electronic Journal of Probability · 2023-01-01 · 7 citations

  articleOpen access

  Two-pointed quantum disks with a weight parameter W>0 are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their boundary lengths gives another quantum disk decorated with an independent chordal SLEκ(ρ−;ρ+) curve. This is the finite-volume counterpart of the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces called quantum wedges, which is fundamental to the mating-of-trees theory. Our results can be used to give unified proofs of the mating-of-trees theorems for the quantum disk and the quantum sphere, in addition to a mating-of-trees description of the weight W=γ22 quantum disk. Moreover, it serves as a key ingredient in our companion work, which proves an exact formula for SLEκ(ρ−;ρ+) using conformal welding of random surfaces and a conformal welding result giving the so-called SLE loop.

  PublisherOA PDFDOI

• Conformal welding of quantum disks and multiple SLE: the non-simple case

  arXiv (Cornell University) · 2023-10-31 · 2 citations

  preprintOpen access

  Two-pointed quantum disks with a weight parameter $W&gt;0$ is a canonical family of finite-volume random surfaces in Liouville quantum gravity. We extend the conformal welding of quantum disks in [AHS23] to the non-simple regime, and give a construction of the multiple SLE associated with any given link pattern for $κ\in(4,8)$. Our proof is based on connections between SLE and Liouville conformal field theory (LCFT), where we show that in the conformal welding of multiple forested quantum disks, the surface after welding can be described in terms of LCFT, and the random conformal moduli contains the SLE partition function for the interfaces as a multiplicative factor. As a corollary, for $κ\in(4,8)$, we prove the existence of the multiple SLE partition functions, which are smooth functions satisfying a system of PDEs and conformal covariance.

  PublisherOA PDFDOI

Frequent coauthors

• Xin Sun

  38 shared

• Ewain Gwynne

  University of Chicago

  17 shared

• Yuval Peres

  Beijing Institute of Mathematical Sciences and Applications

  17 shared

• Christophe Garban

  Centre National de la Recherche Scientifique

  8 shared

• Alex Zhai

  Stanford University

  7 shared

• Morris Ang

  7 shared

• Jason Miller

  University of Cambridge

  6 shared

• Xinyi Li

  6 shared

Labs

• Theory of ProbabilityPI

Education

• B.A.

  University of Oslo, Norway

  2008

• M.S.

  University of Oslo, Norway

  2010

• Ph.D.

  Massachusetts Institute of Technology

  2018