About

Amarjit Budhiraja is a professor whose research focuses on stochastic dynamical systems, large deviations, and probabilistic analysis of complex systems. His work involves the study of asymptotic behaviors, rare events, and the mathematical modeling of stochastic processes, with applications spanning various fields such as ecology, statistics, and network theory. Throughout his career, he has co-advised numerous students and postdoctoral fellows, contributing significantly to the advancement of probability theory and stochastic analysis.

Research topics

• Computer Science
• Data Mining
• Mathematics
• Mathematical optimization
• Mathematical analysis
• Statistics

Selected publications

• Errata to “Many-server asymptotics for join-the-shortest-queue: Large deviations and rare events”

  The Annals of Applied Probability · 2026-02-01

  article1st authorCorresponding
  PublisherDOI

• Level 2.5 large deviations and uncertainty relations for non-Markov self-interacting dynamics

  ArXiv.org · 2026-01-12

  articleOpen access

  We address the general problem of formulating the dynamical large deviations of non-Markovian systems in a closed form. Specifically, we consider a broad class of ``self-interacting'' jump processes whose dynamics depends on the past through a functional of a state-dependent empirical observable. Exploiting a natural separation of timescales, we obtain the exact (so-called ``level 2.5'') large deviation joint statistics of the empirical measure over configurations and of the empirical flux of transitions. As an application of this general framework, we derive explicit general bounds on the fluctuations of trajectory observables, generalising to the non-Markovian case both thermodynamic and kinetic uncertainty relations. We illustrate our theory with simple examples, and discuss potential applications of these results.

  PublisherOA PDF

• Level 2.5 large deviations and uncertainty relations for non-Markov self-interacting dynamics

  arXiv (Cornell University) · 2026-01-12

  preprintOpen access

  We address the general problem of formulating the dynamical large deviations of non-Markovian systems in a closed form. Specifically, we consider a broad class of ``self-interacting'' jump processes whose dynamics depends on the past through a functional of a state-dependent empirical observable. Exploiting a natural separation of timescales, we obtain the exact (so-called ``level 2.5'') large deviation joint statistics of the empirical measure over configurations and of the empirical flux of transitions. As an application of this general framework, we derive explicit general bounds on the fluctuations of trajectory observables, generalising to the non-Markovian case both thermodynamic and kinetic uncertainty relations. We illustrate our theory with simple examples, and discuss potential applications of these results.

  PublisherDOI

• Fluctuations of the Atlas model from inhomogeneous stationary profiles

  UNC Libraries · 2026-04-30

  articleOpen access1st authorCorresponding

  The infinite Atlas model describes the evolution of a countable collection of Brownian particles on the real line, where the lowest particle is given a drift of γ∈[0,∞). The case γ=0 is referred to as the Harris model. In this work we study equilibrium fluctuations for the Atlas model for γ∈[0,∞) when the system of particles starts from an inhomogeneous stationary profile with exponentially growing density. We show that the appropriately centered and scaled occupation measure of the particle positions, with suitable translations, viewed as a space-time random field, converges to a limit that can be characterized in terms of a certain stochastic partial differential equation (SPDE). The initial condition for this equation is given by a Brownian motion, the equation is driven by an additive space-time noise that is white in time and colored in space, and the linear operator governing the evolution is the infinitesimal generator of a geometric Brownian motion. We use this SPDE to also characterize the fluctuations of the ranked particle positions with a suitable centering and scaling. Our results describe the behavior of the particles in the bulk, and one finds that the Gaussian process describing the asymptotic fluctuations has the same Hölder regularity as a fractional Brownian motion with Hurst parameter 1/4. This connection with a fractional Brownian motion becomes even more exact when the inhomogeneous stationary profiles approach the homogeneous stationary profile in a suitable manner. One finds that, unlike the setting of a homogeneous profile (Dembo and Tsai (Ann. Probab.45 (2017) 4529–4560)), the behavior on the lower edge of the particle system is very different from the bulk behavior, and in fact, the variance of the Gaussian limit diverges to ∞ as one approaches the lower edge. Indeed, our results show that, with the gaps between particles given by one of the inhomogeneous stationary distributions, the lowest particle, started from 0, with a linear in time translation, converges in distribution to an explicit non-Gaussian limit as t→∞. In the special case of the Harris model, this limit is given as the law of the difference of two independent Gumbel distributions.

  PublisherDOI

• Long Time Asymptotics for the Stochastic Follow-the-Leader System

  ArXiv.org · 2026-01-05

  articleOpen access

  We introduce and analyze a class of interacting particle systems on the real line that combine features of the stochastic rat race and (deterministic) follow-the-leader models. The particle system evolves as a continuous-time pure jump process: the leading particle moves independently, at Exponential jump times, with constant jump rate and iid jump sizes distributed according to a law $θ$, while each of the remaining particles jumps forward, at Exponential times, at rate equal to its distance from the particle immediately ahead, with jump sizes drawn uniformly from the corresponding gap. The dynamics thus encode competition for leadership together with distance-dependent stochastic interactions. Our main focus is the associated gap process, representing the vector of inter-particle distances. We establish the existence of a unique stationary distribution for the gap process and prove uniform geometric ergodicity. Further, when the leader's jump sizes follow an Exponential distribution, we identify the stationary law explicitly as a product of independent Exponential laws, and show that the associated mixing time scales between $Θ(n)$ and $O(n(\log n)^2)$ for an $n$-particle system. As an application of the mixing time results we establish a functional limit theorem that characterizes fluctuations of particle states at large time, under a suitable spatial and temporal scaling and large particle limit. Finally, when the leader's jumps have heavy but integrable tails, we show that each gap has at least one additional finite moment under stationarity than that of the leader's jump size distribution. The model offers a tractable setting for exploring ergodicity, explicit invariant laws, and mixing behavior in non-diffusive particle systems.

  PublisherOA PDF

• Fluctuations of the Atlas model from inhomogeneous stationary profiles

  The Annals of Probability · 2026-04-21

  preprintOpen access

  The infinite Atlas model describes the evolution of a countable collection of Brownian particles on the real line, where the lowest particle is given a drift of $γ\in [0,\infty)$. We study equilibrium fluctuations for the Atlas model when the system of particles starts from an inhomogeneous stationary profile with exponentially growing density. We show that the appropriately centered and scaled occupation measure of the particle positions, with suitable translations, viewed as a space-time random field, converges to a limit given by a certain stochastic partial differential equation (SPDE). The initial condition for this equation is given by a Brownian motion, the equation is driven by an additive space-time noise that is white in time and colored in space, and the linear operator governing the evolution is the infinitesimal generator of a geometric Brownian motion. We use this SPDE to also characterize the fluctuations of the ranked particle positions with a suitable centering and scaling. Our results describe the behavior of the particles in the bulk and one finds that the Gaussian process describing the asymptotic fluctuations has the same Hölder regularity as a fractional Brownian motion with Hurst parameter $1/4$. One finds that, unlike the setting of a homogeneous profile (Dembo and Tsai (2017)), the behavior on the lower edge of the particle system is very different from the bulk behavior and in fact the variance of the Gaussian limit diverges to $\infty$ as one approaches the lower edge. Indeed, our results show that, with the gaps between particles given by one of the inhomogeneous stationary distributions, the lowest particle, started from $0$, with a linear in time translation, converges in distribution to an explicit non-Gaussian limit as $t\to \infty$.

  PublisherOA PDFDOI

• Large deviation asymptotics for the supermarket model with growing choices

  Quarterly of Applied Mathematics · 2026-01-02

  article1st authorCorresponding

  We consider the Markovian supermarket model with growing choices, where jobs arrive at rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n lamda Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:msub> <mml:mi> λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">n\lambda _{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and each of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> parallel servers processes jobs in its queue at rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Each incoming job joins the shortest among <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript n Baseline element-of StartSet 1 comma ellipsis comma n EndSet"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">d_{n} \in \{1,\dotsc ,n\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> randomly selected queues. Under the assumption <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript n Baseline right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">d_{n} \to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript n Baseline right-arrow lamda element-of left-parenthesis 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false"> → </mml:mo> <mml:mi> λ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda _{n} \to \lambda \in (0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , a large deviation principle (LDP) for the occupancy process is established in a suitable infinite-dimensional path space, and it is shown that the rate function is invariant with respect to the manner in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript n Baseline right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">d_{n} \to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The LDP gives information on the rate of decay of probabilities of various types of rare events associated with the system. We illustrate this by establishing explicit exponential decay rates for probabilities of large total number of jobs in the system. As a corollary, we also show that probabilities of certain rare events can indeed depend on the rate of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript n Baseline right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false"> → </mml

  PublisherDOI

• Long Time Asymptotics for the Stochastic Follow-the-Leader System

  arXiv (Cornell University) · 2026-01-05

  preprintOpen access

  We introduce and analyze a class of interacting particle systems on the real line that combine features of the stochastic rat race and (deterministic) follow-the-leader models. The particle system evolves as a continuous-time pure jump process: the leading particle moves independently, at Exponential jump times, with constant jump rate and iid jump sizes distributed according to a law $θ$, while each of the remaining particles jumps forward, at Exponential times, at rate equal to its distance from the particle immediately ahead, with jump sizes drawn uniformly from the corresponding gap. The dynamics thus encode competition for leadership together with distance-dependent stochastic interactions. Our main focus is the associated gap process, representing the vector of inter-particle distances. We establish the existence of a unique stationary distribution for the gap process and prove uniform geometric ergodicity. Further, when the leader's jump sizes follow an Exponential distribution, we identify the stationary law explicitly as a product of independent Exponential laws, and show that the associated mixing time scales between $Θ(n)$ and $O(n(\log n)^2)$ for an $n$-particle system. As an application of the mixing time results we establish a functional limit theorem that characterizes fluctuations of particle states at large time, under a suitable spatial and temporal scaling and large particle limit. Finally, when the leader's jumps have heavy but integrable tails, we show that each gap has at least one additional finite moment under stationarity than that of the leader's jump size distribution. The model offers a tractable setting for exploring ergodicity, explicit invariant laws, and mixing behavior in non-diffusive particle systems.

  PublisherDOI

• Diffusion limits in the quarter plane and nonsemimartingale reflected Brownian motion

  The Annals of Applied Probability · 2025-10-01

  articleSenior author
  PublisherDOI

• Load Balancing in Parallel Queues and Rank-Based Diffusions

  Mathematics of Operations Research · 2025-05-27 · 2 citations

  article

  Consider a queuing system with K parallel queues in which the server for each queue processes jobs at rate n and the total arrival rate to the system is [Formula: see text], where [Formula: see text] and n is large. Interarrival and service times are taken to be independent and exponentially distributed. It is well known that the join-the-shortest-queue (JSQ) policy has many desirable load-balancing properties. In particular, in comparison with uniformly at random routing, the time asymptotic total queue-length of a JSQ system, in the heavy traffic limit, is reduced by a factor of K. However, this decrease in total queue-length comes at the price of a high communication cost of order [Formula: see text] because at each arrival instant, the state of the full K-dimensional system needs to be queried. In view of this, it is of interest to study alternative routing policies that have lower communication costs and yet have similar load-balancing properties as JSQ. In this work, we study a family of such rank-based routing policies, which we will call Marginal Size Bias Load-Balancing policies, in which [Formula: see text] of the incoming jobs are routed to servers with probabilities depending on their ranked queue length and the remaining jobs are routed uniformly at random. A particular case of such routing schemes, referred to as the marginal JSQ (MJSQ) policy, is one in which all the [Formula: see text] jobs are routed using the JSQ policy. Our first result provides a heavy traffic approximation theorem for such queuing systems in terms of reflected diffusions in the positive orthant [Formula: see text]. It turns out that, unlike the JSQ system, where, due to a state space collapse, the heavy traffic limit is characterized by a one-dimensional reflected Brownian motion, in the setting of MJSQ (and for the more general rank-based routing schemes), there is no state space collapse, and one obtains a novel diffusion limit which is the constrained analogue of the well-studied Atlas model (and other rank-based diffusions) that arise from certain problems in mathematical finance. Next, we prove an interchange of limits ([Formula: see text] and [Formula: see text]) result which shows that, under conditions, the steady state of the queuing system is well approximated by that of the limiting diffusion. It turns out that the latter steady state can be given explicitly in terms of product laws of Exponential random variables. Using these explicit formulae, and the interchange of limits result, we compute the time asymptotic total queue-length in the heavy traffic limit for the MJSQ system. We find the striking result that, although in going from JSQ to MJSQ, the communication cost is reduced by a factor of [Formula: see text], the steady-state heavy traffic total queue-length increases by at most a constant factor (independent of n, K) which can be made arbitrarily close to one by increasing a MJSQ parameter. We also study the case where the system is overloaded—namely, [Formula: see text]. For this case, we show that although the K-dimensional MJSQ system is unstable, unlike the setting of random routing, the system has certain desirable and quantifiable load-balancing properties. In particular, by establishing a suitable interchange of limits result, we show that the steady-state difference between the maximum and the minimum queue lengths stays bounded in probability (in the heavy traffic parameter n). Funding: Financial support from the National Science Foundation [RTG Award DMS-2134107] is gratefully acknowledged. S. Banerjee received financial support from the National Science Foundation [NSF-CAREER Award DMS-2141621]. A. Budhiraja received financial support from the National Science Foundation [Grant DMS-2152577].

  PublisherDOI

Recent grants

• Nonlinear Markov processes, large weakly interacting particle systems, and applications

  NSF · $300k · 2013–2018

• Asymptotics for Particle Systems with Topological Interactions

  NSF · $330k · 2022–2026

• Estimating Probabilities of Rare Events in Interacting Particle Systems

  NSF · $165k · 2019–2023

• RTG: Networks: Foundations in Probability, Optimization, and Data Sciences

  NSF · $1.9M · 2022–2027

• Optimization and Equilibria with Expectation Functions: Analysis, Inference and Sampling

  NSF · $272k · 2018–2022

Frequent coauthors

• Paul Dupuis

  95 shared

• Ruoyu Wu

  31 shared

• Anugu Sumith Reddy

  Applied Mathematics (United States)

  26 shared

• Amit Apte

  26 shared

• Rami Atar

  25 shared

• Sayan Banerjee

  18 shared

• Arnab Ganguly

  15 shared

• Sergio A. Almada Monter

  Princeton University

  14 shared

Labs

• Amarjit BudhirajaPI