arXiv subjects

Chiranjib Mukherjee

Publications and source records attributed to Chiranjib Mukherjee.

At least 19 recordsLinked to original sources

arXiv2026

Hamilton-Jacobi-Bellman Equations in Random Geometries: Homogenization on Continuum Percolation Clusters

We develop a quenched homogenization theory for optimal control problems related to Hamilton--Jacobi--Bellman equations on random geometries arising from continuum percolation. The underlying state space is the infinite connected component of a continuum percolation model conditioned to contain the origin. The relevant law of the environment is no longer translation invariant, and the geometry of the state space becomes part of the homogenization problem. The associated controlled diffusion is allowed to degenerate according to the distance to the random boundary of the cluster. The degeneracy regime is determined by a balance between a negative-moment threshold for the distance-to-boundary function of the cluster and the coercivity of the Hamiltonian. We prove that the rescaled value functions converge, locally in $L^p$ on the rescaled random domains, almost surely to a deterministic limit governed by an effective Hamiltonian. The effective Hamiltonian admits dual variational characterizations involving a class of curl-free gradients satisfying an induced mean-zero condition. The resulting effective theory retains information about the continuum percolation geometry, the degeneracy of the diffusion, and the nonstationarity induced by conditioning on the infinite component. The proof introduces a variational framework for homogenization nonstationary conditioned laws. Its main ingredients are random shifts adapted to the geometry of the cluster, a two-step min--max construction for admissible gradients, and a novel relative entropy structure intrinsic to the stochastic control representation. The latter reveals a new connection between relative entropy and effective theories for nonlinear stochastic control problems and applies equally well in the general setting of stationary ergodic random media on $\mathbb R^d$ and is therefore of independent interest.

math.AP↗

arXiv2026

Haagerup property and group-invariant percolation

Let $\mathcal G$ be the Cayley graph of a finitely generated, infinite group $Γ$. We show that $Γ$ has the Haagerup property if and only if for every $α<1$, there is a $Γ$-invariant bond percolation $\mathbb P$ on $\mathcal G$ with $\mathbb E[\mathrm{deg}_ω(g)]>α\mathrm{deg}_{\mathcal G}(g)$ for every vertex $g$ and with the two-point function $τ(g,h)=\mathbb P\big[g\leftrightarrow h\big]$ vanishing as $d(g,h)\to\infty$. On the other hand, we show that $Γ$ has Kazhdan's property (T) if and only if there exists a threshold $α^*<1$ such that for every $Γ$-invariant bond percolation $\mathbb P$ on $\mathcal G$, $\mathbb E[\mathrm{deg}_ω(o)]>α^*\mathrm{deg}(o)$ implies that the two-point function is uniformly bounded away from zero. These results in particular answer questions raised by Lyons (J. Math. Phys. 41. 1099-1126 (2000)) about characterizations of properties of groups beyond amenability through group-invariant percolations. The method of proof is new and is based on a construction of percolations with suitable dependence structures built from invariant point processes on spaces with measured walls. This construction furthermore leads to quantitative bounds on the two-point functions, exhibiting in particular exponential decay of the two-point function in several prominent examples of Haagerup groups, including co-compact Fuchsian groups, co-compact discrete subgroups of $\mathrm{Isom}(\mathbb H^n)$ and lamplighters over free groups. This method also allows us to extend the aforementioned characterization of property (T) to the setting of relative property (T) and provide an application to Bernoulli percolation at the uniqueness threshold.

math.GR↗

arXiv2026

Degenerate Diffusions on Continuum Percolation and Hamilton-Jacobi-Bellman Equations

We study degenerate controlled diffusions and Hamilton--Jacobi--Bellman equations posed on genuine continuum percolation clusters. The diffusion is constrained to evolve inside the infinite cluster and degenerates according to the distance to the irregular random boundary. Under suitable regularity and degeneracy assumptions on the diffusion matrix, we prove that the corresponding controlled diffusion never reaches the boundary and therefore admits a unique global strong solution. Using this result, we establish a stochastic control representation formula for viscosity solutions of degenerate Hamilton--Jacobi--Bellman equations posed on the random cluster, without imposing boundary conditions. We further verify that the structural assumptions introduced in this work, including quantitative integrability properties of the distance-to-the-boundary function and the associated degeneracy, hold for concrete continuum percolation models such as the Boolean model. In particular, although the diffusion never reaches the boundary, the boundary geometry remains a fundamental ingredient of the theory through the admissible degeneracy regime. The results establish a quenched framework for stochastic control and degenerate partial differential equations on genuine continuum percolation geometries, and identify a link between the analytic structure of the diffusion and the stochastic geometry of the underlying cluster. They also provide the foundational framework for the homogenization theory developed in the companion article [BMM26].

math.PR↗

arXiv2026

Coarse embeddability, $L^1$-compression and Percolations on General Graphs

We show that a locally finite, connected graph has a coarse embedding into a Hilbert space if and only if there exist bond percolations with arbitrarily large marginals and two-point function vanishing at infinity. We further show that the decay of the two-point function is stretched exponential with stretching exponent $α\in[0,1]$ if and only if the $L^1$-compression exponent of the graph is at least $α$, leading to a probabilistic characterization of this exponent. These results are new even in the particular setting of Cayley graphs of finitely generated groups. The proofs build on new probabilistic methods introduced recently by the authors to study group-invariant percolation on Cayley graphs [28,29], which are now extended to the general, non-symmetric situation of graphs to study their coarse embeddability and $L^1$-compression exponents.

math.PR↗

arXiv2026

Randomized Geodesic Flow on Hyperbolic Groups

Motivated by Gromov's geodesic flow problem on hyperbolic groups $G$, we develop in this paper an analog using random walks. This leads to a notion of a harmonic analog $Θ$ of the Bowen-Margulis-Sullivan measure on $\partial^2 G$. We provide three different but related constructions of $Θ$: 1) by moving the base-point along a quasigeodesic ray 2) by moving the base-point along random walk trajectories 3) directly as a push-forward under the boundary map to $\partial^2 G$ of a measure inherited from studying all bi-infinite random walk trajectories (with no restriction on base-point) on $G^\mathbb{Z}$. Of these, the third construction is the most involved and needs new techniques. It relies on developing a framework where we can treat bi-infinite random walk trajectories as analogs of bi-infinite geodesics on complete simply connected negatively curved manifolds. Geodesic flow on a hyperbolic group is typically not well-defined due to non-uniqueness of geodesics. We circumvent this problem in the random walk setup by considering \emph{all} trajectories. We thus get a well-defined discrete flow that we call the \emph{randomized geodesic flow}, given by the $\mathbb{Z}-$shift on bi-infinite random walk trajectories. The $\mathbb{Z}-$shift is the random analog of the time one map of the geodesic flow. As an analog of ergodicity of the geodesic flow on a closed negatively curved manifold, we establish ergodicity of the $G$-action on $(\partial^2G, Θ)$. As a consequence of our construction, we prove that the randomized geodesic flow is exponentially mixing of all orders and establish a functional CLT.

math.PR↗

arXiv2025

Self-repellent Brownian Bridges in an Interacting Bose Gas

We consider a model of $d$-dimensional interacting quantum Bose gas, expressed in terms of an ensemble of interacting Brownian bridges in a large box and undergoing the influence of all the interactions between the legs of each of the Brownian bridges. We study the thermodynamic limit of the system and give an explicit formula for the limiting free energy and a necessary and sufficient criterion for the occurrence of a condensation phase transition. For $d\geq 5$ and sufficiently small interaction, we prove that the condensate phase is not empty. The ideas of proof rely on the similarity of the interaction to that of the self-repellent random walk, and build on a lace expansion method conducive to treating {\it paths} undergoing mutual repellence within each bridge.

math.PR↗

arXiv2025

Subcritical Gaussian Multiplicative Chaos in the Wiener Space: Construction, Moments and Volume Decay

We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos \cite{K85}. Namely, if $H_T(ω)$ is a random field defined w.r.t. space-time white noise $\dot B$ and integrated w.r.t. Brownian paths in $d\geq 3$, we consider the renormalized exponential, weighted w.r.t. the Wiener measure $\mathbb P_0$. We construct the almost sure limit $μ_γ$ in the {\it entire weak disorder (subcritical)} regime and call it {\it subcritical GMC} on the Wiener space. We show that $$ μ_γ\Big\{ω: \lim_{T\to\infty} \frac{H_T(ω)}{T(ϕ\starϕ)(0)} \ne γ\Big\}=0 \qquad \mbox{almost surely,} $$ meaning, $μ_γ$ is supported only on $γ$-{\it thick paths}, and consequently, the normalized version is singular w.r.t. the Wiener measure. We characterize uniquely the limit $μ_γ$ w.r.t. the mollification scheme $ϕ$ in the sense of Shamov \cite{S14} and the random {\it rooted} measure $\mathbb Q_{μ_γ}(d\dot B dω)= μ_γ(dω,\dot B)P(d\dot B)$. We then determine the fractal properties of the measure around $γ$-thick paths: $-C_2 \leq \liminf_{r\to 0} r^2 \log \widehatμ_γ(\|ω\| < r) \leq \limsup_{r\to 0}\sup_ηr^2 \log \widehatμ_γ(\|ω-η\| < r) \leq -C_1$ w.r.t a weighted norm $\|\cdot\|$. Here $C_1>0$ and $C_2<\infty$ are the uniform upper (resp. pointwise lower) Hölder exponents which are {\it explicit} in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and $L^p$ ($p>1$) moments for the total mass of $μ_γ$ in the weak disorder regime.

math.PR↗

arXiv2025

Effective Mass of the Fröhlich Polaron and the Landau-Pekar-Spohn Conjecture

We prove that there is a constant $\overline C\in (0,\infty)$ such that the effective mass $m(α)$ of the Fröhlich Polaron satisfies $m(α) \geq \overline C α^4$, which is sharp according to a long-standing prediction of Landau-Pekar [19] from 1948 and of Spohn [36] from 1987. The method of proof, which demonstrates how the sharp quartic divergence rate of $m(α)$ appears in a natural way, is based on analyzing the Gaussian representation of the Polaron measure and that of the associated tilted Poisson point process developed in [26]. Additionally, our technique here leads to accompanying results including, 1) an explicit identification of local interval process from [26] in the strong coupling limit in terms of functionals of the Pekar process [27], 2) strict monotonicity of the effective mass $m(α)$ for all $α>0$ and 3) the quartic divergence of $m(α)$ for a generalized class of Polaron type interactions.

math-ph↗

arXiv2025

Additive-multiplicative stochastic heat equations, stationary solutions, and Cauchy statistics

We study long-term behavior and stationary distributions for stochastic heat equations forced simultaneously by a multiplicative noise and an independent additive noise with the same distribution. We prove that nontrivial space-time translation-invariant measures exist for all values of the parameters. We also show that if the multiplicative noise is sufficiently strong, the invariant measure has Cauchy-distributed marginals. Using the same techniques, we prove a similar result on Cauchy-distributed marginals for a logarithmically attenuated version of the problem in two spatial dimensions. The proofs rely on stochastic analysis and elementary potential theory.

math.PR↗

arXiv2024

Schur Multipliers of $C^*$-algebras, group-invariant compactification and applications to amenability and percolation

Let $Γ$ be a countable discrete group. Given any sequence $(f_n)_{n\geq 1}$ of $\ell^p$-normalized functions ($p\in [1,2)$), consider the associated positive definite matrix coefficients $\langle f_n, ρ(\cdot) f_n\rangle$ of the right regular representation $ρ$. We construct an orthogonal decomposition of the corresponding {\it Schur multipliers} on the reduced group $C^*$-algebra or the uniform Roe algebra of $Γ$. We identify this decomposition explicitly via the limit points of the orbits $(\widetilde f_n)_{n\geq 1}$ in the group-invariant compactification of the quotient space constructed by Varadhan and the first author in [14]. We apply this result and use positive-definiteness to provide two (quite different) characterizations of amenability of $Γ$ -- one via a variational approach and the other using group-invariant percolation on Cayley graphs constructed by Benjamini, Lyons, Peres and Schramm [1]. These results underline, from a new point of view to the best of our knowledge, the manner in which Schur multipliers capture geometric properties of the underlying group $Γ$.

math.OA↗

arXiv2022

Positive and negative moments for directed polymers in random environment in weak disorder

Very recently, Junk [11] showed that for directed polymers in bounded random environments, the weak disorder (uniform integrable) phase implies that the polymer martingale is bounded in $L^p$ for some $p>1$ and also in $L^q$ for some $q<0$. Here, we establish this characterization of the weak disorder phase without requiring the boundedness assumption on the environments.

math.PR↗

arXiv2022

Geometry of the Gaussian multiplicative chaos in the Wiener space

We develop an approach for investigating geometric properties of Gaussian multiplicative chaos (GMC) in an infinite dimensional set up. The base space is chosen to be the space of continuous functions endowed with Wiener measure, and the random field is a space-time white noise integrated against Brownian paths. In this set up, we show that in any dimension $d\geq 1$ and for any inverse temperature, the GMC-volume of a ball, uniformly around all paths, decays exponentially with an explicit decay rate. The exponential rate reflects the balance between two competing terms, namely the principal eigenvalue of the Dirichlet Laplacian and an energy functional defined over a certain compactification developed earlier in [MV14]. For $d\geq 3$ and high temperature, the underlying Gaussian field is also shown to attain very high values under the GMC -- that is, all paths are "GMC-thick" in this regime. Both statements are natural infinite dimensional extensions of similar behavior captured by $2d$ Liouville quantum gravity and reflect a certain "atypical behavior" of the GMC: while the GMC volume decays exponentially fast uniformly over all paths, the field itself attains atypically large values on all paths when sampled according to the GMC. It is also shown that, despite the exponential decay of volume for any temperature, for small enough temperature, the normalized overlap of two independent paths tends to follow one of only a finite number of independent paths for most of its allowed time horizon, allowing the GMC probability to accumulate most of its mass along such trajectories.

math.PR↗

arXiv2022

On Null-homology and stationary sequences

The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly appears quite naturally in ergodic theory in the study of measure-preserving transformations arising from various group actions or, equivalently, the study of stationary sequences when adopting a probabilistic perspective as in this paper. Our purpose is to give a new and relatively short proof of the coboundary theorem due to Schmidt (1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same \emph{null-homology equivalence class}. We also discuss various aspects of null-homology within the class of Markov random walks, compare null-homology with a formally stronger notion which we call {\it strict-sense null-homology}. Finally, we also discuss some concrete cases where the notion of null-homology turns up in a relevant manner.

math.PR↗

arXiv2021

Identification of the Polaron measure I: Fixed coupling regime and the central limit theorem for large times

We consider the Fröhlich model of the Polaron whose path integral formulation leads to the transformed path measure $$ \widehat{\mathbb P}_{α,T}(\mathrm dω)= Z_{α,T}^{-1}\,\, \exp\bigg\{\fracα{2}\int_{-T}^T\int_{-T}^T\frac{e^{-|t-s|}}{|ω(t)-ω(s)|} \, d s \, d t\bigg\}\,\mathbb P(\mathrm dω) $$ with respect to $\mathbb P$ which governs the law of the increments of the three dimensional Brownian motion on a finite interval $[-T,T]$, and $ Z_{α,T}$ is the partition function or the normalizing constant and $α>0$ is a constant. The Polaron measure reflects a self attractive interaction. According to a conjecture of Pekar that was proved in [DV83] $$ g_0=\lim_{α\to\infty}\frac{1}{α^2}\bigg[\lim_{T\to\infty}\frac{\log Z_{α,T}}{2T}\bigg] $$ exists and has a variational formula. In this article we show that for any $α>0$, the infinite-volume limit $\widehat{\mathbb P}_α=\lim_{T\to\infty}\widehat{\mathbb P}_{α,T}$ exists which is also identified explicitly. As a corollary, we deduce the central limit theorem (for any $α>0$ and as $T\to\infty$) for the distribution of $\frac{ω(T)-ω(-T)}{\sqrt{2T}}$ both under the finite-volume Polaron measure $\widehat{\mathbb P}_{α,T}$ and its infinite-volume counterpart $\widehat{\mathbb P}_α$, and obtain an expression for the limiting variance.

math.PR↗

arXiv2021

Quenched and averaged large deviation rate functions for random walks in random environments: the impact of disorder

In 2003, Varadhan [V03] developed a robust method for proving quenched and averaged large deviations for random walks in a uniformly elliptic and i.i.d. environment (RWRE) on $\mathbb Z^d$. One fundamental question which remained open was to determine when the quenched and averaged large deviation rate functions agree, and when they do not. In this article we show that for RWRE in uniformly elliptic and i.i.d. environment in $d\geq 4$, the two rate functions agree on any compact set contained in the interior of their domain which does not contain the origin, provided that the disorder of the environment is sufficiently low. Our result provides a new formulation which encompasses a set of sufficient conditions under which these rate functions agree without assuming that the RWRE is ballistic (see [Y11]), satisfies a CLT or even a law of large numbers ([Zer02,Ber08]). Also, the equality of rate functions is not restricted to neighborhoods around given points, as long as the disorder of the environment is kept low. One of the novelties of our approach is the introduction of an auxiliary random walk in a deterministic environment which is itself ballistic (regardless of the actual RWRE behavior) and whose large deviation properties approximate those of the original RWRE in a robust manner, even if the original RWRE is not ballistic itself.

math.PR↗

arXiv2021

Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation

We consider a class of Gibbs measures defined with respect to increments $\{ω(t)-ω(s)\}_{s 0$) and unbounded (singular) interactions (including singularities of the form $x\mapsto 1/|x|^p$ in $d\geq 3$ or $x\mapsto δ_0(x)$ in $d=1$) attached to the space variables. These assumptions on the interaction seem to be sharp and cover quantum mechanical models like the Nelson model and the polaron problem with ultraviolet cut off (both carrying bounded spatial interactions with power law decay in time) as well as the Fröhlich polaron with a short range interaction in time but carrying Coulomb singularity in space. In this set up, we develop a unified approach for proving a central limit theorem for the rescaled process of increments for any coupling parameter and obtain an explicit expression for the limiting variance which is strictly positive. As a further application, we study the solution of the multiplicative-noise stochastic heat equation in spatial dimensions $d\geq 3$. When the noise is mollified both in time and space, we show that the averages of the diffusively rescaled solutions converge pointwise to the solution of a diffusion equation whose coefficients are homogenized in this limit.

math.PR↗

arXiv2021

Identification of the Polaron measure in strong coupling and the Pekar variational formula

The path measure corresponding to the Fröhlich Polaron appearing in quantum statistical mechanics is defined as the tilted measure $$ \widehat{\mathbb P}_{\varepsilon,T}= \frac{1}{Z(\varepsilon,T)}\exp\bigg(\frac{1}{2}\int_{-T}^T\int_{-T}^T \frac{\varepsilon\mathrm e^{-\varepsilon |t-s|}}{|ω(t)-ω(s)|} \mathrm d s \,\mathrm d t\bigg)\mathrm d\mathbb P. $$ Here $\varepsilon>0$ is the Kac parameter (or the inverse-coupling), and $\mathbb P$ is the law of $3d$ Brownian increments. In [13] it was shown that the (thermodynamic) limit $\lim_{T\to\infty}\widehat{\mathbb P}_{\varepsilon,T}=\widehat{\mathbb P}_\varepsilon$ exists as a process with stationary increments and this limit was identified explicitly as a mixture of Gaussian processes. In the present article, the strong coupling limit or the vanishing Kac parameter limit $\lim_{\varepsilon\to 0} \widehat{\mathbb P}_\varepsilon$ is investigated. It is shown that this limit exists and coincides with the increments of the Pekar process, which is a stationary diffusion process with generator $\frac 12 Δ+ (\nablaψ/ψ)\cdot \nabla$, where $ψ$ is the unique (modulo shifts) maximizer of the Pekar variational problem $$ g_0=\sup_{\|ψ\|_2=1} \Big\{\int_{\mathbb R^3}\int_{\mathbb R^3}\,ψ^2(x) ψ^2(y)|x-y|^{-1}\mathrm d x\mathrm d y -\frac 12\|\nabla ψ\|_2^2\Big\}. $$ As shown in [12,6,1], the Pekar process is itself approximated by the limiting "mean-field Polaron measures", and thus, the present identification of the strong coupling Polaron is a rigorous justification of the "mean-field approximation" (on the level of path measures) conjectured by Spohn in [15]. This approximation in the vanishing Kac limit ($\varepsilon\to 0$) is also shown to hold for a general class of Kac-Interaction of the form $H(t,x)=\varepsilon \mathrm e^{-\varepsilon|t|} V(x)$ where $V$ is any continuous function vanishing at infinity.

math.PR↗

arXiv2021

Space-time fluctuation of the Kardar-Parisi-Zhang equation in $d\geq 3$ and the Gaussian free field

We study the solution $h_\varepsilon$ of the Kardar-Parisi-Zhang (KPZ) equation for $d \geq 3$: $$ \frac{\partial}{\partial t} h_{\varepsilon} = \frac12 Δh_{\varepsilon} + \bigg[\frac12 |\nabla h_\varepsilon |^2 - C_\varepsilon\bigg]+ β\varepsilon^{\frac{d-2}2} ξ_{\varepsilon} $$ with $h_\varepsilon(0,x)=0$. Here $ξ_\varepsilon=ξ\star ϕ_\varepsilon$ is a spatially smoothened (at scale $\varepsilon$) Gaussian space-time white noise and $C_\varepsilon$ is a divergent constant as $\varepsilon\to 0$. When the disorder $β$ is sufficiently small and $\varepsilon\to 0$, $h_\varepsilon(t,x)- \mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)\to 0$ in probability where $\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)$ is the {\emph stationary solution} of the KPZ equation - more precisely, $\mathfrak h^{\mathrm{st}}_{\varepsilon}$solves the above equation with a random initial condition (that is independent of the driving noise $ξ$) and its law is constant in $(\varepsilon,t,x)$. In the present article we quantify the rate of the above convergence in this regime and show that the fluctuation {\emph about} the stationary solution $$ (\varepsilon^{1-\frac d2} [h_\varepsilon(t,x) - \mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)])_{x,t} $$ converges pointwise (with finite dimensional distributions in space and time) to a Gaussian free field (GFF) evolved by the deterministic heat equation. We also identify the fluctuations {\it of} the stationary solution itself and show that the rescaled averages $\int_{\mathbb R^d} {\mathrm d} x φ(x) \varepsilon^{1-\frac d2} [\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)- \mathbb E(\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x))]$ converge to that of the {\emph stationary solution} of the stochastic heat equation with additive noise, but with (random) {\emph GFF marginals} (instead of flat initial condition).

math.PR↗