Two manifestations of rigidity phenomena in
random point sets :forbidden regions and maximal degeneracy
Subhro Ghosh
National University of Singapore

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Point processes and rigidity
The most popular model of random point sets is perhaps the
Poisson point process,

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Point processes and rigidity
The most popular model of random point sets is perhaps the
Poisson point process, which is characterized by spatial
independence.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Point processes and rigidity
The most popular model of random point sets is perhaps the
Poisson point process, which is characterized by spatial
independence.
But some of the most scientifically interesting models of
random point sets are strongly correlated,

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Point processes and rigidity
The most popular model of random point sets is perhaps the
Poisson point process, which is characterized by spatial
independence.
But some of the most scientifically interesting models of
random point sets are strongly correlated, and in fact many ofthem exhibit repulsion.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Point processes and rigidity
The most popular model of random point sets is perhaps the
Poisson point process, which is characterized by spatial
independence.
But some of the most scientifically interesting models of
random point sets are strongly correlated, and in fact many ofthem exhibit repulsion. E.g., GUE eigenvalues, zeros ofrandom polynomials, etc.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Point processes and rigidity
The most popular model of random point sets is perhaps the
Poisson point process, which is characterized by spatial
independence.
But some of the most scientifically interesting models of
random point sets are strongly correlated, and in fact many ofthem exhibit repulsion. E.g., GUE eigenvalues, zeros ofrandom polynomials, etc.
The question of spatial conditioning, therefore, becomes a
non-trivial one in these models.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Point processes and rigidity
The most popular model of random point sets is perhaps the
Poisson point process, which is characterized by spatial
independence.
But some of the most scientifically interesting models of
random point sets are strongly correlated, and in fact many ofthem exhibit repulsion. E.g., GUE eigenvalues, zeros ofrandom polynomials, etc.
The question of spatial conditioning, therefore, becomes a
non-trivial one in these models.
Namely, given a domain D, how does the point configuration
outside of D impact the distribution of the points inside D ?

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Point processes and rigidity
The most popular model of random point sets is perhaps the
Poisson point process, which is characterized by spatial
independence.
But some of the most scientifically interesting models of
random point sets are strongly correlated, and in fact many ofthem exhibit repulsion. E.g., GUE eigenvalues, zeros ofrandom polynomials, etc.
The question of spatial conditioning, therefore, becomes a
non-trivial one in these models.
Namely, given a domain D, how does the point configuration
outside of D impact the distribution of the points inside D ?
It turns out that such spatial conditioning leads to remarkablesingularities in the distribution of the points inside thedomain.
Subhro Ghosh National University of Singapore

Rigidity Phenomena

Point processes and rigidity
The most popular model of random point sets is perhaps the
Poisson point process, which is characterized by spatial
independence.
But some of the most scientifically interesting models of
random point sets are strongly correlated, and in fact many ofthem exhibit repulsion. E.g., GUE eigenvalues, zeros ofrandom polynomials, etc.
The question of spatial conditioning, therefore, becomes a
non-trivial one in these models.
Namely, given a domain D, how does the point configuration
outside of D impact the distribution of the points inside D ?
It turns out that such spatial conditioning leads to remarkablesingularities in the distribution of the points inside thedomain. Roughly speaking, this is what we refer to as rigidity.
Subhro Ghosh National University of Singapore

Rigidity Phenomena

Instances of rigidity

The most basic instance of rigidity is the rigidity of particle
numbers.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Instances of rigidity

The most basic instance of rigidity is the rigidity of particle
numbers.
Rigidity of particle numbers basically means that the numberof particles in a bounded domain is a (deterministic) functionof the particle configuration outside the domain.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Instances of rigidity

The most basic instance of rigidity is the rigidity of particle
numbers.
Rigidity of particle numbers basically means that the numberof particles in a bounded domain is a (deterministic) functionof the particle configuration outside the domain.
So, this amounts to a local law of conservation of mass : weare not allowed to perturb the point configuration in waysthat create new particles or delete existing ones !

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Instances of rigidity

The most basic instance of rigidity is the rigidity of particle
numbers.
Rigidity of particle numbers basically means that the numberof particles in a bounded domain is a (deterministic) functionof the particle configuration outside the domain.
So, this amounts to a local law of conservation of mass : weare not allowed to perturb the point configuration in waysthat create new particles or delete existing ones !
This has implications in the study of stochastic geometry onthese point processes,

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Instances of rigidity

The most basic instance of rigidity is the rigidity of particle
numbers.
Rigidity of particle numbers basically means that the numberof particles in a bounded domain is a (deterministic) functionof the particle configuration outside the domain.
So, this amounts to a local law of conservation of mass : weare not allowed to perturb the point configuration in waysthat create new particles or delete existing ones !
This has implications in the study of stochastic geometry onthese point processes, notably in the use of Burton and Keane
type arguments, or the “finite energy” property.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Instances of rigidity

Rigidity of particle numbers has been shown to occur for the
Dyson log gas [G.] and the critical 2D Coulomb gas [G. Peres].

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Instances of rigidity

Rigidity of particle numbers has been shown to occur for the
Dyson log gas [G.] and the critical 2D Coulomb gas [G. Peres].
Rigidity of particle numbers was also established for the zerosof the planar Gaussian analytic function [G. - Peres]f (z) =

Subhro Ghosh National University of Singapore∞Xzkξk √ .
k!
k=0

Rigidity Phenomena

Instances of rigidity

In subsequent works, rigidity of particle numbers wasestablished for a variety of determinantal point processes(with projection kernels), particularly in the works of Bufetov,
Qiu, Osada, Shirai ...

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Instances of rigidity

In subsequent works, rigidity of particle numbers wasestablished for a variety of determinantal point processes(with projection kernels), particularly in the works of Bufetov,
Qiu, Osada, Shirai ... These include the Airy, Bessel andGamma kernel processes, determinantal processes associatedwith generalized Fock spaces, and so forth.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Instances of rigidity

In subsequent works, rigidity of particle numbers wasestablished for a variety of determinantal point processes(with projection kernels), particularly in the works of Bufetov,
Qiu, Osada, Shirai ... These include the Airy, Bessel andGamma kernel processes, determinantal processes associatedwith generalized Fock spaces, and so forth.
Projection kernel in the above is necessary ! [G.-Krishnapur]

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Rigidity of general obervables
In general, for a point process Π and a bounded domain D, let
us denote by Πin the point configuration inside D, and by
Πout the point configuration outside D.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Rigidity of general obervables
In general, for a point process Π and a bounded domain D, let
us denote by Πin the point configuration inside D, and by
Πout the point configuration outside D.
The observable χ(Πin ) is said to be rigid if χ(Πin ) is adeterministic function of Πout .

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Rigidity of general obervables
In general, for a point process Π and a bounded domain D, let
us denote by Πin the point configuration inside D, and by
Πout the point configuration outside D.
The observable χ(Πin ) is said to be rigid if χ(Πin ) is adeterministic function of Πout .
An important class of examples are linear statistics:X
χ(Πin ) =
ϕ(λ)
λ∈Πin

for some function ϕ.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Rigidity of general obervables
In general, for a point process Π and a bounded domain D, let
us denote by Πin the point configuration inside D, and by
Πout the point configuration outside D.
The observable χ(Πin ) is said to be rigid if χ(Πin ) is adeterministic function of Πout .
An important class of examples are linear statistics:X
χ(Πin ) =
ϕ(λ)
λ∈Πin

for some function ϕ. Setting ϕ = 1D gives the number ofpoints in D.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Rigidity of general obervables
In general, for a point process Π and a bounded domain D, let
us denote by Πin the point configuration inside D, and by
Πout the point configuration outside D.
The observable χ(Πin ) is said to be rigid if χ(Πin ) is adeterministic function of Πout .
An important class of examples are linear statistics:X
χ(Πin ) =
ϕ(λ)
λ∈Πin

for some function ϕ. Setting ϕ = 1D gives the number ofpoints in D.
Natural to ask about rigidity of more general functionals of apoint process (other than the particle count), particularly
higher moments of the points in D.
Subhro Ghosh National University of Singapore

Rigidity Phenomena

Rigidity of general obervables

Consider zero process the family of Gaussian analytic functions
fα (z) =∞X
k=0

Subhro Ghosh National University of Singapore

ξk

zk
.
(k!)α/2

Rigidity Phenomena

Rigidity of general obervables

Consider zero process the family of Gaussian analytic functions
fα (z) =

∞
X
k=0

ξk

zk
.
(k!)α/2

α = 1 recovers the standard planar case.
1], the first m moments of the zero process
For α ∈ ( m1 , m−1
are rigid. [G.-Krishnapur]

Subhro Ghosh National University of Singapore

Rigidity Phenomena

General picture

Rigidity of particle numbers is connected with suppressed
fluctuation of particle numbers (o(Volume)).

Subhro Ghosh National University of Singapore

Rigidity Phenomena

General picture

Rigidity of particle numbers is connected with suppressed
fluctuation of particle numbers (o(Volume)).
Rigidity of general observables connected with suppressed
fluctuation of other linear statistics.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

General picture

Rigidity of particle numbers is connected with suppressed
fluctuation of particle numbers (o(Volume)).
Rigidity of general observables connected with suppressed
fluctuation of other linear statistics.
Rigidity is also connected with faster decay of holeprobabilities

Subhro Ghosh National University of Singapore

Rigidity Phenomena

General picture

Rigidity of particle numbers is connected with suppressed
fluctuation of particle numbers (o(Volume)).
Rigidity of general observables connected with suppressed
fluctuation of other linear statistics.
Rigidity is also connected with faster decay of holeprobabilities and singularity of Palm measures

Subhro Ghosh National University of Singapore

Rigidity Phenomena

General picture(Moment-matching) [G.] Consider a point process Π having
precisely the first m moments rigid, and two collections ofpoints ζ = (ζ1 , · · · , ζk ) and η = (η1 , · · · , ηl ). Then Palm
measures [Π]ζ and [Π]η are mutually absolutely continuous iff
the first m moments of ζ and η match,

Subhro Ghosh National University of Singapore

Rigidity Phenomena

General picture(Moment-matching) [G.] Consider a point process Π having
precisely the first m moments rigid, and two collections ofpoints ζ = (ζ1 , · · · , ζk ) and η = (η1 , · · · , ηl ). Then Palm
measures [Π]ζ and [Π]η are mutually absolutely continuous iff
the first m moments of ζ and η match, and the two Palm
measures are mutually singular otherwise.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

General picture(Moment-matching) [G.] Consider a point process Π having
precisely the first m moments rigid, and two collections ofpoints ζ = (ζ1 , · · · , ζk ) and η = (η1 , · · · , ηl ). Then Palm
measures [Π]ζ and [Π]η are mutually absolutely continuous iff
the first m moments of ζ and η match, and the two Palm
measures are mutually singular otherwise.
However, very few rigorous theorems establishing generalimplications like the above between these concepts.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Conditioning on a large hole

We say that the disk D is a hole if there are no particles insideD.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Conditioning on a large hole

We say that the disk D is a hole if there are no particles insideD.
We look at the conditional distribution of points outside D
given that D is hole.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Conditioning on a large hole

We say that the disk D is a hole if there are no particles insideD.
We look at the conditional distribution of points outside D
given that D is hole.
When radius(D) → ∞, how does the outside configuration
behave ?

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Conditioning on a large hole

We say that the disk D is a hole if there are no particles insideD.
We look at the conditional distribution of points outside D
given that D is hole.
When radius(D) → ∞, how does the outside configuration
behave ?
In other words, what causes a large hole (a rare event) tooccur ?

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Conditioning on a large hole: the Ginibre ensemble
This question was investigated by Jancovici, Lebowitz andManificat for the 2D Coulomb gas.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Conditioning on a large hole: the Ginibre ensemble
This question was investigated by Jancovici, Lebowitz andManificat for the 2D Coulomb gas. What they showed was :Ginibre Ensemble

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Conditioning on a large hole: the GUE processThis question was investigated by Majumdar, Nadal,
Scardicchio and Vivo for the Dyson log gas.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Conditioning on a large hole: the GUE processThis question was investigated by Majumdar, Nadal,
Scardicchio and Vivo for the Dyson log gas. What they
showed was :GUESubhro Ghosh National University of Singapore

Rigidity Phenomena

Appearance of a “Forbidden region” in Gaussian zeros
We consider this problem for the zeros of the standard planar
Gaussian analytic function.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Appearance of a “Forbidden region” in Gaussian zeros
We consider this problem for the zeros of the standard planar
Gaussian analytic function.
We show that :Gaussian Zeros
Subhro Ghosh National University of Singapore

Rigidity Phenomena

Appearance of a “Forbidden region” in Gaussian zeros

Theorem (G.- Nishry)
The conditional intensity for zeroes of Gaussian randompolynomials has the following behaviour:

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Appearance of a “Forbidden region” in Gaussian zeros

Theorem (G.- Nishry)
The conditional intensity for zeroes of Gaussian randompolynomials has the following behaviour:
There is a singular component at the edge of the hole

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Appearance of a “Forbidden region” in Gaussian zeros

Theorem (G.- Nishry)
The conditional intensity for zeroes of Gaussian randompolynomials has the following behaviour:
There is a singular component at the edge of the hole
There is subsequent “forbidden region”, namely, in the√annulus R < r < eR, the conditional intensity → 0 as
R → ∞.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Appearance of a “Forbidden region” in Gaussian zeros

Theorem (G.- Nishry)
The conditional intensity for zeroes of Gaussian randompolynomials has the following behaviour:
There is a singular component at the edge of the hole
There is subsequent “forbidden region”, namely, in the√annulus R < r < eR, the conditional intensity → 0 as
R → ∞.
√
Beyond eR, the conditional intensity behaves, in the limit
R → ∞, like the equilibrium intensity.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Appearance of a “Forbidden region” in Gaussian zeros

Theorem (G.- Nishry)
The conditional intensity for zeroes of Gaussian randompolynomials has the following behaviour:
There is a singular component at the edge of the hole
There is subsequent “forbidden region”, namely, in the√annulus R < r < eR, the conditional intensity → 0 as
R → ∞.
√
Beyond eR, the conditional intensity behaves, in the limit
R → ∞, like the equilibrium intensity.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Forbidden region

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Forbidden region

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics

Large deviations for (empirical measures of) zeros of (thepolynomial truncations of) the Gaussian analytic function(inspired by Zelditch-Zeitouni)

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics

Large deviations for (empirical measures of) zeros of (thepolynomial truncations of) the Gaussian analytic function(inspired by Zelditch-Zeitouni) If Z is a (the empiricalmeasure of) a configuration of zeros, thenP(Z ) ≈ exp(−R 4 I (Z )).

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics

Large deviations for (empirical measures of) zeros of (thepolynomial truncations of) the Gaussian analytic function(inspired by Zelditch-Zeitouni) If Z is a (the empiricalmeasure of) a configuration of zeros, thenP(Z ) ≈ exp(−R 4 I (Z )). I is the LDP rate function.
No zeros in the hole D is the same as Z (D) = 0.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics

Large deviations for (empirical measures of) zeros of (thepolynomial truncations of) the Gaussian analytic function(inspired by Zelditch-Zeitouni) If Z is a (the empiricalmeasure of) a configuration of zeros, thenP(Z ) ≈ exp(−R 4 I (Z )). I is the LDP rate function.
No zeros in the hole D is the same as Z (D) = 0.
To find the “most likely configuration” given that there is holeis roughly the same as minimizing the rate functional I overthe space of probability measures (on C) under the constraintthat there is zero mass on D.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics
Constrained optimization problem on the space of probability
measures.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics
Constrained optimization problem on the space of probability
measures.
The functional to be optimized is highly non-smooth :|z|2− Σ(µ) − C ,I (µ) = 2 sup Uµ (z) −2z∈C
where Uµ is the logarithmic potential and Σ(µ) is thelogarithmic energy of the measure µ and C is a constant.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics
Constrained optimization problem on the space of probability
measures.
The functional to be optimized is highly non-smooth :|z|2− Σ(µ) − C ,I (µ) = 2 sup Uµ (z) −2z∈C
where Uµ is the logarithmic potential and Σ(µ) is thelogarithmic energy of the measure µ and C is a constant. No
clear variational method available.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics
Constrained optimization problem on the space of probability
measures.
The functional to be optimized is highly non-smooth :|z|2− Σ(µ) − C ,I (µ) = 2 sup Uµ (z) −2z∈C
where Uµ is the logarithmic potential and Σ(µ) is thelogarithmic energy of the measure µ and C is a constant. No
clear variational method available. Tackled by “guessing” thesolution and then establishing that it is indeed the minimizer
using potential theoretic methods.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics
Constrained optimization problem on the space of probability
measures.
The functional to be optimized is highly non-smooth :|z|2− Σ(µ) − C ,I (µ) = 2 sup Uµ (z) −2z∈C
where Uµ is the logarithmic potential and Σ(µ) is thelogarithmic energy of the measure µ and C is a constant. No
clear variational method available. Tackled by “guessing” thesolution and then establishing that it is indeed the minimizer
using potential theoretic methods.
Heuristics made rigorous by obtaining “effective” versions oflarge deviation estimates and approximating the analytic
function zeros by those of the polynomials.
Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Recently, stealthy particle systems (and more generally,stealthy random fields) have gained significant attention incondensed matter physics, c.f. works of Torquato, Stillinger,
Batten, Zhang, Chertkov, Car, DiStasio ...

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Recently, stealthy particle systems (and more generally,stealthy random fields) have gained significant attention incondensed matter physics, c.f. works of Torquato, Stillinger,
Batten, Zhang, Chertkov, Car, DiStasio ...
Stealthy ⇐⇒ the spectrum of the process

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Recently, stealthy particle systems (and more generally,stealthy random fields) have gained significant attention incondensed matter physics, c.f. works of Torquato, Stillinger,
Batten, Zhang, Chertkov, Car, DiStasio ...
Stealthy ⇐⇒ the spectrum of the process (i.e., the Fouriertransform of the two-point correlation)

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Recently, stealthy particle systems (and more generally,stealthy random fields) have gained significant attention incondensed matter physics, c.f. works of Torquato, Stillinger,
Batten, Zhang, Chertkov, Car, DiStasio ...
Stealthy ⇐⇒ the spectrum of the process (i.e., the Fouriertransform of the two-point correlation) has a “gap”, namely itvanishes in a neighbourhood of the origin.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Recently, stealthy particle systems (and more generally,stealthy random fields) have gained significant attention incondensed matter physics, c.f. works of Torquato, Stillinger,
Batten, Zhang, Chertkov, Car, DiStasio ...
Stealthy ⇐⇒ the spectrum of the process (i.e., the Fouriertransform of the two-point correlation) has a “gap”, namely itvanishes in a neighbourhood of the origin.
Nomenclature “stealthy” because such systems are invisible todiffraction experiments with waves having frequency inside the“gap”.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Recently, stealthy particle systems (and more generally,stealthy random fields) have gained significant attention incondensed matter physics, c.f. works of Torquato, Stillinger,
Batten, Zhang, Chertkov, Car, DiStasio ...
Stealthy ⇐⇒ the spectrum of the process (i.e., the Fouriertransform of the two-point correlation) has a “gap”, namely itvanishes in a neighbourhood of the origin.
Nomenclature “stealthy” because such systems are invisible todiffraction experiments with waves having frequency inside the“gap”.
Stealthy particle systems conjectured to have deterministicallybounded holes [Zhang-Stillinger-Torquato].

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Theorem (G.-Lebowitz)
Stealthy random fields (i.e., random fields with a spectralgap) exhibit maximal rigidity : namely, the process inside abounded domain is a deterministic function of the process
outside the domain.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Theorem (G.-Lebowitz)
Stealthy random fields (i.e., random fields with a spectralgap) exhibit maximal rigidity : namely, the process inside abounded domain is a deterministic function of the process
outside the domain.
Same conclusion holds if, instead of having a gap, the spectral
density decays fast enough (faster than any polynomial) at theorigin.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Theorem (G.-Lebowitz)
Stealthy random fields (i.e., random fields with a spectralgap) exhibit maximal rigidity : namely, the process inside abounded domain is a deterministic function of the process
outside the domain.
Same conclusion holds if, instead of having a gap, the spectral
density decays fast enough (faster than any polynomial) at theorigin.
Special case : Guassian process with a gap (or fast decay) in the
spectrum

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Theorem (G.-Lebowitz)(Bounded holes) Holes in a stealthy particle system arebounded deterministically

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Theorem (G.-Lebowitz)(Bounded holes) Holes in a stealthy particle system arebounded deterministically with a universal upper bound that isinversely proportional to the size of the spectral gap.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Theorem (G.-Lebowitz)(Bounded holes) Holes in a stealthy particle system arebounded deterministically with a universal upper bound that isinversely proportional to the size of the spectral gap.
(Anti-concentration) The particle number in a domain isbounded deterministically

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Theorem (G.-Lebowitz)(Bounded holes) Holes in a stealthy particle system arebounded deterministically with a universal upper bound that isinversely proportional to the size of the spectral gap.
(Anti-concentration) The particle number in a domain isbounded deterministically by (a constant multiple of) the
expected number of points in the domain.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Stealthy random fields

Theorem (G.-Lebowitz)(Bounded holes) Holes in a stealthy particle system arebounded deterministically with a universal upper bound that isinversely proportional to the size of the spectral gap.
(Anti-concentration) The particle number in a domain isbounded deterministically by (a constant multiple of) the
expected number of points in the domain.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Heuristics

The existence of a gap / fast decay in the spectrum can be
exploited to construct linear functionals of the process whichhave low variance.
A linear functional with a low variance is approximately
constant, so this gives an approximate linear constraint
Sufficiently rich class of such constraints can be exploited todeduce degenerate behaviour.

Subhro Ghosh National University of Singapore

Rigidity Phenomena

Thank you !!

Subhro Ghosh National University of Singapore

Rigidity Phenomena