This document is a dissertation proposal by Rishideep Roy at the University of Chicago in November 2014. The proposal is to generalize results on extreme values and entropic repulsion for two-dimensional discrete Gaussian free fields to a more general class of Gaussian fields with logarithmic correlations. Specifically, the proposal plans to find the convergence in law of the maximum of these log-correlated Gaussian fields under minimal assumptions, as well as obtain finer estimates on entropic repulsion which relates to the behavior of these fields near hard boundaries. The proposal provides background on related works and outlines the key steps to be taken, including proving expectations and tightness of maxima, invariance of maximum distributions under perturbations, approximating the fields, and

1. Dissertation Proposal: Extreme values of log-correlated Gaussian
fields
Rishideep Roy
University of Chicago
November 11, 2014
Abstract
Extreme values and entropic repulsion for two-dimensional discrete Gaussian free fields are
of significant interest and have been a subject of many recent works. I plan on generalizing
results on extreme values for a general class of Gaussian fields with logarithmic correlations, of
which the Gaussian membrane model at the critical dimension is a particular example. I also
plan on finding finer estimates on entropic repulsion of GFF which has previously computed
upto the level of exponent of highest order term.
1 Introduction
The convergence in law for the maximum of various log-correlated Gaussian fields (including branch-
ing Brownian motion, branching random walk, two-dimensional discrete Gaussian free field, etc.)
have been topics intensive research. The question arises from dealing with entropic repulsion of
the discrete Gaussian free field as talked about in [3], which deals with the behavior of the field
when pressed against a hard wall. The maxima of the discrete Gaussian free field arises as a
natural statistics to look into while dealing with this question. In fact the tails of the maxima
are directly related to the entropic repulsion of the discrete Gaussian free field leading to further
research into the asymptotics of it’s maxima. An important work towards this direction is the work
of [7] where they establish the relationship between the covariance structure of the discrete Gaus-
sian free field(DGFF) with that of the Modified Branching Random walk(MBRW) which helped
to make further studies into the maxima of the DGFF using results of [12] about Gaussian fields.
Of greatest relevance to the work are [5, 10, 1, 6, 11]. The papers [5, 10] obtained the convergence
in law for the maximum of Brancing Brownian Motion where [5] made use of a connection with
the KPP-equation [8], and [10] relied on the notion of derivative martingale introduced therein.
These results helped in establishing the tightness for the maxima of the DGFF after appropriate
centering by obtaining bounds on the maxima and the order of the right tail of the same. Recently,
the convergence for general Branching Random Walk under very mild assumptions was established
in [1]. Even more recently, the convergence for two-dimensional DGFF was proved in [6] suing
finer estimates on the right tail of the maxima of DGFF. For a general class of log-correlated fields
(whose covariances admit a certain kernel representation) was established in [11]. In the current
work, we aim at the convergence in law for the maximum of discrete log-correlated Gaussian fields
under minimal assumptions possible, which will include all previous examples like the Gaussian
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2. free field and the class of fields talked about in [11] as well as the Gaussian Membrane model in
the critical dimension four whose maxima and entropic repulsion has been researched upon in [9].
The second part of the work would be to look into the question of entropic repulsion of the
DGFF where we try to approximate the probability that the field is positive everywhere on a
discrete box in two dimensions. The exponent of the highest order term has been established in [3]
but we aim at establishing that it is infact slightly smaller in order, by looking at the exponent of
the second order term. The path that we wish to tread is first trying to solve this for a Branching
Random Walk and then try to extend our result for a Modified Branching Random Walk, which
can be further modified for the DGFF in dimension two by using approximations arising out of the
similarity in their covariance structures.
1.1 Assumptions for the log-correlated gaussian field
Fix d ∈ N and we let VN = Zd
N be the d-dimensional box of side length N with the left bottom
corner located at the origin. For convenience, we will consider a suitably normalized version of
Gaussian fields, under which we say a sequence of Gaussian fields {ϕN
v : v ∈ VN } is log-correlated
if and only if there exists a constant α > 0 such that
(A.1) | Cov(ϕN
v , ϕN
u ) − (log N − log(|u − v| ∨ 1))| ≤ α where | · | denotes the Euclidean norm.
It is clear that (A.1) on its own cannot ensure the convergence in law for the maximum, since
the oscillation for the covariances in the O(1) term will result in oscillation for the maximum even
for its expectation. Therefore, it is plausible to further pose assumptions on the convergence of
covariances as follows.
(A.2) There exist a continuous function f : (0, 1)d → R and a function g : Zd × Zd → R such that
the following holds. For all > 0, there exists K = K( ) such that for all L ∈ N, x ∈ (0, 1)d,
u, v ∈ [0, L]d and N ≥ LK we have
| Cov(ϕN
xN+v, ϕN
xN+u) − log N
L − f(x) − g(u, v)| < .
(A.3) There exists a continuous function h : ((0, 1)d × (0, 1)d) {(x, x) : x ∈ (0, 1)d} → R such that
the following holds. For all > 0, there exists L = L( ) > 0 such that for all |x − y| ≥ 1
K and
N ≥ LK we have
| Cov(ϕN
xN , ϕN
yN ) − h(x, y)| < .
The assumptions (A.2) and (A.3) appear to be technical, but essentially they are saying that after
suitable centering (if necessary) the covariances for the fields in the macroscopic level (where the
distance between two vertices is linear in N) and microscopic level (where the distances between two
vertices is O(1)) converge to continuous functions. Note that we allow additive constant oscillations
for covariances of two vertices u, v whose distance is in the middle level (i.e., 1 |u − v| N).
While it is the case that one does not exactly need (A.2) and (A.3) in order to have convergence
in law for the maximum, we remark that if the covariances in the macroscopic or microscopic level
oscillate across the field, then in generic situations the law of the maximum will not converge.
Furthermore, we note that to our best knowledge the Gaussian fields satisfying (A.1), (A.2) and
(A.3) cover all the instances of discrete log-correlated Gaussian fields that have been studied so far,
including two-dimensional DGFF as well as the log-correlated fields studied in [11].
Denote by MN = maxv∈VN
φN
v and mN = EMN . Our main result is the following theorem.
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3. Theorem 1.1. Under Assumption (A.1), we have that
mN =
√
2d log N −
3
2
√
2d
log log N + O(1) , (1)
and (MN − mN ) is tight. Furthermore, with additional Assumptions (A.2) and (A.3), we have that
(MN − mN ) converges in law.
In addition, one could go through our proof and find that (MN − mN ) converges to a mixture
of Gumble. Alternatively, a very nice machinery employed in [2] will rigorously and immediately
give the Gumble description of the limiting law.
2 Expectation and tightness for the maximum
This section is devoted to the proof of (1) and the tightness of (MN − mN ). In [7], the expected
maximum for 2D DGFF was computed up to additive constant. The only assumption that was
used about the DGFF is that the covariance is logarithmic up to additive constant, which is exactly
(after normalization) equivalent to our assumption (A.1). With covariances being logarithmic, the
arguments in [7] carries out to any dimension automatically, thereby yielding (1). The tightness
is somewhat trickier, as [7] builds upon [4] in which a more subtle property about DGFF was
employed. To this end, we compute the order of the right tail for the maximum, which will be used
later for other purposes.
Theorem 2.1. Under Assumption (A.1), there exists a constant C > 0 depending only on (α, d)
such that for all λ ∈ [1,
√
log N],
C−1
λe−
√
2dλ
≤ P(MN > mN + λ) ≤ Cλe−
√
2dλ
.
As a complimentary result that will be useful later, we also give an upper bound on the left
tail. We note that it is impossible to give a sharp estimate on the left tail in the general setup
under consideration, as witnessed by drastically different left tails for the maximum of a Branching
random walk and the discrete Gaussian free field. As such, our goal is merely to show that the left
tail decays at least at an exponential rate.
Lemma 2.2. There exist constants C, c > 0 (depending only on α, d) so that for all n ∈ N and
0 λ (log n)2/3
P(max
v∈VN
ϕN
v mN − λ) Ce−cλ
.
Our proofs will rely on Slepian’s comparison lemma for Gaussian process [12] which is stated
below:
Lemma 2.3. Let A be an arbitrary finite index set and let {Xa : a ∈ A} and {Ya : a ∈ A}
be two centered Gaussian processes such that: E(Xa − Xb)2 ≥ E(Ya − Yb)2, for all a, b ∈ A and
Var(Xa) = Var(Ya) for all a ∈ A. Then P(maxa∈A Xa ≥ λ) ≥ P(maxa∈A Ya ≥ λ) for all λ ∈ R.
The tightness of (MN − mN ) follows immediately from (1) and Theorem 2.1.
The main reason that log-correlated Gaussian fields are tractable is that it admits a tree struc-
ture, at least approximately with high precision. In light of this, it is useful in the analysis is to
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4. compare a log-correlated field with a branching random walk, or a modified branching random
walk introduced in [7]. The usual Branching Random Walk cannot be directly used and needs to
be modified as the mutli-scale analysis of the Guassian fields which gives the tree structure tends
to place points placed very close by, in different boxes leading to them being almost independent.
Modified Branching Random walk settles that issue, thereby allowing for comparison of right tails
using 2.3. For the upper bound on the Left tail too, we use 2.3 after computing a bound on Left
tail of MBRW, which is done using tightness for recentered MBRW and it’s approximation by a
Gaussian field which has block diagonal covariance structure, the blocks coming from independent
MBRWs.
3 Invariance for law of the maximum under suitable perturbation
The next step we follow in order to achieve our goal is looking at the stability of the distribution of
the maxima under perturbation. We can show that under perturbations using Gaussian variables
at fine and coarse levels leave the distribution unchanged. This invariance property will facilitate us
in constructing a new field that approximates our target field in Section 4. So we can work with this
modified log-correlated Gaussian field instead of the original which we denote as { ˜ϕN
v : v ∈ VN }.
We start with a number of notations.In order to show this invariance we make use of another result
that large peaks of the field are either located very close to each other or are very far from each
other. Another geometric property of the maxima is that the chance that the maxima of the field
in the box lies inside a very small box of size negligible as compared to the original box, is in turn
negligible.
4 An approximation of the log-correlated Gaussian field
In this section, we approximate the log-correlated Gaussian field. More precisely, we will approxi-
mate the modified log-correlated Gaussian field, as the modification facilitates the approximation.
To this end, we divide the field into three scales: (1) The top scale which mainly features covariances
between ˜ϕN
u and ˜ϕN
v with |u − v| N; (2) The bottom scale which mainly features covariances
between ˜ϕN
u and ˜ϕN
v with |u − v| 1; (3) The middle scale which mainly features covariances
between ˜ϕN
u and ˜ϕN
v with 1 |u − v| N. By Assumptions (A.2) and (A.3), the covariances
of { ˜ϕN
u : u ∈ VN } converge in the top and bottom scale. So in the top and bottom scales, we
approximate { ˜ϕN
u } by its corresponding “limit” fields. In the middle scale, we simply approximate
{ ˜ϕN
u } by an MBRW. One can then expect that our approximation gives an additive o(1) error for
covariances in top and bottom scale, and an additive O(1) error in the middle scale. It turns out
that this guarantees that the limiting laws of the re-centered maxima coincide.
5 The Limit Process
In order to prove our required convergence in distribution we first construct a limit process. In
order to construct this we consider a hypercube of finite boundary, say one. From our assumption
about convergence of the covariance functions at coarser levels which is the top scale we divide the
hypercube into smaller hypercubes. Corresponding to each of these hypercubes we construct an
independent random variable whose distribution depends on the distribution of the maxima of the
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5. finer part of the constructed Gaussian field in the previous section. The finer part implies the part
removing the top level. We prove the following result about the right tail of the maxima of the
finer part, which we call as {Xf
v : v ∈ VN } :
Proposition 5.1. There exists a constant α∗ > 0 such that
lim
z→∞
lim sup
L→∞
lim sup
N→∞
|z−1
e
√
2dz
P(max
v∈VN
Xf
v ≥ m ˜N + z) − α∗
| = 0.
This helps us in precisely estimating the fine field and thereby constructing our new random
variables inside the smaller hypercubes. To each of this we add the part arising out of the top level
convergence of covariance functions. Then taking the maxima of all of these random variables we
get our required limit process.
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