Sagyn Omirbekov

Hossein Davarzani

Stéfan Colombano

A Ahmadi-Senichault

Non-Newtonian behaviour of foam flow in porous media: experiments and upscaling

Keywords: foam flow, porous media, non-Newtonian fluid, yield stress, upscaling

teaching and research institutions in France or abroad, or from public or private research centers.

Foam injection in porous media has been investigated for a variety of applications in the oil industry, especially, for Enhanced Oil Recovery (EOR). Foam injection technics developed for EOR can also be applied to soil remediation processes to remove Non-Aqueous Phase Liquids (NAPL) from aquifers. The primary purpose of foam injection in soil remediation operations is to control the permeability of porous media to divert the flow from high to low permeable zones in aquifers. There are some major differences between oil reservoirs and aquifers. For instance, porous media in oil reservoirs are mainly low permeable and consolidated, and under high pressure and temperature conditions. While, polluted aquifers are mainly unconsolidated and highly permeable. As a consequence of these differences, in-situ foam generation in aquifers is questionable, and also foam flow in aquifers calls upon further investigation.

Generally, understanding of foam flow in porous media is hampered due to complex behaviour of foam and apparent contradictions in foam studies. For example, some authors indicated Newtonian [START_REF] Persoff | A Laboratory Investigation of Foam Flow in Sandstone at Elevated Pressure[END_REF] behaviour of foam flowing in porous media, others shear-thinning [START_REF] Hirasaki | Mechanisms of Foam Flow in Porous Media: Apparent Viscosity in Smooth Capillaries[END_REF], shear-thickening [START_REF] Patzek | Kern River Steam-Foam Pilots[END_REF], a mixture of Newtonian and shear-thinning [START_REF] Vassenden | Experimental Foundation for Relative Permeability Modeling of Foam[END_REF] or dependent on foam quality (foam gas volume fraction) [START_REF] Alvarez | Unified Model for Steady-State Foam Behavior at High and Low Foam Qualities[END_REF]. In addition, it must be noted that most studies presented in the literature concern low permeable media for EOR applications.

The objective of this study is to determine, experimentally, the rheological behaviour of bulk foam and to investigate foam flow in high permeable unconsolidated porous media by performing numerous laboratory experiments and finally to scale-up, theoretically and/or numerically, the foam flow from pore to Darcy scales.

At first, the rheological parameters of the bulk foam, pre-generated using a fine sand-packed column, were measured by a rheometer. The bulk foam for 85 % foam quality was found to have a non-Newtonian yield stress fluid behaviour, which could be correctly represented by the Herschel-Bulkley model. Then, the pregenerated foam has been injected into fine sand and also calibrated glass beads packed columns (4 cm internal diameter, 40 cm height). In each experiment, a fixed quality foam is injected for different total flow rates, where gas and surfactant solution rates are changed simultaneously. Pressure drops are measured over the 1D column for each flow rate, thus allowing to investigate the macroscopic flow rate (Q) -pressure gradient (P) relationship. The macroscopic flow of 85% and below foam quality for various unconsolidated porous media clearly leads to Q(P) featuring a yield stress behaviour.

The experimental results are the basis to validate foam upscaling in porous media, where foam was considered as a single-phase Herschel-Bulkley fluid. This assumption is valid when the bubble size is much smaller than the pore size. The closure problem obtained by homogenization from the general case of non-Newtonian fluids flow in porous media [START_REF] Orgéas | Upscaling the flow of generalised Newtonian fluids through anisotropic porous media[END_REF] is solved numerically for yield stress fluids for different 2D and 3D unit cell geometries.

In this talk, experimental and numerical results will be presented, and the experimental data will be compared to the upscaled results.


Dan Betea

Jérémie Bouttier

Peter Nejjar

Mirjana Vuletić

New edge asymptotics of skew Young diagrams via free boundaries

We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik-Deift-Johansson and Baik-Rains in resolving Ulam's problem on longest increasing subsequences of random permutations and the last passage percolation (corner growth) discrete versions thereof. Moreover they interpolate between said measures and the uniform measure on partitions. In the new KPZ-like 1/3 exponent edge scaling limit with logarithmic corrections, we find new probability distributions generalizing the classical Tracy-Widom GUE, GOE and GSE distributions from the theory of random matrices.

Introduction and main results

Background. The poissonized Plancherel [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF] and the discrete corner growth [START_REF] Johansson | Shape fluctuations and random matrices[END_REF] measures are two probability measures on integer partitions coming from the study of longest increasing subsequences of random permutations, respectively directed last passage percolation in an N × N square-LPP-models with iid geometric weights. Both (cf. op. cit.) are known to exhibit KPZ N 1/3 fluctuation behavior at the edge, with the Tracy-Widom GUE distribution [START_REF] Tracy | Level-spacing distributions and the Airy kernel[END_REF] from random matrix theory as the limiting distribution. Baik and Rains [START_REF] Baik | Algebraic aspects of increasing subsequences[END_REF][START_REF] Baik | The asymptotics of monotone subsequences of involutions[END_REF] have considered symmetrized versions of both, with similar results except now the limiting distributions are the Tracy-Widom GOE and GSE distributions [START_REF] Tracy | Matrix kernels for the Gaussian orthogonal and symplectic ensembles[END_REF] from random matrix theory-and some interpolating ones. In all cases the weight of a partition is proportional to the (possibly square of) number of (semi-) standard Young tableaux of said shape. Such measures are part of a large class of determinantal/pfaffian measures called Schur measures [START_REF] Okounkov | Infinite wedge and random partitions[END_REF]-see [START_REF] Borodin | Eynard-Mehta theorem, Schur process, and their Pfaffian analogs[END_REF] for the pfaffian case.

In this note we generalize the above poissonized Plancherel and discrete Schur measures to skew Young diagrams-i.e. counting (semi-) standard Young tableaux of skew shape-coming from pairs (or triples) of partitions with free ends. In the N1/3 scaling limit we find that the behavior of the first part of one (any) of the partitions is governed by new generalizations of the classical Tracy-Widom distributions. We finally give a directed LPP interpretation of our models in terms of directed polymers in a reflecting strip. We call these models LPP on a tie.

Outline. Below we describe the main result; its relation to LPP in Section 2; the new distribution functions we obtain as limits in Section 3; and a sketch of proof in Section 4. Full details will appear elsewhere [START_REF] Betea | The Free Boundary Schur Process and Applications II[END_REF].

Main results. For two integer partitions µ ⊂ λ (i.e. µ i ≤ λ i ∀i), let us denote by f λ/µ , respectively f n, λ/µ , the number of standard Young tableaux of skew shape λ/µ filled with numbers 1, 2, . . . , |λ/µ| := |λ| -|µ|, respectively the number of semi-standard Young tableaux of shape λ/µ filled with 1, . . . , n. Let or(λ) (respectively oc(λ)) denote the number of odd rows (respectively odd columns) of a partition λ. Fix positive real parameters u, q < 1, a 1 , a 2 , b 1 , b 2 , v ≤ 1 1 ,2 , and consider the following measures on pairs/triples of partitions:

M , x (µ, λ) ∝ ∆ x (µ, λ) • u |µ| • |λ/µ| f λ/µ |λ/µ|! , M , x (µ, λ, ν) ∝ ∆ x (µ, ν) • u |µ| v |ν| • |λ/µ|+|λ/ν| f λ/µ f λ/ν |λ/µ|! • |λ/ν|! , M , x (µ, λ) ∝ ∆ x (µ, λ) • u |µ| • q |λ/µ| f n, λ/µ , M , x (µ, λ, ν) ∝ ∆ x (µ, ν) • u |µ| v |ν| • q |λ/µ|+|λ/ν| f n, λ/µ f n, λ/ν (1.1)
where x ∈ {aa, ab, bb, -} is a boundary label for the boundary partitions (-stands for absence of boundary parameters); where

∆ x (µ, λ) =            a oc(µ) 1 a oc(λ) 2 , x = aa, a oc(µ) 1 b or(λ) 2 , x = ab, b or(µ) 1 b or(λ) 2 , x = bb, 1, x = -; (1.2)
and where the partition functions making each into a probability measure can be explicitly computed using the methods of [START_REF] Betea | The Free Boundary Schur Process and Applications I[END_REF].

They will be referred to as the upwards (for ) and up-down (for ) free-boundary poissonized Plancherel (for M) and free-boundary geometric corner growth (for M) measures.

The upwards measures M become the "symmetrized" poissonized Plancherel measures studied in [START_REF] Baik | Algebraic aspects of increasing subsequences[END_REF][START_REF] Baik | The asymptotics of monotone subsequences of involutions[END_REF] when u = 0. The up-down measures M become the classical poissonized Plancherel measure from [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF] when u = v = 0. Moreover they all become the uniform measure on partitions when → 0. Thus they interpolate poissonized Plancherel ↔ uniform 3 . A similar remark holds for the M measures, replacing "poissonized Plancherel" with Johansson's corner growth [START_REF] Johansson | Shape fluctuations and random matrices[END_REF] (for M , ••• when u = v = 0) and the respective symmetrized versions [START_REF] Baik | Algebraic aspects of increasing subsequences[END_REF] (for M , ••• when u = 0).

Our main result is a generalization of those in [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF][START_REF] Baik | The asymptotics of monotone subsequences of involutions[END_REF], and can be seen as a pfaffian analogue of [START_REF] Betea | The Periodic Schur Process and Free Fermions at Finite Temperature[END_REF]Theorem 1.1]. We concentrate on the large scale behavior of λ 14 as , n → ∞ while all other parameters go to 1 in a suitable critical regime.

Theorem 1.1. Fix η, α i , β i , i = 1, 2 positive reals. Let M := 1-u 2 → ∞ and set u = v = exp(-ηM -1/3 ) and (a i , b i ) = u α i /η , u β i /η i = 1, 2 all going to 1 as M → ∞. (In particular, ∼ M 2/3 → ∞.) We have: lim M→∞ M , x λ 1 -2M M 1/3 ≤ s + 1 η log M 1/3 η = F 1;d x (s), lim M→∞ M , x λ 1 -2M M 1/3 ≤ s + 1 2η log M 1/3 2η = F 2;d x (s) (1.3)
where x ∈ {aa, ab, bb, -}, the distributions F ••• are defined in Section 3, and

d x =      α 1 , α 2 ; η, x = aa, α 1 , -; η, x = ab, η, x = bb or -. (1.4) Theorem 1.2. Fix η, α i , β i , i = 1, 2 positive reals. As n → ∞ (n a positive integer), let u = v = exp(-ηn -1/3 ), (a i , b i ) = u α i /η , u β i /η i = 1, 2
all going to 1 and set q = 1u 2 → 0. We have: andd x is as in (1.4).

lim n→∞ M , x λ 1 -χn n 1/3 ≤ s + 1 η log n 1/3 η = F 1;d x (s), lim n→∞ M , x λ 1 -χn n 1/3 ≤ s + 1 2η log n 1/3 2η = F 2;d x (s) (1.5) where χ = 2q ∑ ≥0 u 2 1-u 2 q n→∞ ---→ 2, x ∈ {aa, ab, bb, -},
Notice the logarithmic corrections in all cases. Also notice the unusual scaling q = O(n -1/3 ) in Theorem 1.2, different from the usual q = O(n -1 ) [START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF].

We can also show convergence of the first k parts of λ to the first k parts of the ensembles given by the corresponding kernels of Section 3, a result in the spirit and generalizing those of [START_REF] Borodin | Asymptotics of Plancherel measures for symmetric groups[END_REF][START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF]. We omit the statement for brevity.

We further emphasize we concentrate here on the new interesting "crossover" regime u, v → 1. I.e. the case u → u 0 ∈ [0, 1) leads, up to deterministic shift, to the same asymptotics as u = 0-e.g. for the x =label, the limiting distributions are the Tracy-Widom F GOE distribution in the case of the upwards measure and the Tracy-Widom F GUE distribution for the up-down measure. We also omit this for brevity.

Finally, the new limiting distributions-defined in Section 3-contain all the classical Tracy-Widom distributions as limits.

Theorem 1.3. We have:

lim η→∞ F 1;α 1 ,α 2 ;η (s) = F (s; α 2 ), lim η→∞ F 2;α 1 ,α 2 ;η (s) = F GUE (s) where F GUE is the Tracy-Widom GUE distribution [15] and F (s; α 2 ) is the Baik-Rains [3] Tracy-Widom GOE/GSE [16] crossover-F (s; 0) = F GOE (s) while F (s; ∞) = F GSE (s).

A corner growth interpretation

In this section we describe directed last passage percolation on a tie. More precisely, we show how the measures M ,and M , -, and in particular the observables λ 1 , come from certain LPP models on an infinite reflecting strip-the above mentioned tie models. In the case of u = v = 0 (for the up-down measure) and u = 0 (for the upwards measure), the models become the usual LPP models of Johansson [START_REF] Johansson | Shape fluctuations and random matrices[END_REF] and the Baik-Rains symmetrized versions [START_REF] Baik | Algebraic aspects of increasing subsequences[END_REF] respectively. For brevity, we will restrict the discussion to the measure M ,and make remarks about the other one. First we fix parameters x 1 , . . . , x n , y 1 , . . . , y n . Note in the end we take x i = y i = q ∀i. Suppose we have an infinite strip-or tie-on the discrete square lattice constructed from big n × n adjacent squares and triangles like in Figure 1 (left). The strip has reflecting boundaries (red lines in fig. cit.). Each big square, sitting centrally in the strip, contains n 2 unit squares, and each big triangle n(n -1)/2 unit squares and n unit triangles. In each unit square/triangle there is a geometric random variable Geom(z) 5 -independent from the others-of a certain parameter z chosen as follows. Associate our x i , y j parameters with the north-east (NE) and north-west (NW) boundaries of the strip (ends of the tie) as depicted in Figure 1 (left). Pick a unit square from a big n × n square. The number inside has distribution Geom((uv) 2s x i y j ) where s = 0, 1, 2, . . . is the vertical position of the big square in the strip (starting from the top s = 0) and to figure out i, j send two rays of light from said square to the top, one in the NE and the other in the NW directions. The rays reflect off each boundary. They will intersect the top NE and NW borders in an x i and y j parameter respectively, which are our sought variables. For a unit square inside a big n × n triangle, the number inside is either Geom(u 2 (uv) 2s x i x j ) or Geom(v 2 (uv) 2s y i y j ) with s and determining i, j as above. Note in this case, the two rays will hit the same top boundary, either labeled x or y. Finally, in the unit boundary triangles, the numbers inside are Geom(u(uv) s x i ) or Geom(v(uv) s y i )) with s and determining i as above. See Figure 1 (left) for precise examples.

By Borel-Cantelli, almost surely only finitely many numbers in this strip will be nonzero-say those outside the green area in Figure 1 (left). Look at the longest polymer (path) with south-east (SE) or south-west (SW) steps starting from the top unit square in the strip and going down, reflected by the two boundaries if need be. Here by length we mean the sum of the integers encountered by the path. Call this length L. It equals 199 in Figure 1 (left).

Pick now a uniformly distributed partition κ with parameter uv-i.e. Prob(κ) = (uv; uv) ∞ (uv) |κ| -and place it at the nodes of the south-eastern and south-western-most bottom boundaries separating the infinite region of 0's inside the strip-without loss of generality it can be positioned on the SE and SW sides of a big square. Using the Fomin growth rule description of the Robinson-Schensted-Knuth (RSK) correspondence-like described in e.g. [START_REF] Betea | Perfect sampling algorithm for Schur processes[END_REF]-inductively "flip", starting from the bottom, every unit square and triangle to produce, from three (or two for the triangles) partitions on its boundary and the integer inside, a fourth partition-placed at the top vertex. Call the final partition sitting at the top of the tie λ, and µ, ν the ones sitting at the top ends of the reflecting boundaries. The properties of the RSK correspondence imply Prob(µ, λ, ν) ∝ M , -(λ, µ, ν) (upon taking x i = y i = q ∀i). Greene's theorem [START_REF] Greene | An extension of Schensted's theorem[END_REF] yields L + κ 1 = λ 1 . We thus obtain:

Theorem 2.1. It holds that Prob(L + κ 1 ≤ k) = M , -(λ 1 ≤ k).
The corner growth construction corresponding to the measure M , -is depicted in Figure 1 (right; L = 130; there are no big squares and hence no x parameters; v = 1). Theorem 2.1 holds mutatis-mutandis. A similar construction holds for the M , -, M ,measures if one replaces the geometric random variables with appropriate Poisson point processes. For the measures with boundary a and/or b parameters, a more involved construction generalizing that of [START_REF] Baik | Algebraic aspects of increasing subsequences[END_REF] exists. E.g. for the M , aa measure, one just puts another distribution in the unit triangles of Figure 1 (left): Geom(a 1 u(uv) s x i ) respectively Geom(a 2 v(uv) s y i ) for the two boundaries. For a b parameter-say on the right boundary-one replaces Geom(a 2 v(uv

) s y i ) with Geom (b 2 ) (v(uv) s y i ) where X ∼ Geom (b) (z) if Prob(X = k) = (1 -z 2 )(1 + bz) -1 b k mod 2 z k ∀k ≥ 1.
Because of the mod 2, one heuristically sees that the b parameters do not matter in the limits of Theorem 1.2, whereas the a's do and can be taken as "strengths" of the respective boundaries.

Geom((uv) 4 x3y2) Geom(v 2 y1y2) Geom(u 2 (uv) 2 x1x2) Geom(vy3) Geom(u(uv)x3) Geom(x2y4) Geom((uv) 2 x4y2) Geom((uv) 4 x2y4) Geom(u 2 x2x4) Geom(v 2 (uv) 2 y2y4) Geom(ux2) Geom(v(uv)y2)

Definition of distribution functions

Fix parameters α 1 , α 2 , η > 0. Our new limit distributions are defined as Fredholm pfaffians. For k = 1, 2 let

F k;α 1 ,α 2 ;η (s) := pf J -A k;α 1 ,α 2 ;η L 2 s+ log 2 k•η ,∞ , F k;α 1 ,-;η := lim α 2 →0 F k;α 1 ,α 2 ;η , F k;η := lim α 1 →0 F k;α 1 ,-;η (3.1)
where J is the anti-symmetric kernel J(x, y) = δ x,y 0 1 -1 0 and where the A kernels (operators) are defined as follows. Let τ, τ satisfy 0 < τ, τ < 1 6 min(η, α 1 , α 2 ). In case α 1 (or α 2 or both) is zero, the corresponding factor is absent from the minimum conditions. Define the following pairs of contours, oriented bottom-to-top:

(C 1,1 ζ , C 1,1 ω ) = (τ + iR, τ + iR), (C 1,2 ζ , C 1,2 ω ) = (τ + iR, -τ + iR), (C 2,2 ζ , C 2,2 ω ) = (-τ + iR, -τ + iR).
Define these auxiliary products of Euler Gamma functions 6 :

γ (1) (ζ) := Γ 1 2 + α 1 -ζ 2η , 1 + α 2 -ζ 2η Γ 1 2 + α 1 +ζ 2η , α 2 +ζ 2η , γ (2) (ζ) := Γ 1 4 + α 1 -ζ 4η , 3 4 + α 2 -ζ 4η Γ 3 4 + α 1 +ζ 4η , 1 4 + α 2 +ζ 4η . (3.2)
The desired 2 × 2 hypergeometric Airy matrix kernels are below. The ζ integral is always over C i,j

ζ , the ω always over C i,j ω for corresponding i, j, d ζω := dζdω(2πi) -2 , and

d ζ := dζ(2πi) -1 . A 1;α 1 ,α 2 ;η 1,1 (x, y) = Γ ζ η , ω η γ (1) (ζ)γ (1) (ω) sin π(ζ-ω) 2η sin π(ζ+ω) 2η e ζ 3 3 -xζ+ ω 3 3 -yω d ζω , A 1;α 1 ,α 2 ;η 1,2 (x, y) = Γ ζ η , 1 - ω η γ (1) (ζ) γ (1) (ω) sin π(ζ+ω) 2η sin π(ζ-ω) 2η e ζ 3 3 -xζ-ω 3 3 +yω d ζω , (3.3) A 1;α 1 ,α 2 ;η 2,2 (x, y)
= Γ 1 - ζ η , 1 - ω η 1 γ (1) (ζ)γ (1) (ω) sin π(ζ-ω) 2η sin π(ζ+ω) 2η e -ζ 3 3 +xζ-ω 3 3 +yω d ζω 4η 2
sgn(xy) and A 2;α 1 ,α 2 ;η 1,1

(x, y) = Γ 1 2 + ζ 2η , 1 2 + ω 2η γ (2) (ζ)γ (2) (ω) sin π(ζ-ω) 4η cos π(ζ+ω) 4η e ζ 3 3 -xζ+ ω 3 3 -yω d ζω 4η , A 2;α 1 ,α 2 ;η 1,2 (x, y) = Γ 1 2 + ζ 2η , 1 2 - ω 2η γ (2) (ζ) γ (2) (ω) cos π(ζ+ω) 4η sin π(ζ-ω) 4η e ζ 3 3 -xζ-ω 3 3 +yω d ζω 4η , (3.4)
A 2;α 1 ,α 2 ;η 2,2 (x, y) = Γ 1 2 - ζ 2η , 1 2 - ω 2η 1 γ (2) (ζ)γ (2) (ω) sin π(ζ-ω) 4η cos π(ζ+ω) 4η e -ζ 3 3 +xζ-ω 3 3 +yω d ζω 4η
with the remark that everywhere

A ••• 2,1 (x, y) := -A ••• 1,2 (
y, x). We note that, upon using the Gamma duplication formula, we can write, for k = 1, 2,

F k;η s - log 2 k • η = pf J -A k;η L 2 (s,∞) (3.5)
where the kernels A k;η have a much simpler form:

A 1;η 1,1 (x, y) = Γ 1 - ζ η , 1 - ω η sin π(ζ-ω) 2η sin π(ζ+ω) 2η
+ Γ ζ η e -ζ 3 3 +xζ d ζ η -Γ ω η e -ω 3 3 +yω d ω η -sgn(x -y)
and

A 2;η 1,1 (x, y) = Γ 1 2 - ζ 2η , 1 2 - ω 2η sin π(ζ-ω) 4η cos π(ζ+ω) e ζ 3 3 -xζ+ ω 3 3 -yω d ζω 4η , A 2;η 1,2 (x, y) = Γ 1 2 - ζ 2η , 1 2 + ω 2η cos π(ζ+ω) 4η sin π(ζ-ω) 4η e ζ 3 3 -xζ-ω 3 3 +yω d ζω 4η , A 2;η 2,2 (x, y) = Γ 1 2 + ζ 2η , 1 2 + ω 2η sin π(ζ-ω) 4η cos π(ζ+ω) 4η e -ζ 3 3 +xζ-ω 3 3 +yω d ζω 4η (3.7)
where again

A ••• 2,1 (x, y) := -A ••• 1,2 (y, x
) and the contours are as before with one important exception: in the case of A 1; η 1,2 , the ω contour C 1,2 ω passes locally to the right of 0, but is otherwise as stated (to account for the pole at ω = 0 whose residue was not taken). This happens because-up to inessential conjugation and taking the appropriate residues in

A 1; ••• 2,2 as α 1 , α 2 → 0-we have A k;0,0;η x + 2 log 2 k•η , y + 2 log 2 k•η = A k;η (x, y), k = 1, 2.

Sketch of proof

The first tool we use to prove our results is the Schur measure with two free boundaries from [START_REF] Betea | The Free Boundary Schur Process and Applications I[END_REF]. Recall the definition of skew Schur functions evaluated at a specialization ρ via the Jacobi-Trudi formula: s λ/µ (ρ) = det 1≤i,j≤n h λ i -i-(µ j -j) (ρ) for n large enough. Here a specialization ρ is just a sequence of numbers (h n (ρ)) n≥0 -its values on the complete symmetric functions-assembled into the generating series H(ρ; z) := ∑ n≥0 h n (ρ)z n . On sequences of partitions µ ⊂ λ ⊃ ν, consider the weights

W x (λ, µ, ν) := ∆ x (µ, ν) • u |µ| v |ν| • s λ/µ (ρ + )s λ/ν (ρ -) (4.1)
with x ∈ {aa, ab, bb, -}, ρ ± two specializations and ∆ x as in (1.2)-note after proper normalization [START_REF] Betea | The Free Boundary Schur Process and Applications I[END_REF], these become probability measures. We observe our original measures (1.1) are of the form given in equation (4.1) as follows. For the up-down measures, ρ + = ρ -and both are the poissonized Plancherel pl specialization for M and ρ q (a specialization in n variables all = q) for M respectively; for the upwards measures , set v = 1 and ρ + = pl for M, = ρ q for M while in both cases ρ -= 0, the empty specialization; recall ρ , ρ q are defined by H(pl ; z) = exp( z), H(ρ q ; z) = (1qz) -n ; and finally note parameters a 1 , a 2 , b 1 , b 2 , u, v, q satisfy the inequalities from the Introduction. The reason for the above is as follows: Schur functions, specialized in variables, are generating series for semi-standard Young tableaux. As such, s λ/µ (ρ q ) = s λ/µ (q, . . . , q) = q |λ/µ| f n, λ/µ (there are n q's inside the parentheses) and a limiting argument shows s λ/µ (pl ) = |λ/ u| f λ/µ |λ/µ|! . Define the point configuration associated with λ coming from the above measures as S x (λ) := {λ jj + 1/2} ⊂ Z + 1/2. It is a simple point process. Consider the shifted version S x t (λ) := S x (λ) + 2D t where for t a parameter, D t is an integer independent of everything else having theta distribution Prob(D t = d) = t 2d (uv) 2d 2 θ 3 (t 2 ;(uv) 4 ) . A slightly more general version of [START_REF] Betea | The Free Boundary Schur Process and Applications I[END_REF]Theorem 2.5] shows that, for x ∈ {aa, ab, bb, -}, the shifted point process S x t (λ) is pfaffian with 2 × 2 matrix correlation kernel K x of the form: with F(z) = H(ρ + ;z) H(ρ -;z -1 ) ∏ n≥1 H(u 2n v 2n-2 ρ -;z)H(u 2n v 2n ρ + ;z) H(u 2n-2 v 2n ρ + ;z -1 )H(u 2n v 2n ρ -;z -1 ) ; with κ x 1,1 (z, w), κ x 1,2 (z, w) and κ x 2,2 (z, w) given7 respectively by v 2 tz 1/2 w 3/2 • ((uv) 2 ; (uv) 2 ) 2

K x 1,1 (k; k ) = 1 (2iπ)

Figure 1 :

 1 Figure 1: construction described in Section 2, with longest polymer in orange. n = 4, L = 199 (left) and L = 130 (right).

e ζ 3 3 -xζ+ ω 3 3 -yω d ζω 4 , 8 De -ζ 3 3 +xζ-ω 3 3 +yω d ζω η 2

 338332

2

 2 |z|=r dz z k+1 |w|=r dw w k +1 F(z)F(w)κ x 1,1 (z, w), K x 1,2 (k; k ) = -K x 2,1 (k ; k)

The restrictions on the a i , b i can be somewhat relaxed.

Here u, v, a i , b j are boundary parameters, is a poissonization parameter keeping track of the size of the skew Young diagrams, and q a geometrization parameter for the same purpose.

Another such interpolating measure has been studied in[START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF] Example 3.4] (for the bulk) and in[4, Theorem 1.1] (for the edge). It is an instance of the periodic Schur process, while the measures M considered in this paper are instances of the free boundary Schur process.

[START_REF] Betea | The Periodic Schur Process and Free Fermions at Finite Temperature[END_REF] We obtain essentially the same result if we consider µ or ν instead of λ.

We say X ∼ Geom(z) if Prob(X = k) = (1z)z k , ∀k ≥ 0.

We use the notation Γ(a, b, c, . . . ) = Γ(a)Γ(b)Γ(c) • • • .

We use the notation (x; q) ∞ := ∏ ≥0 (1xq ), θ q (x) := (x; q) ∞ (q/x; q) ∞ , θ 3 (z; q) := (q; q) ∞ θ q (-z √ q) and (a, b, . . . ; q) ∞ := (a; q) ∞ • (b; q) ∞ • • • .

Acknowledgements

D.B. is especially grateful to Patrik Ferrari for suggesting simplifications in Section 3 and to Alessandra Occelli for suggesting the name for the models of Section 2.

* D.B. was partially supported by the German Research Foundation as part of the CRC 1060-B04 project. † J.B. was partially supported by the "Combinatoire à Paris" project funded by the City of Paris, and by the grant ANR-14-CE25-0014 "GRAAL".

‡ P.N. was partially supported by ERC Advanced Grant No. 338804 and ERC Starting Grant No. 716117.

where g x (z) =

where r, r satisfy max(v, q) < r, r < min(q -1 , u -1 ) (q does not appear for measures M) and r < r for K x 1,2 ; and where the choice of ρ ± was described in the previous paragraph. To that point, we have

(for all M). In this shifted process, the distribution of λ 1 , in each case, is a Fredholm pfaffian for its respective kernel.

Using steepest descent asymptotic analysis, we can show that the the latter kernels converge to the ones defining our distributions in the limits prescribed in Theorems 1.1 and 1.2. We write F(z)z -k as exp(MS(z)) (for M) and exp(nS(z)) (for M) in the scaling prescribed with S having a double critical point at z = 1; we thus scale the integration variables (z, w) = (exp(ζ M -1/3 ), exp(ωM -1/3 )) (replace M by n for M) so that only an M 1/3 (or n 1/3 ) neighborhood of 1 makes a non-zero contribution-see [4, Section 5] for a related computation. For κ, one uses the estimate log(q c ; q) ∞ = -

for q = e -r , r → 0 + (c ∈ C -Z ≤0 ), along with (-x; q) ∞ = (x 2 ;q 2 ) ∞ (x;q) ∞ . The logarithmic shifts in our results come from the log r terms above.

This together yields the result modulo the shift we have made. It (2D t ) can be removed in the end 8 as 2D t /M 1/3 (or 2D t /n 1/3 ) converges to 0 in probability. Moreover, a slightly improved version of the above argument can be used to show convergence of multi-point correlation functions to the desired ensembles-one again has to remove the shift at the end. This finishes the proof.


Preface: the everyday politics of public space

Mattias De Backer, Claske Dijkema and Kathrin Hörschelmann In: Space andCulture, 2019, DOI: 10.1177/1206331219830080 Pay attention to the inventive arenas that exert political power outside that stage or change the contents of the drama onstage. From the places that you have been instructed to ignore or rendered unable to see come the stories that change the world, and it is here that culture has the power to shape politics and ordinary people have the power to change the world. You can see the baffled, upset faces of the actors on stage when the streets become a stage. Solnit (2016: 26) Over the last two decades or so, a rich literature has emerged about the politics of public space, including Iveson's (2013) work on the appropriation of public spaces by active/activist citizens involved in 'DIY urbanism, Hou's (2010) work on guerrilla urbanism, Holston's (2008) research on practices of everyday resistance and insurgent citizenship and Isin's (2008) and Massey et al.'s (2000) interventions on claiming rights to the city. Others, such as [START_REF] Mitchell | The right to the city. Social justice and the fight for public space[END_REF] and Low and Smith (2006), have considered public space's potential as a locus for (counter-)politics and activism, drawing on Lefebvre's trialectics.

At the same time, numerous critics have warned about the consequences of increasing surveillance, commercialisation, privatisation, militarisation and the sanitisation of public space (Atkinson, 2003;zukin, 1998;[START_REF] Lofland | The public realm: exploring the city's quintessential social territory[END_REF]Low 2001;Sorkin, 1992;Zukin, 2000), which limit possibilities for political expression in public space (publicity in space) and citizenship. These different forms of control over public space stand in tense relationship with the democratic ideal of a space that is open to all, acknowledged by all and in the interests of all (Carr et al., 1994;Henaff andStrong, 2001: 35, Sennett 1998;[START_REF] Mauger | L'émeute de novembre 2005 : une révolte protopolitique (l'association Raisons d'agir)[END_REF][START_REF] Bilge | Beyond subordination vs. resistance: An intersectional approach to the agency of veiled Muslim women[END_REF]. While Hou (2010: 3) is still optimistic that citizens can hold the state accountable through opportunities of assembly and public discourses, i.e. political expressions in public space, , Holston finds that erosions of citizenship come with the increased urge of the state to control public spheres, leaving no place for conflict (Holston, 1999;[START_REF] Himada | From noun to verb: The micropolitics of making collective[END_REF].

However, many urban theorists, when talking about politics, refer to those phenomena of political participation that involve representative and deliberative systems, including "new forms of social organization-transnational labor organizing, indigenous rights and environmental justice movements (…)-

[which] are always creating alternative new spaces of and for public political expression" (Low and Smith, 2006: 16). In this special issue, we want to zoom in on the everyday and the micro-level as a site of political expression. For this we turn to the lens of performative, prefigurative and affective politics of the everyday, agreeing with Himada and Manning (2009: 5) that "Politics" with a capital 'P' is much less the "real deal" than it presents itself. While "Politics"

operates in the sphere of representation, where precomposed bodies are already circulating, "the micropolitical is that which subverts this tendency in the political to present itself as already formed."

Interestingly, as Brighenti (2008) notes, political philosophical reflection on the public sphere, such as that of Jurgen Habermas or Hannah Arendt, is almost exclusively focused on the dimension of political participation and deliberative procedures, while micro-sociological, interactionist studies of the public realm are mainly concerned with the cognitive frameworks and registers of interpersonal interaction. Both approaches, the author argues, "miss the spatial and material constraints that constitute the public" (p. 12). In tying these two interests together, we ask how micro-and everyday politics unfold along contested lines of publicness and privacy and whether conceptions of shared space need to be rethought when they are compared and contrasted with different articulations in diverse socio-cultural contexts and from intersectional perspectives. Empirical micro-sociological studies indicate how publicness and privacy are negotiated through embodied spatial practices, discursive constructions and the negotiation of access to public space, in relation to questions of visibility/invisibility, intimacy, fear, freedom, belonging, exposure, and recognition. This inevitably leads to a critical discussion of the central assumptions regarding private/public space and its generational, gendered, classed and 'culturalised' construction.

Beyond the revolutionary images of people claiming their rights during mass demonstrations at Tahrir Square, Maidan, Gezi Park, and an occupied Wall Street, we are interested in everyday appropriations of public space by marginalized groups and the ways in which they claim spaces of intimacy, privacy and freedom, while seeking to shape the city to their needs. The spatiality of intimacy versus exposure, of visibility versus invisibility, and of recognition versus misrecognition puts an emphasis on the different ways in which spaces are experienced by individuals and groups. For instance, research in Palestine has demonstrated that the much cherished European ideal of open and shared public spaces contrasts with the practices of women who want to traverse public space in a more concealed fashion [START_REF] Abbas | Thinking Arab women's spatiality: the case of 'mutanazahat' in Nablus, Palestine. Gender, Place[END_REF].

However, visibility also exposes individuals to risks that a focus on the importance of claiming public space and gaining notable presence perhaps leads us to underestimate. Further, the value of public space may, for some, consist exactly in the opportunities that it offers for becoming less visible and for escaping restrictions that may be experienced in domestic spaces [START_REF] Wilson | The sphinx in the city: Urban life, the control of disorder, and women[END_REF]. Anonymity has been prized by many urban theorists as an urban characteristic that can provide freedoms not afforded by close-knit social contexts, where surveillance can be seemingly omnipresent [START_REF] Wirth | Urbanism as a Way of Life[END_REF], Benjamin 1935[START_REF] Rancière | Disagreement: Politics and Philosophy[END_REF].

The boundaries between visibility and invisibility are thus frequently blurred and neither is unproblematic nor do they furnish a progressive politics per se, as gendered constructions and experiences of in/visibility in particular show. As De Backer argues in this issue, visibility and invisibility can "easily transform into the other or present themselves simultaneously" (cf. De Backer, 2018: 1). Visibility in itself is also far from unproblematic as a claim to urban citizenship. It can become a ground for contesting the presence of subaltern groups in public space and is not guaranteed to translate into achieving citizen rights to the city (cf. Dijkema, 2018, also see Hörschelmann and van Blerk, 2011).

This perspective on the micro and everyday can be understood against the backdrop of several theoretical axes: Katz's notion of minor theory (1995,1996,2017), the work of Foucault, de Certeau and Lefebvre on the production of power, knowledge and place, and prefigurative spatial politics in anarchist theory.

1) Cindy Katz' notion of minor theory is a first cornerstone bringing together empirical work on everyday politics of public space. She argues that the 'cultural turn' sparked an interest in those subaltern groups excluded from knowledge production. However, the rules of the game of knowledge production have remained the same, which makes it nearly impossible to dismantle it and rebuild an alternative. Instead of continuing to focus on the absences, exclusions and silences, the author develops a conceptual framework to alter the ways knowledge is produced and shared. Instead of describing oppression in the lexicon of major theory, she advocates for the development of minor theory:

The differences are about performance; different ways of 'doing theory', different ways of reading, writing,talking that apparently remain veiled to those still bent on 'mastery'. Maybe these practices are not so much veiled as unappetizing. They assert differences that are relentlessly material-embodied, positioned, sensual-that do not ignore the alternative material conditons (sic) under which knowledge is produced and shared. (Katz, 1996: 488) Intriguingly, she adds that constituting the minor is about consciously working in a vocabulary in which one is not at home, in the analogy of the Czech writer Franz Kafka struggling with German while writing masterpieces such as Metamorphosis.

It refers to the process of writing a minor literature by subverting a major language from within, forcing it to express something different. It is in this capacity that the minor is transformative, reworking the major from within: "[b]ecoming minor sets up an imbricated or interstitial politics; a way of negotiating and reworking a space of betweenness to produce something new" (p. 496).

2) This new space of knowledge refers strongly to the novel social and spatial realm which Henri [START_REF] Lefebvre | The production of space[END_REF] called 'representational space'. Relevant as a theoretical backdrop for this special issue is the idea in Lefebvre's trialectics that the traditional opposition between the common and the powerful, or between routine navigations of space and top-down planning procedures, can be overcome in a thirdspace where true representation can occur, akin to a politics of contestation without a capital P. This is in line with [START_REF] Foucault | Discipline, toezicht en straf[END_REF] conceptualisations of the microphysics of power and with de Certeau's (1984) twin concept of 'strategy' vs. 'tactic'. Essential in the latter's analysis is the idea that being powerful or powerless is inextricably linked to the occupation of space. While the powerful can claim a permanent ownership over the central territories of our world-be it a medieval castle or the Chrysler Building-the powerless can only poach on this space [START_REF] Fielder | Poaching on public space: urban autonomous zones in French Banlieue Films[END_REF], claiming it temporarily in an act of defiance, protest and resistance. As illuminating as this double metaphor is, what is missing from de Certeau's analysis is the subtle in-betweenness and trialectics essential to authors such as Lefebvre and Deleuze, where the oppressed, while temporarily claiming a space for themselves, can simultaneously be the powerful, excluding others from their territory.

3) These struggles for power and knowledge do not only take place in space, they are of space. Space has political potential in the sense that it is in perpetual development: it cannot be separated from time (Massey, 1993). The present therefore holds the promise of the future and this promise is (pre)figured in space.

As Ince (2012, p. 9) argues: "if territory is constituted and reconstituted over time, there is scope for intervention." This idea of openness and possibility does not foreclose the importance and political potential of boundary-making. Anarchist theory calls attention to the role that the bounding of spaces can play in the everyday "constitution and mobilization of political subjectivities and identities" which are "located in particular places, demarcating certain physical or symbolic territories" (Ince, 2012(Ince, , 1650)). The bounding of space plays an important role in prefigurative politics. Prefiguration refers to a beginning, a start of something that will be continued in the future and will shape the contours of this future. The creation of new forms of boundaries which are formed independently or in interaction with those imposed through territorial governance thus have the function to protect the self but also to build anew and enter in relation. Beyond protection, they thus create possibilities to speak out, to formulate political statements through translating/transforming everyday practice and individual opinions into collective discourse.

Inspired by these three theoretical axes, this special issue begs the question of what can be considered political in the public realm. The everyday uses of public space that this special issue deals with, have in common that they take as a starting point those at the lower end of asymmetric power relations: young girls/men (Wijntuin and Koster); racialised youth in marginalised social-housing neighbourhoods (De Backer, Dijkema); children in cities dominated by cars (Caroll et al.); and finally those speaking minority languages (Polese et al.). On what basis can we qualify these groups' behaviour as political?

Much ink has been spilled on the question "what is political?" and where to draw the line between the political and the apolitical: the 'not-quite political' (as in politics with a capital 'P'), the pre-political [START_REF] Mauger | L'émeute de novembre 2005 : une révolte protopolitique (l'association Raisons d'agir)[END_REF], the proto-political or infrapolitical [START_REF] Scott | Weapons of the weak: everyday forms of peasant resistance[END_REF][START_REF] Scott | Domination and the Arts of Resistance: Hidden Transcripts[END_REF][START_REF] Scott | Infrapolitics and Mobilizations: A Response by James C. Scott[END_REF] and the postpolitical (Wilson and Swyngedouw, 2014). However, not all authors agree on their definition of the political. Rancière argues that "political activity makes visible what had no business being seen, and makes heard a discourse where once there was only place for noise" (Rancière, 1999, p. 30; translation modified in Dikeç 2012, p. 674).

Politics implies "the refusal of an identity imposed by others" (Davis, 2010: 88). It has to be understood not as a goal to be reached in the future but as an "open set of practices driven by the assumption of equality between any and every speaking being and by the concern to test this equality" (Rancière, 1999: 30). An act functions politically if, through it, 'those that have no part' achieve to be taken into account. To those authors, the political is disruptive [START_REF] Dikeç | Disruptive politics[END_REF] and consists of an antagonistic moment (Dikeç, 2012, p. 674).

Others have argued that this definition of what constitutes politics is too narrow [START_REF] Beveridge | The post-political trap? Reflections on politics, agency and the city[END_REF]. In their critique of the post-political city, which draws attention to the depoliticization of cities, they insist on the contingency between politicisation and depoliticization and on political agency beyond the antagonistic moment. This is closer to Scott's concept of infrapolitics (2012) which refers to a kind of politics that avoids direct confrontation but opens up to a "subterranean world of political conflict". His analysis is limited to contexts in which direct confrontation would be too costly and in which direct claims of those at the lower end would not be heard. For Scott, the political lies in the accumulation of everyday acts-and not in the individual acts themselves-that, together, can have a massive effect. [START_REF] Solnit | Hope in the dark: untold histories, wild possibilities[END_REF], in her passionate analysis of activism, refers to the importance of hope, imagination and determination of ordinary people in the shadows, long before a movement is formed and becomes recognised as political.

Where Beveridge and Koch's conception of the political differs though, is in the function of visibility of conflict as a condition for the political (2017). Papers in this issue claim that tactics of invisibility may just as much be a part of the political as those tactics of visibility more often researched (De Backer; Wijntuin and Koster this issue). Rather than deciding on one specific definition of the (a)political, we argue with Beveridge and Koch (2017, p. 10) that "drawing the boundary between the two is an integral part of (de)politicization and the normative and institutional ordering of politics which emerges from it." If we want urban research to be a rampart against depoliticisation and contribute to equal access to city spaces we need to make an effort to recognize the political beyond the antagonistic moment and beyond visibility [START_REF] Davidson | Recovering the politics of the city: From the 'post-political city' to a 'method of equality' for critical urban geography[END_REF].

The papers in this special issue are empirical studies into agency in public space by those who claim their part in the city through disobeying norms that may be patriarchal (Wijntuin and Koster this issue); racist (Dijkema, this issue) or intersectional (De Backer, this issue) in nature; or by those whose access to public space is limited as a result of the priority given to cars (Caroll et al., this issue) or-in a more symbolic sense-through the imposition of language (Polese et al. this issue). The city hereby is not a stage for political action, for making a claim, but access to the city itself is the claim.

Contributions

The special issue brings together work that elaborates these theoretical concerns about the significance of everyday negotiations of, and micro-political challenges to, public/private space divides further through contextualised readings and analyses of diverse case studies. To prepare the special issue, we asked our contributors to reflect on the following questions:

• How are publicness and privateness produced performatively?

• How can embodied and affective interaction between people and place also be political? • How does this construct as well as problematize public/private space and the divides between them? And what role does (in)visibility play here? • How should we reformulate insights about the political function of public space gained by authors such as [START_REF] Mitchell | The right to the city. Social justice and the fight for public space[END_REF] or Low and Smith (2006), when we include the 'politicalness' of everyday interaction, especially taking into account the gendered, generational, culturalised and classed nature of public space? • In which cases can claims to space be interpreted as a form of prefigurative politics? Can Schatzki's idea of prefigurative politics helps us to further extend our understanding of public space beyond Enlightenment ideals of the public sphere as a space for deliberation and active citizenship? • How do radical approaches to public space respond to diverse articulations and performances of publicness and privacy?

In their contribution to this special issue, Penelope Carroll, Octavia Calder-Dawe and Karen Witten answer these questions through an analysis of data on children's practices of play, collected in nine inner city and suburban neighbourhoods in Auckland, New Zealand. They identify some interesting contradictions in how we, as a society and as adults, approach and "territorialise" children. For instance, since the UN Convention on the Rights of Children (1989), children's rights to protection have been acknowledged to a much greater extent than their rights to agency and self-realisation. Similarly, children have increasingly been given a formal role in politics, through youth parliaments, councils and advisory groups, but their being political actors in everyday practices has not received the same recognition. These everyday practices are at the heart of Carroll et al.'s paper, in which they analyse their children's behaviour as political, prefiguring a desired future including new forms of sociality and a fairer, more child-friendly and playful city. While adult citizen-activist initiatives can generally be considered overt examples of prefigurative politics, children's play is much subtler; it is experimental and experiential rather than deliberate and goaloriented, it prefers doing over saying and feeling over reasoning.

Polese deals with contestations in and over public space, including institutions such as public schools. He is interested in "actors of a country" that negotiate and compete over its regulation. The author explores everyday resistance to state policy through informal practices, namely the choice of speaking a language other than that sanctioned by the state. The empirical evidence comes from the city of Odessa, where Russian is the primary language for a large part of the population while Ukrainian is the official language. When public servants (teachers) discuss non-work related themes in Russian they, in some ways, bring into the public a language that is supposed to remain in the private sphere. They play with the distinction between private and public spaces. For example, while courses will take place in Ukrainian, informal exchanges between teachers and pupils or between colleagues in public schools may take place in Ukrainian, thereby trespassing state law and carving out a space in public for what are supposed to be private practices. Through speaking Russian, they de facto "minimize state influence in a given situation" (Polese 2018, p.3). This form of political action is based on a Foucauldian conceptualization of power as diffuse, present everywhere, embodied in discourses and knowledge, and which transcends politics with a capital P (Polese 2018, p.4), allowing observation of subaltern responses to power.

Claske Dijkema and Mattias de Backer both consider how young people from marginalized ethnic communities contest racism, control and exclusion from urban citizenship through micro-political practices of becoming in/visible. Claske Dijkema asks whether such micro-political practices of claiming space and of developing noticeable presence in marginalized urban spaces, which are qualified as uncivil behaviour by those in charge of public order, can be understood as claims to citizenship. Her paper attends to the ways in which citizenship is negotiated through localized practices and everyday forms of confrontation in postcolonial France. Analysing both the sites and the acts through which racialized young men in a social housing estate in Grenoble exercise 'insurgent citizenship', she comes to the conclusion that their localised claims to space qua visibility don't address a wider public and therefore can only be partially understood as claims of citizenship. Access to public space is the claim.

Mattias de Backer examines similar practices of negotiating in/visibility in the city. He focuses particularly on the ways in which young men and women from migrant communities in Brussels deal with the paradoxical relationship between visibility as control (surveillance) and visibility as recognition/a way of claiming citizenship through access to public space. He attends to the role of gender in the different negotiations of in/visibility in private and public space, while also drawing attention to transgression as a way of staking belonging to the city. De Backer's analysis brings to light the potential of transforming public space by playing with visibility and invisibility, such as when young men intentionally occupy hypersurveilled spaces in central parts of the city and contest the exclusions that such surveillance might otherwise institute. He concludes that regimes of visibility, which include ambient features and material structures, can easily transform into each other and that public space is changed through practices of playing with in/visibility in turn.

Echoing several observations of De Backer's analysis, Wijntuin and Koster present findings from research undertaken in Utrecht, The Netherlands. They find that young men and women of Moroccan origin follow remarkably different spatial patterns: the former are stationary and claim a space while hanging out, while the latter wander around. However, this more fleeting use of space also enables them to claim it, but in a radically different manner. Wandering is considered as a legitimate way for girls to be outside, it enables them to escape social control inside and outside the house and it make it possible for these girls to interact with boys in a way that is both socially acceptable and negotiates the norm. Interestingly, while De Backer sheds light on the practices of these young men, using tactics against dominant control infrastructures while dominating female behaviour in their home neighbourhoods from a position of power, Wijntuin and Koster analyse the young women's behaviour as combining strategic (intentional) and tactical (improvisational) navigation.

Wijntuin and Koster point at the notion of "resistance", used by studies such as that of [START_REF] Peleman | Power and territoriality: A study of Moroccan women in Antwerp[END_REF] and [START_REF] Ehrkamp | I've had it with them!' Younger migrant women's spatial practices of conformity and resistance[END_REF], to analyse subaltern groups' practices in public space, with female teens of North-African origin depicted as agents who purposefully or unintentionally resist the male spatial dominance. This binary logic of domination and resistance has been criticised by feminist Muslim scholars [START_REF] Abu-Lughod | A community of secrets: The separate world of Bedouin women[END_REF][START_REF] Ahmed | Women and gender in Islam[END_REF][START_REF] Bilge | Beyond subordination vs. resistance: An intersectional approach to the agency of veiled Muslim women[END_REF]Bracke and Fadil, 2012;Ilahi, 2009) who convincingly show that such a western perspective on gender and feminism does not hold in other regions. This subtler power relation between men and women in the East is mirrored by more complex distinctions of public and private (Mazumdar and Marumdar, 2001;[START_REF] Mohammad | Making gender, ma(r)king place: Youthful British Pakistani Muslim women's narratives of urban space[END_REF].

Perhaps not everything that is part of the sidewalk ballet is necessarily a prelude to a political movement or a revolution, but many visible and invisible minor acts are part of a power negotiation that has political value. If we can continue to work on the necessary language to describe it, not only will our understanding grow, it will also contribute to developing a vocabulary that enables change. Moreover, developing a new language can also serve as an impetus to unlock the implicit knowledge which Foucault called savoir, holding a mirror to the academic world:

In a society, different bodies of learning, philosophical ideas, everyday opinions, but also institutions, commercial practices and police activities, mores -all refer to a certain implicit knowledge [savoir] special to this society. This knowledge is profoundly different from the bodies of learning [des connaisances] that one can find in scientific books, philosophical theories, and religious justifications." (Foucault, 1998: 261).

In order to tap into this practical wisdom, we think, it is necessary to consider both grand political theory and a sharp eye for the micro-sociology of the public realm, for the routine navigation of bodily movement, the negotiation of identity and the coping with the ever-present conflicts which Soenen (2006) has so aptly called "the small encounter."

  The political nature of space resides notably in related bordering practices. Bordering refers to "the spatial strategic representation of the making and claiming of difference" (Berg and Van Houtum 2003:2 in Ince 2012, p.6). It is a tool of territorialisation, which Ince understands as "ongoing, uneven and contested processes associated with a range of powerful forces and constituted by diverse territorial practices infused with multiple political, cultural, economic and social trajectories and intersections"(cf. Marston 2000; Valentine 2007 in Ince 2012, p. 5-6).